Surface Science 440 (1999) 279–289 www.elsevier.nl/locate/susc

Lattice vibrations and stability of reconstructed (110) noble metal surfaces I. Vilfan a, *, F. Lanc¸on b, E. Adam b a J. Stefan Institute, P.O. Box 3000, SI-1001 Ljubljana, Slovenia b De´partement de Recherche Fondamentale sur la Matie`re Condense´e, CEA-Grenoble, F-38054 Grenoble cedex 9, France Received 9 January 1999; accepted for publication 9 July 1999

Abstract The interplay between the surface phonons and different missing-row reconstructions of (110) noble-metal surfaces is investigated. The phonon dispersion curves and densities of states are calculated for a slab model with 24 layers of atoms, using the Sutton–Chen many-body potential. The phonon contribution to the surface free energy is evaluated for Au, Pt and Ag surfaces. The results show that surface phonons play an important role in stabilizing a surface reconstruction when the energy difference between different reconstructions is of the order 1 meV/(surface unit cell ) or smaller, and in particular in the case of Au(110) when the (1×2) and (1×3) reconstructions are close in energy. We also observe that the Sutton–Chen potential reproduces the phonon dispersion curves better for Ag (4d metal ) than for Au and Pt (5d metal ). © 1999 Elsevier Science B.V. All rights reserved. Keywords: Gold; Platinum; Silver; Surface energy; Surface relaxation and reconstruction; Surface thermodynamics; Surface waves: phonons

1. Introduction In the past, the stability of missing-row reconstructed noble metal (110) surfaces has been investigated with semi-empirical as well as firstprinciples energy calculations [1–8]. The calculations for Au and Pt show that the energies of subsequent reconstructions: (1×2), (1×3), … are very close to one another. In some models the (1×3) … reconstructions were found even lower in energy than the (1×2) structure [1,2,6 ], which would lead to facetting — not quite in agreement with the experiments for Au or Pt. The experiments tell us that for clean 5d fcc metals (Au, Pt, Ir), the (1×2) reconstructed surface is more stable than * Corresponding author. E-mail address: [email protected] (I. Vilfan)

the unreconstructed structure, whereas the clean 3d and 4d fcc metals (such as Ni, Cu, Pd, Ag) do not reconstruct. The origin of missing-row reconstructions was found in more pronounced relativistic effects for the 5d metals, and in the fact that the 5d electron wavefunctions are more delocalized [9,10]. In the above theoretical papers, conclusions about the relative stability of different reconstructions have been drawn on the basis of the surface energy alone and not from the surface free energy. The difference between the two is related to the entropy, i.e. thermal disordering. Two main contributions to the surface entropy come from surface lattice vibrations and from meandering of steps and of bound pairs of opposite steps, i.e. ‘Ising domain walls’ [11]. In this paper we shall concentrate on the first contribution, we shall investigate

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the role of surface phonons on the stability of totally flat, ideally missing-row reconstructed noble metal surfaces. We expect that phonons play an important role, in particular when different reconstructions are very close in energy, as is the case with (1×2), (1×3), … reconstructions. The calculations will be carried out on a slab model using the semi-empirical Sutton–Chen many-body potential [12,13]. For comparison, two different approaches will be used in calculating the phonon frequencies. In the first approach we will treat the lattice vibrations as collective modes in the harmonic approximation, and in the second they will be approximated by local ( Einstein) modes [14,15] treated in the quasi-harmonic approximation.

2. The potential

potential is truncated with a smooth, fifth-order polynomial cut-off between r(i,j)/r =2 and 0 r(i,j)/r =3, where r =a/앀2 is the nearest-neigh0 0 bour atom separation, in order to get the potential and its first and second derivatives continuous. In this way spurious forces, which could emerge from discontinuities in the derivative of U at the cutoff, are eliminated. In the calculations of the surface modes we use a slab model [17] with 2880 atoms [3456 atoms in case of the (1×3) reconstruction] distributed among ≥24 (110)-oriented planes of an fcc lattice, see Fig. 1. Periodic boundary conditions are ˚ of applied in all three directions and about 20 A empty space is added on top of the slab in the z direction to avoid interaction between the opposite surfaces across the empty space and to allow the lattice to relax.

