Relation between surface stress and (1 2) reconstructionfor (1 1 0) fcc transition metal surfaces

Stephane Olivier*, Andres Saul, Guy TregliaCentre de Recherche sur les Mecanismes de la Croissance Cristalline, CNRS, Campus de Luminy, Case 913, 13288 Marseille Cedex 9, France

Abstract

In this work we question the validity of surface stress relief as a driving force for surface reconstruction in the particular case

of the (1 1 0) face of fcc transition metals. To this aim, we calculate within tight-binding second moment potential, both surface

energy and surface stress before and after the (1 2) missing row reconstruction. Doing that, one indeed checks thatreconstruction is energetically favourable for Au and Pt and not for Ni and Cu as experimentally seen, but we also find that it

enhances surface stress along the dense direction for all the elements and along the less dense direction for those which

reconstruct. Therefore, surface stress relief is definitely not the driving force for this reconstruction.

# 2003 Elsevier Science B.V. All rights reserved.

PACS: 62.20.Dc; 68.35.Bs; 68.35.Md

Keywords: Surface stress; Surface relaxation and reconstruction; Molecular dynamics; Low index single crystal surfaces; Transition metals

1. Introduction

It is well known that clean surfaces of some pure

materials undergo reconstructions, which can be clas-

sified into three categories: those which just involve

displacements of the surface atoms (e.g. Si(1 0 0)-

(1 2) [1], Ge(1 0 0)-(1 2), Si(1 1 1)-(7 7),W(1 0 0)-(1 2) [24]), and those for which surfacedensity either increases due to the addition of atoms

(e.g. Ir, Pt, Au(1 0 0)-(1 5) [5] or Au(1 1 1)-(22 3p ) [6,7]) or decreases due to the removal ofatoms (e.g. Ir, Pt, Au(1 1 0)-(1 2)). Obviously, allthese reconstructions only occur if they lead to a

decrease of the surface energy, but their physical

origins can be manifold. For Si and Ge(1 0 0), for

example, the (1 2) reconstruction minimises the

number of surface atom dangling bonds, the buckling

of the surface dimers being closely related to details of

the electronic structure. For the (1 5)-like recon-structions of Ir, Pt and Au(1 0 0) the reconstruction

corresponds to the formation of a quasi-hexagonal

surface layer that increases the compactness of out-

ermost surface layer. A general argument concerning

the stress release produced by the surface rearrange-

ment has been sometimes used. Although this argu-

ment is probably sound for the kind of reconstructions

associated to an increase of the number of surface

atoms, recent experiments performed by Bach et al.

[8] have shown that the stress relief for the (1 0 0)

surface of Au is not large enough to cause reconstruc-

tion. Previously, Needs and Mansfield [9] have argued

that the surface stress alone does not contain sufficient

information to determine whether a surface is stable to

a change in the density of surface atoms. In the same

way, Filippetti and Fiorentini [10] have supported the

Applied Surface Science 212213 (2003) 866871

* Corresponding author.

E-mail address: olivier@crmc2.univ-mrs.fr (S. Olivier).

0169-4332/03/$ see front matter # 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0169-4332(03)00018-7

view that no unique quantity can be invoked as a

general driving force for surface reconstruction of

metals. In order to analyse the applicability of a

surface stress relief criterion to a different kind of

reconstruction, we present in this paper calculations of

surface energy and surface stress for the missing row

(1 2) reconstruction which occurs on the (1 1 0)surface of fcc 5d transition metals Ir, Pt and Au.

The (1 1 0) surface of fcc is anisotropic with two

inequivalent main directions. Along the [1 1 0] x-direction there is a higher density of atoms than along

the [0 0 1] y-direction (see Fig. 1(a)). Themissing

row reconstruction consists in removing one over

two [1 1 0] dense row, leading to a succession of(1 1 1) microfacets (see Fig. 1(b)). This reconstruction

has been extensively studied experimentally [1114]

and theoretically [1518].

