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