We use the semi-empirical many-body Sutton– Chen interatomic potential [12,13]: U=

e

A B a

∑ ∑ 2 i j≠i r(i, j )

n

−eC∑ i

C A BD a

∑ j≠i r(i, j )

m 1/2

3. Lattice relaxation .

(1) The first term describes the core repulsion, and the second term the bonding mediated by the electrons. e and C, together with the exponents m and n, are the parameters of the model, fitted to bulk elastic quantities and are listed in Table 1 for Au, Pt and Ag. The bulk lattice constant is a and the distance between the atoms i and j is r(i,j). The advantage of this model is that it carries ingredients of many-body interactions, it is analytic and suitable for computer modelling. Its drawback is that it neglects the angular forces, i.e. the forces which depend on the angle between two bonds attached to a given atom [8,16 ]. In this paper the

The lattice relaxation was investigated by minimizing the potential energy at constant volume of the simulated box. The resulting lattice relaxations are collected in Table 2. The magnitude of the lattice relaxation depends strongly on the difference in the exponents n and m and is therefore substantially smaller for the Ag(110) surfaces than it is for the Au or Pt surfaces. As expected, the topmost layers always relax inwards. The atoms in the second layer of (1×2) and (1×3) reconstructions and in the third layer of the (1×3) reconstruction relax also laterally. When the Ag(110) slabs are simulated with the Sutton–Chen potential, the atoms in the second layer relax towards the missing rows. This direction is in agreement with LEED analysis for Cs-induced

Table 1 Parameters of the Sutton–Chen potential [13] and the surface energies e of relaxed lattices, calculated with this potential 0 e(meV )

Au Pt Ag

12.793 19.833 2.5415

C

34.408 34.408 144.41

m

8 8 6

n

10 10 12

˚) a (A

4.08 3.92 4.09

e [meV/(1×1) unit cell ] 0 (1×1) (1×2)

(1×3)

473.9 734.6 680.3

468.8 726.7 704.1

467.1 724.1 697.3

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Fig. 1. Side view on the simulated slabs. The top and bottom surfaces are reconstructed.

(1×2) reconstruction on Ag(110) [18]. In case of the (1×2) reconstructed Au(110) and Pt(110), the atoms in the second layers relax away from the missing rows, in disagreement with the experiments [19]. This shows that wrong direction of relaxation in the cases of Au and Pt is not a generic shortcoming of the Sutton–Chen potential but more an indication that this potential is less suitable for Au and Pt than for Ag. The deeper layers (third layer in case of the (1×2) reconstruction and third and fourth layers in case of (1×3) reconstructions) buckle in such a way that the atoms in the bottom of the grooves relax outwards, whereas the atoms below the topmost rows are pushed inwards, in agreement with the experiments [19].

4. Normal modes The normal modes are obtained as the solution of the eigenvalue equation of the dynamical matrix. The dynamical matrix between the atoms i and j is: D

a,b

(i, j )=

∂2U ∂R (i ) ∂r ( j ) a b

,

(2)

where a and b are the cartesian coordinates of the lattice. The advantage of the Sutton–Chen potential is that the dynamical matrix is an analytic function of the atomic coordinates. It allows, therefore, treatment of relatively large systems. For a slab, the translational symmetry in the

Table 2 Lattice relaxations calculated with the Sutton–Chen model at T=0 K. Dd =(d −d )/d is the change in the interlayer spacing. ij i j 0 d =a/앀8 is the bulk interlayer spacing and if the layer is buckled, we take the medium value (for the (1×3) structure, e.g. 0 d =(z +z )/2). b is buckling of the plane i (for the (1×3) structure, e.g. b =(z −z )/2d ) and p the pairing of the atoms in the 3 3 5 i 4 4 6 0 i plane i ( p =2 Dx /a, where x is the horizontal shift of the i-th atom from its equilibrium position. The directions of buckling and i i i pairing are displayed in Fig. 2. All values are in %

Au, Pt

Ag

(1×1) (1×2) (1×3) (1×1) (1×2) (1×3)