We have considered in this work the seven fcc

transition metals: two 5d metals Pt and Au which

reconstruct and three 4d metals Rh, Pd and Ag, and

two 3d metals Cu and Ni, which do not. Ir which

reconstructs experimentally has not been considered

because we were unable to get a satisfactory parame-

terisation with our semi-empirical potential. The cal-

culations have been performed using a semi-empirical

tight-binding potential. In order to get a theoretical

criterion for the stability of reconstruction, alternative

to the experimental one, we compare the surface energy

of systems before and after reconstruction. We then

calculate the surface stress in the same conditions, in

order to see if there exists some correlation between the

signs of the variations in surface energy and surface

stress induced by reconstruction.

1.1. Method

The semi-empirical potential used here is derived

from a tight-binding description of the electronic

structure within the so-called second moment approx-

imation (SMA) [19,20]. In this scheme the contribu-

tion of the i atom to the total energy writes as a sum of

a pair-wise repulsive part and a many-body attractive

band contribution:

Ei AX

j

epdij=d01 xX

j

e2qdij=d01s

(1)

j runs over the neighbours of the atom i, dij is the

interatomic distance, and d0 is the first neighbour bulk

equilibrium distance. The four parameters p, q, A and xare determined (see Table 1) by fitting the experimen-

tal cohesive energy Ec, lattice parameter a, and some

elastic constants (bulk modulus B, shear constants C44or C0). The calculations have been performed withinteractions calculated up to third neighbours using (1)

and then linked to zero at the distance of the fourth

neighbour using a fifth-order polynomial.

We have used a slab with periodic boundaries along

the two surface directions [1 1 0] and [0 0 1]. The slabthickness of 15 layers has been chosen in order to

avoid interactions between the two surfaces. All the

atoms of the slab were allowed to move and we have

fully relaxed the system using a quenched mole-

cular dynamics procedure.

Fig. 1. (1 1 0) fcc surface before (a) and after (b) the missing-row

reconstruction.

Table 1

Parameters (p, q, A (eV) and x (eV)) of the SMA potential used in this work

Ni Cu Rh Pd Ag Pt Au

p 11.34 10.55 14.92 10.90 10.85 10.47 10.145

q 2.27 2.43 2.38 3.72 3.18 3.935 4.03

A 0.0958 0.08938 0.09357 0.17099 0.10309 0.29453 0.20967

x 1.5624 1.2799 1.9159 1.7134 1.1895 2.6929 1.8184

S. Olivier et al. / Applied Surface Science 212213 (2003) 866871 867

1.2. Surface energy and surface stress

Using a slab geometry the surface energy can be

calculated as:

g 12A

EN NEb (2)

N is the number of atoms in the slab unit cell, A the

surface area, EN the total energy of the slab, and Ebthe energy of bulk atom (i.e. the opposite of the

cohesive energy). The surface stress is a symmetric

tensor with elements:

sij 1A

@Ag@Eij

(3)

where Eij is element of the strain tensor andi; j x; y; z. Note that a positive (negative) value ofs means a tensile (compressive) surface stress. Afterfull energy relaxation all the szj components are zero.The simple development of this equation leads to the

Shuttleworth [21] relation:

sij gdij @g@Eij

; i; j x; y (4)

There are two contributions to the surface stress, the

first one is the surface energy and the second one is its

strain derivative.

2. Results

2.1. Surface energy

In Table 2, we give the calculated surface energies

for the unreconstructed (1 1) and reconstructed(1 2) surfaces of fcc transition metals, both beforeand after atomic relaxation. As can be seen, influence

of relaxation is minor (from 0.5% for Ni to 9% for Pt).

We then plot in Fig. 2 the energy balance Dg in boththe not relaxed (a) and relaxed cases (b). Note that a

negative Dg means that the reconstructed structure isenergetically favorable. Satisfactorily we do find, in

agreement with experiments, that the 5d metals Au

and Pt reconstruct (Dg < 0) whereas the 3d metals Niand Cu do not (Dg > 0). Moreover, the behaviour ofthe 4d metals is found intermediate between the 3d and

5d ones sinceDg is negative for Ag and Pd and positivefor Rh. This is also consistent with experiments since

these elements, even though they do not reconstruct

under perfect conditions, are found to reconstruct

when submitted to external constraints (epitaxial

stress [22] or adsorption of alkali atoms [23,24]).