Dd 12

Dd

−12.1 −11.1 −10.9 −3.8 −5.6 −4.9

2.4 −3.3 −2.6 0.3 −0.7 −1.7

23

Dd 34 −0.8 1.2 −5.8 −0.4 −0.4 −1.5

Dd 45 −0.8 2.6

Dd

b 3

b 4

b 5

−1.0

3.6 2.1

4.2

2.4

−1.0

0.6 1.3

0.7

0.4

56

p

p 3

p 4

1.6 1.0

1.0

1.6 1.5

−0.4 0.2

0.6

0.8

2

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of the eigenvalue equation: (3) ∑ D (l,l∞;q)u (l∞;q,p)=4p2Mn2(q,p)u (l;q,p), a,b b a ∞ l where M is the atomic mass and D (l,l∞;q) is the a,b two-dimensional Fourier transform of D (i,j): a,b 1 (4) D (l,l∞;q)= ∑ ∑ D (i, j ) eiq r(i,j). a,b N jµl∞ iµl a,b As an illustration, for the unreconstructed slab with 24 layers, D (l,k;q) is a 72 ×72 matrix, and a,b for the (1×3) reconstruction, it is a 214×214 matrix. In an alternative approach, we will, for comparison, also present the results of a local-harmonic description ( Einstein model ) in which the local frequencies are [15]:

1

n= i 2p

S

3 ∑ D i,a a=1 , 3M

(5)

where D is the a-th eigenvalue of the local 3×3 i,a dynamical matrix D (i,i) at the site i. a,b 5. Dispersion curves Fig. 2. Lattice relaxations in noble metal (110) surfaces, calculated with the Sutton–Chen potentials: (a) unreconstructed surface; (b) (1×2); and (c) (1×3) reconstructed surfaces. Vertical arrows indicate the direction of buckling and horizontal arrows the direction of pairing for Au or Pt. The displacements are collected in Table 3.

vertical (z) direction is broken. As a consequence, normal modes can only propagate in the (x, y) plane. For each wavevector q in the (x, y) plane, there are in total 3N R ‘standing waves’ perpendicz ular to the slab, each of them corresponding to a discrete normal mode. N is the number of planes z in the slab and R is the reconstruction parameter (R=1 for unreconstructed and R=3 for the (3×1) reconstructed slab). Following Allen et al. [17], the normal-mode frequencies v (q) and their eigenp vectors u(l;q,p), where l is the index of the plane and p the polarization, are obtained as a solution

The dispersion curves along the symmetry directions of the two-dimensional surface Brillouin zone, Fig. 3, for three different reconstructions of Au and Ag slabs are shown in Figs. 4 and 5. Not surprisingly, at low frequencies ( long wavelength acoustic modes) the phonon dispersion curves calculated with the Sutton–Chen potentials fit well to the neutron scattering data on bulk Au [20] since the potential parameters are fitted to the experimental bulk modulus and elastic constants. At the top of the bulk modes, the agreement with the experiments is better for Ag than for Au and Pt, the calculated bulk phonon frequencies in the latter cases are about 15% too low [20]. Although it is not our aim to present a comprehensive analysis of surface modes on reconstructed surfaces, some comment is in place here. For all unreconstructed (110) surfaces, we observe at least two (depending on q) surface modes below the

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conjecture is also confirmed by viewing the Einstein, local-mode frequencies, collected in Table 4. The local high frequency modes originate in the atoms in the third rows (below the topmost rows) which are compressed by strong inward relaxation of the topmost rows of atoms. The S H band cannot be caused by pairing of the atoms in the second layer in the direction away from the missing rows since it is also seen on unreconstructed surfaces where there is no pairing. Since it appears above the optical band for Au and Pt and not for Ag, we conclude that the details of the interatomic potential are responsible for the S band. H Another consequence of the finite slab thickness are resonances of modes with the same symmetry. This causes many small singularities in the density of states, discussed in Section 6. The singularities, of course, disappear for semi-infinite systems.

Fig. 3. Brillouin zones of unreconstructed (total area), (1×2) reconstructed ( light+dark shaded areas), and (1×3) recon: is at q=(p/a, 0) and Y : structed (dark shaded area) surfaces. X at q=(0, 앀2 p/a).

bulk acoustic band. The polarizations of these two modes in the symmetry points are collected in Table 3. When the surface-phonon penetration depth becomes comparable to the slab thickness, the modes from opposite surfaces start to interact and the frequency splits. This is the case for the low frequency surface modes close to the C9 point. We would also like to comment on the high frequency surface band (S ). In the calculation, H the S band appears above the bulk optical band H of Au and Pt when the interatomic potential is approximated by a semi-empirical potential like the Sutton–Chen or the glue-model [3]. In the experiments, this mode has not been found — at least not above the optical band [21]. The polarizations of the S modes are collected in Table 3. The H band corresponds to strong vibrations of the bondlength connecting the topmost-row atoms with the atoms beneath (in the third layer). This

6. Density of states The density of states was calculated following a modified method of Gilat and Raubenheimer [22] in which the phonon frequencies and their gradients were first calculated on an approximately square mesh of wavevectors q in the first surface j Brillouin zone [ for unreconstructed (110) surface the mesh consisted of 40×28 points in one quadrant of the surface Brillouin zone]. The density of states is then equal to: g(n)=

앀2a2

1 ∑ S (n) . 6p2N i,j i,j |Vn | z i,j

(6)

Table 3 Polarizations of some surface modes of unreconstructed Au and Pt slabs in the symmetry points and half-way between them. S and S are two acoustic-branch surface modes, S is the surface mode above the bulk optical mode. X, Y and Z are the directions 1 2 H of polarization. X and Y lie in the plane of the surface so that X is perpendicular and Y parallel to the missing rows. Z is perpendicular to the surface. (X )Z means that the corresponding mode is predominantly Z polarized with a small component in the X direction

S 1 S 2 S H

Topmost layer Second layer Topmost layer Second layer Topmost layer Second layer

C9

: C9 [X

: X

: [S: X

S:

: S: [Y

: Y

: [C9 Y

XZ XZ Z Z Z Z

XZ XZ XZ XZ (X )Z X(Z )

Z X X Z Z X

YZ X X YZ (Y )Z X

Y X X Y Z 0

Y XZ XZ Y (X )Z Y

Y Z X 0 Z Y

Y Z X X (Y )Z (Y )Z

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I. Vilfan et al. / Surface Science 440 (1999) 279–289

(a)

(b)

(c)

Fig. 4. Phonon dispersion curves of an Au(110) slab with 24 layers of atoms (12 atoms in a column), calculated with the Sutton– Chen potential in the harmonic approximation: (a) unreconstructed surface; (b) (1×2); and (c) (1×3) reconstructed surfaces. The phonon dispersion curves of Pt(110) have the same shape, they are only scaled by the factor 1.30, originating in 앀e/(Ma2).

Fig. 5. Phonon dispersion curves of unreconstructed Ag(110) slab with 24 layers of atoms, calculated with the Sutton–Chen potential.

The sum S runs over all the rectangles on the j square mesh in one quadrant of the Brillouin zone. The sum over i is over all the frequencies n i,j calculated at the centres of these rectangles. S (n) is the length of the n=n line segment in i,j i,j the rectangle centered at q . g(n) is normalized so j that ∆2 g(n) dn=1 . The densities of states for the 0 slab with different reconstructions of Au(110) are shown in Fig. 6. The density of states of Pt(110) has equal structure but extends to higher energies, because the two Sutton–Chen potentials differ only in e. The calculated densities of states of ‘bulk’ and unreconstructed Ag(110) are shown in Fig. 7. A word on the ‘bulk’ density of states is relevant here. In the slab model, the bulk density of states is calculated in such a way that the vacuum above

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I. Vilfan et al. / Surface Science 440 (1999) 279–289 Table 4 Au(110) local-mode vibrational frequencies (THz). The atoms are numbered as in Fig. 2 Atom No.

1

2

3

4

5

6

7

8

(1×1) (1×2) (1×3)

2.084 2.167 2.143

2.346 2.186 2.206

2.472 2.185 2.114

2.380 2.617 2.159

2.411 2.640

2.418 2.508

2.360 2.411

2.389 2.416

the slab is removed and periodic boundary conditions are applied. However, as a finite system with periodic boundary conditions is not an infinite system, the bulk wavevector component q can z only have discrete values which are multiples of

9

2.389

10

Bulk

2.369

2.390 2.390 2.390

p/L, where L is the slab thickness. As a consequence of finite slab thickness, the slab and ‘bulk’ densities of states show many small singularities. The effect of these singularities, however, is integrated out in the free energy.