Let us recall that our aim here is to correlate surface

stress and reconstruction in the framework of a com-

mon potential. Therefore, in view of the overall con-

sistency between our energetic calculations and

experimental observations, and being aware of the

limitations related to the use of semi-empirical

many-body potentials such as SMA, but also EAM

[25] or others, we will define in the following as

reconstructing elements those which do it according

to the energetic criterion (Ag, Pd, Au and Pt) and as

not reconstructing ones the others (Ni, Rh and Cu).

2.2. Surface stress

The next step was the calculation of surface stress

before and after reconstruction. The results are given

in Table 3 along the two directions. Let us first remark

that, as expected with such a potential from the effect

of bond breaking, we find that the surface undergoes a

Table 2

Surface energy of the not reconstructed (g11) and the recon-structed (g12) surfaces in erg/cm2 for a not relaxed (NRel) and afully relaxed (Rel) slab

Ni Rh Cu Ag Pd Au Pt

g11 NRel 1943 2250 1370 766 981 699 1347g11 Rel 1933 2242 1358 744 923 611 1222g12 NRel 2005 2305 1405 763 957 668 1299g12 Rel 1992 2295 1391 741 904 589 1186

Table 3

Surface stress along the dense x- and less dense y-directions with

and without reconstruction in erg/cm2 for a not relaxed (NRel) and

a fully relaxed (Rel) slab

Ni Rh Cu Ag Pd Au Pt

sxx;11 NRel 1620 1787 1417 1490 2895 3114 5116sxx;11 Rel 1183 1329 980 872 1453 1194 2218sxx;12 NRel 1630 1798 1426 1493 2892 3106 5105sxx;12 Rel 1268 1403 1036 884 1462 1236 2258syy;11 NRel 2171 2342 1739 1424 2470 2499 4189syy;11 Rel 1433 1538 1045 606 805 580 1126syy;12 NRel 1697 1861 1421 1329 2448 2556 4241syy;12 Rel 1108 1183 825 532 791 665 1205

868 S. Olivier et al. / Applied Surface Science 212213 (2003) 866871

tensile stress (positive value) for all the elements along

both directions. We will come back on this point in

Section 3.

From these values, we are then able to calculate the

variation of surface stress induced by reconstruction

along one or the other direction: Dsxx and Dsyy. Theselead to the general trends which appear in Fig. 2. The

first general result is that, whatever the elementDsxx ispositive at least in the relaxed situation. It means that

the stress along the dense direction increases under

reconstruction, which could be guessed from simple

bond breaking elements, when going from 3D bulk to

2D surface and now to a dense 1D row. The situation

along the less dense direction is more complex.

Indeed, one finds that syy decreases for the elementswhich do not reconstruct whereas it increases for those

who do it. In other words, the occurrence of recon-

struction is linked to an enhancement of the tensile

stress along this non dense direction. These results

invalidate definitely any argument based on surface

stress relief to understand the physical origin of the

missing row reconstruction.

It is worth noticing that, contrary to what occurs for

surface energy and in agreement with what could be

guessed from elastic properties, the atomic relaxation

quantitatively modifies the absolute value of surface

stress. Thus, in the case of Ni for instance, relaxation

modifies surface stress by about 30% along the dense

Fig. 2. Reconstruction induced variations of surface energy (solid line) and surface stress along the dense x-direction (dashed line) and less

dense y-direction (dotted line) for the seven fcc metals, for a not relaxed (a) and for a fully relaxed slab (b), and also as a function of the unique

parameter q (c). In (c) the energetic and stress quantities were divided by the energetic parameter x [20].

S. Olivier et al. / Applied Surface Science 212213 (2003) 866871 869

direction and by about 50% along the less dense one.

Moreover, the effect is sufficiently important to mod-

ify qualitatively the trend followed by Dsxx along thetransition series (see Fig. 2(a) and (b)).

3. Discussion

Our main result is that, within SMA, elements

which reconstruct in the missing row model undergo

a general increase of surface stress, which means that

this reconstruction is by no means driven by a ten-

dency to release this stress. However, things are

different along the two directions. Let us recall that

the (1 2) reconstruction consists in removing onedense [1 1 0] row over two in the surface. Therefore,the atoms in the remaining rows have lost neighbours

with respect to the unreconstructed surface so that they

would like to increase the in-row bonding to

compensate this. This could be done by enhancing

their orbital overlap by increasing close-packing along

the row. Unfortunately, these atoms are strongly

bonded to the substrate which forbids them to do that.