(a)

(b)

(c)

(d)

Fig. 6. Harmonic-approximation phonon density of states of an Au(110) slab with 12 atoms in each column: (a) bulk, i.e. a slab without free surfaces; (b) unreconstructed surface; (c) (1×2) and (d ) (1×3) reconstructed surfaces. The density of states of Pt(110) slabs is wider and lower by the factor 1.30. Many singularities can be seen clearly. They are the consequences of finite thickness of the slab.

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I. Vilfan et al. / Surface Science 440 (1999) 279–289

Fig. 7. Phonon density of states of an unreconstructed Ag(110) slab compared with the bulk density of states. The slab is composed of 24 layers, i.e. 12 atoms in a column.

7. Free energy The phonons of the slab form a set of 3N (where N is the total number of atoms in the slab) normal modes. The total surface-phonon free energy is then the difference of slab and bulk onemode free energies, summed over all normal modes [16 ]: F =3Nk T s B

P C A BD 2

hn

Dg(n)dn, (7) 2k T 0 B where Dg(n) is the difference between the densities of states of the slab with (two) free surfaces and the bulk. The surface free energy per one (unreconstructed) surface unit cell of size a×a/앀2 is: ln 2 sinh

F s , f =e + (8) s 0 2N N x y where N and N are the numbers of atoms in x y rows parallel to the x and y axes respectively, and e is the surface potential energy from Table 1. 0 The temperature dependence of the surface free energy of Au(110) in the harmonic approximation is shown in Fig. 8. For Pt(110) the effect of surface phonons on the relative stability of different reconstructions is less pronounced. In case of Ag(110), the energy difference between the unreconstructed

Fig. 8. Free energy of unreconstructed (solid line), (1×2) reconstructed (dashed line), and (1×3) reconstructed (dash-dotted line) Au(110) surfaces. All free energies are per unreconstructed surface unit cell.

and reconstructed structures is high and the surface phonons do not play a decisive role in stabilizing the (1×1) reconstruction, at least not in the present model. We also investigated the free energy of gold surfaces with local modes in the quasi-harmonic approximation to take into account the variation of the atomic relaxations with temperature. To determine the equilibrium atomic positions we performed molecular-dynamics (MD) simulations using the Sutton–Chen potential. First, the bulk lattice constant was determined for three different temperatures (150, 300 and 600 K ) by calculating the evolution of bulk configurations at constant temperature and pressure ( p=0) using the constrained equations of motion technique [23–25]. Discarding the first initial transitory period of 2 ps, an average of over 8 ps MD time was taken to get the temperature-dependent atomic volume and thus the lattice constant. The thermal expansion coefficient obtained from the simulations at 150 and 300 K is 2.65×10−5 K-1 and exceeds the experimental value [26 ] by a factor of two. Thus, the details of the Sutton–Chen potential are such that it does not reproduce the anharmonic properties of the lattice very accurately. In the next step we performed MD simulations

I. Vilfan et al. / Surface Science 440 (1999) 279–289

of slabs with reconstructed and unreconstructed surfaces at 150, 300 and 600 K. The temperature was set by using the constrained equations of motion and keeping the temperature of two atomic layers at the middle of the slabs constant. The initial configurations were constructed using the lattice constants from the above bulk MD simulations and then the total volume was kept constant. The simulated total time was 300 ps for each reconstruction and each temperature, only the last 200 ps being used for the thermodynamic averages. We observed that the (1×1) and (1×2) reconstructed slabs were unstable at 600 K, whereas the (1×3) slab was stable at that temperature during the whole simulation run. This is in agreement with our free energy calculations shown in Fig. 8 — the fundamental structure is stable while the metastable structures are not. At lower temperature, the simulated time was too short to see the escape from the metastable states. The equilibrium atomic positions, obtained by averaging over 200 MD configurations, were used to calculate the potential energy, the phonon frequencies and free energies, again in the local-mode approximation. The surface free energy is:

C

D

sinh(hn /2k T ) i B , (9) F =3k T∑ ln s B sinh(hn /2k T ) i bulk B where the local-mode frequencies in the slab, n , i are given by Eq. (5) and n is the vibrational bulk frequency in the bulk. The phonon free energy is very weakly affected by anharmonicity. The main difference between the harmonic and quasi-harmonic approximations is in the potential energy, which increases with temperature in the quasi-harmonic approximation because of thermal expansion, whereas it is temperature independent in the harmonic approximation. As a consequence, the total free energy has weaker temperature dependence in the quasi-harmonic approximation.

8. Results and discussion In this work we have discussed the interplay between the surface phonons and reconstruction

287

on (110) surfaces of noble metals. The calculations were carried out with the Sutton–Chen potential which has analytic derivatives (forces and force constants). It therefore allows calculations of dispersion curves and phonon density of states on relatively large systems. Together with the slab model, it is very suitable for the surface investigations. The calculated directions of lattice relaxation along the z ([110]) axis of all three metals considered (Au, Pt, Ag) are in agreement with the experiments. The topmost atoms of Au and Pt relax inwards more than the Ag atoms. In the simulations, the second-layer atoms of the (1×2) missing-row reconstructed Ag surfaces relax horizontally towards the missing rows, in agreement with the experiments on Cs doped Ag(1×2) [18]. In the case of Au or Pt, these atoms relax away from the missing rows, in disagreement with the experiments. This shows that the direction of relaxation depends on the parameters of the Sutton– Chen potential and is not a generic property of the Sutton–Chen or other semi-empirical potentials. In general we found that the Sutton–Chen potential is better suited for the 4d than for the 5d transition metals. The reason lies in the shape of the potential. Strong inward relaxation of the topmost-layer atoms causes a stiffening of the potential acting on the atoms in the third layer below the topmost rows. In case of reconstructed Au and Pt, the potential is additionally stiffened by the lateral relaxation of the second-layer atoms away from the missing rows. Stiffening is most clearly seen in the local-mode approximation: the vibrational frequencies of these atoms increase above the bulk value. In the collective-mode description stiffening causes a surface-phonon band (S ) above the bulk H optical band of Au and Pt. The existence of the S band was not confirmed by the experiments for H Au(1×2) [21]. Such a band also causes an increase in the surface-phonon free energy. With a more realistic potential the upper phonon band should not be present and the role of surface phonons in stabilizing reconstructed surfaces should be more pronounced. In this paper we considered the phonon contribution to the surface entropy and free energy.

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I. Vilfan et al. / Surface Science 440 (1999) 279–289

Other contributions which could influence the stability of reconstructed surfaces come from meandering of thermally excited steps and from the conduction electrons. Instead of step meandering entropy let us make a crude estimate of the Ising domain contribution to the entropy of 1×2 reconstructed surfaces. Close to (but below) the deconstruction transition temperature T the antiC phase domain free energy is small and therefore S# E /T , where E  is the domain-wall energy. C For a crude estimate we remind that 10 atomic bonds, each having an energy of the order 1 eV, are broken for a minimal domain. Because of anisotropy in the domain-wall energy, the domains are strongly elongated in the direction of the missing rows. For a typical domain size of 3×10 surface unit cells, this gives the entropic contribution to the free energy of the order 0.1 eV/(surface unit cell ) or smaller. This is smaller than (but of the same order as) the phonon contribution. It is not trivial to see how the difference between the step free energies of different reconstructions behave, but we cannot exclude the possibility that the step free energy also plays a role in stabilizing different reconstructions. The electronic contribution to the surface entropy is small, as can be seen from the following estimate. The entropy of free electrons is: S=

p2 3

g(e )k2 T, F B

(10)

where e is the density of electrons at the Fermi F energy. In the bulk at room temperature, TS is smaller than 1 meV/(unit cell ). The electronic contribution to the surface free energy is thus negligible. One difference between the phonon and electronic contributions should be emphasized. Whereas for phonons, all modes in the Brillouin zone contribute to the entropy, only the modes close to k (short wavelengths) make the electronic F contribution to the entropy. Therefore it is possible that the electronic contribution to the entropy, although small, is more sensitive to reconstruction than the phonon part. In conclusion, the free energy of surface phonons must be included in the stability analysis of reconstructed surfaces when the energy difference

between different reconstructions is of the order 1 meV/(surface unit cell ) or smaller. This is the case with the (1×2) and (1×3) reconstructions in Au(110). The surface phonons do influence the relative stability of reconstructed surfaces in particular at higher temperatures. For a more detailed quantitative analysis, however, a detailed knowledge of the surface phonon densities of states for the relevant reconstructions were crucial.

Acknowledgements We would particularly like to thank T. Deutsch for many fruitful discussions. The work was in part supported by the French-Slovenian programme PROTEUS.

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