This competition, which is already at the origin of the

tensile character of surface stress, obviously enhances

this character for the in-row stress.

The behaviour along the [0 0 1] less dense direction

is more difficult to understand. We have shown in

Fig. 2(a) that the more the element reconstructs (Dgdecreases), the more the difference of surface stress

along the less dense [0 0 1] direction increases. In

order to get more insight on this argument, we have

extended to the case of surface stress, the simple

model previously used to interpret the evolution of

surface energy under reconstruction [20]. The main

ingredient is to assume that the reconstruction is

mainly driven by the q parameter of the SMA poten-

tial, which characterises the range of its attractive part

(see Section 1.1). We have then plotted in Fig. 2(c) the

variation of surface energy and stress, normalised to x,as a function of this unique q parameter, having fixed p

(which drives the repulsive part) to the mean value of

10.55, d0 (the first neighbour bulk equilibrium dis-

tance) to 2A(see Section 1.1). As can be seen, we

indeed recover the essential features of Fig. 2(a).

The intermediate behaviour of the 4d elements

previously mentioned is quite consistent with this

analysis. Indeed, Guillope and Legrand [20] have

shown that a small compression (1%) can favour

the reconstruction in Ag. This is confirmed from

the experimental point of view in the case of Ag/

Cu(1 1 0) [26]. In the same spirit, Filhol et al. [22]

have shown that deposition of Pd on Ni(1 1 0), with a

compression of Pd to the lattice parameter of Ni, leads

to a reconstruction of Pd.

Finally, we can wonder if is it possible to get some

local view of the microscopic mechanism responsible

of this puzzling variation of stress under reconstruc-

tion along the less dense direction. To answer this

question, we give in Fig. 3 the local contributions to

the variation of this stress induced by reconstruction,

for two elements presenting opposite behaviours: Au

which reconstructs and Cu which does not. The main

feature is that, in both cases, if one averages stress

variation per plane parallel to the surface, one finds

that reconstruction has some compressive effect in the

outer layer, a tensile one just below, and almost no

effect in the third plane. Note that when adding all

these contributions one indeed recovers the variations

derived from the values given in Table 3:85 erg/cm2for Au and 225 erg/cm2 for Cu. There is a smalldifference coming from the lower planes not shown in

Fig. 3. If one now looks more quantitatively to these

values, one sees that the results are almost reversed

between both elements: the decrease of the tensile

stress at the surface is more pronounced than the

increase in the first underlayer (1013 erg/cm2 versus778 erg/cm2) for Cu, whereas the opposite is observed

for Au (718 erg/cm2 versus 872 erg/cm2). The netresult is that tension increases for Au and decreases for

Cu, due to the different balance between the two layers

for both elements.

Fig. 3. Local contributions to the variation of stress along the less

dense direction y induced by the reconstruction for Cu (left) and Au

(right).

870 S. Olivier et al. / Applied Surface Science 212213 (2003) 866871

In conclusion, using a semi-empirical SMA poten-

tial, we find as expected that reconstruction is ener-

getically favorable for Pt and Au but not for Ni and Cu,

in agreement with experiments. Moreover, we have

shown that surface stress is enhanced due to recon-

struction along the dense direction whatever the ele-

ment, and along the less dense one only for elements

which do reconstruct, relaxation having to be taken

into account to get reliable results. This behaviour,

which definitely invalidates a surface stress relief

criterion for this reconstruction, has been analysed

locally and interpreted in terms of the essential para-

meters of the interatomic potential.

Acknowledgements

We wish to thank P. Muller for fruitful discussions.

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S. Olivier et al. / Applied Surface Science 212213 (2003) 866871 871

Relation between surface stress and (1x2) reconstruction for (1 1 0) fcc transition metal surfacesIntroductionMethodSurface energy and surface stress

ResultsSurface energySurface stress

DiscussionAcknowledgementsReferences