Relaxation and reconstruction on (111) surfaces of Au, Pt, and Cu

8 Pages·0.17 MB·English

by  Z. Crljen

Relaxation and reconstruction on (111) surfaces of Au, Pt, and Cu
Zˇ. Crljen, P. Lazi´c, D. Sˇokˇcevi´c, and R. Brako
R. Boˇskovi´c Institute, P.O. Box 180, 10002 Zagreb, Croatia
(Dated: February 2, 2008)
Wehavetheoreticallystudiedthestabilityandreconstructionof(111)surfacesofAu,Pt,andCu.
Wehavecalculatedthesurfaceenergy,surfacestress,interatomicforceconstants,andotherrelevant
quantities by ab initio electronic structure calculations using the density functional theory (DFT),
in a slab geometry with periodic boundary conditions. We have estimated the stability towards
a quasi-one-dimensional reconstruction by using the calculated quantities as parameters in a one-
dimensionalFrenkel-Kontorovamodel. Onallsurfaceswehavefoundanintrinsictensilestress. This
stress is large enough on Au and Pt surfaces to lead to a reconstruction in which a denser surface
layerisformed,inagreementwithexperiment. Theexperimentallyobserveddifferencesbetweenthe
3 densereconstructionpatternonAu(111)andasparsestructureofstripesonPt(111)areattributed
0
tothe details of theinteraction potential between the first layer of atoms and thesubstrate.
0
2
PACSnumbers: 71.15.Mb;68.35.Bs;68.47.Dc
n Keywords: Density functional calculations; Surface energy; Surface relaxation and reconstruction; Surface
a stress;Lowindexsinglecrystalsurface;Copper;Gold;Platinum
J

0
I. INTRODUCTION to estimate the stability of the first layer of atoms with
3
respecttoaquasi-one-dimensionalreconstruction,which

1 can be treated theoretically by the Frenkel-Kontorova
The properties of close-packed noble and transition
v model. In the Sec. IV we discuss the results.
metal surfaces have been extensively studied in recent
8
years. Theresearchhasmadegreatadvancessincethein-
9
5 troduction of new experimentaltechniques, in particular
II. CALCULATIONS OF SURFACE
1 scanningtunneling microscopy(STM), andthe improve-
PROPERTIES
0 ment of first-principle computational methods based on
3 the density functional approach. These surfaces show a
0 The reconstructed phase of some close-packed metal
wide variety of behavior with respect to reconstruction,
/ surfacesmayhaveaperiodicitywhichinvolvesmanysur-
t preferred site and strength of chemisorption of reactive
a face atoms. For example, the reconstruction of Au(111),
m species, adsorbate diffusion, etc. one of the surfaces considered in this paper, involves a
- Owing to the abrupt change of the electronic struc- uniaxially compressedfirst atomic layer with a period of
d ture on metal surfaces the first layer of atoms may have around 70 ˚A. Full first-principle calculations with such
n rather different properties from the bulk. Often, a large large supercells are not feasible. One instead evaluates
o
intrinsic tensile surface stress appears,which is the driv- the stability by calculating the relevant quantities of the
c
: ing force leading to compressive reconstruction on many unreconstructed surface, and uses them in phenomeno-
v close-packedmetalsurfaces. The stabilityofaparticular logical models of reconstruction.
i
X surface is the result of the interplay of several physical There have been a number of papers in which the
r quantities characteristic of surface, such as surface en- stability of (111) surfaces of noble metals was evalu-
a ergy,surfacestress,interatomicforceconstants,etc. The ated using the Frenkel-Kontorova (FK) model8,9,10 of
reconstruction is more likely to occur the larger the in- uniaxially compressive reconstruction. Ravelo and El-
trinsic stress and the smaller the energy for sliding the Batanouny(Ref.11,seealsoreferencestherein)havecon-
atomsofthefirstlayersintopositionsoutofregistrywith structed effective potentials between surface atoms and
the substrate. performed molecular-dynamics simulations of the recon-
Inthispaperweconsider(111)surfacesofcopper,plat- structedphase. MansfiledandNeeds23 havederivedcon-
inum, and gold. The Au(111) surface reconstructs form- ditions of stability of (100) and (111) surfaces towards a
ing a uniaxially compressed layer, which can be seen in compressive reconstruction in the FK model, and eval-
STM experiments.1,2,3 The Pt(111) reconstructs only at uated it for several metals. Apparently, their values of
high temperatures,4,5 or in the presence of saturated Pt the parametersare not good enough, as they have found
vapour,6,7 and Cu(111) does not reconstruct. We report no reconstruction of Pt(111) and Au(111). In particu-
on first-principle numerical calculations of the proper- lar, the strength of the interatomic potentials seems to
ties of these surfaces using the density functional theory be overestimated. Takeuchi et al.21 have constructed a
(DFT). In Sec. II we first describe the generalitiesof the two-dimensionalpotentialbetweenfirst-andsecond-layer
computationprocedure. Wethencalculatetheimportant atoms of a Au(111) surface using the results of first-
physical quantities, such as the effective force constants principle calculations. They have calculated the recon-
in the first layer, the surface energies, and the intrinsic struction pattern using simulation techniques in a two-
surfacestress. InSec.IIIweusethesequantitiesinorder dimensional FK model. Using the two-dimensional po-
2
tential in a molecular-dynamics simulation, Narasimhan louin zone were used. In some calculations described in
and Vanderbilt22 have concluded that the herringbone the following subsections, e.g., when alternate rows of
patternofthereconstructionstripesobservedonAu(111) first-layer atoms were displaced in opposite directions,
isfavoredbythelong-rangeelasticinteractionsmediated in order to probe the restoring forces within the first
by the substrate. Recently, Narasimhan and Pushpa25 layer, unit cells with four atoms in each plane were
havestudiedatwo-dimensionalFKmodelofthePt(111) used, and the number of k-points was reduced to 18. In
surfaceforwhichtheparametershavebeenobtainedfrom the z-direction, perpendicular to the layers, only k = 0
ab initio calculations. They have obtained simulated was considered, consistent with the assumption that the
STM images of the reconstruction stripes recombining slabs, which repeat periodically because of the compu-
and intersecting in various ways, similarly to the struc- tational method, did not couple to each other. In most
tures observed in STM experiments. calculations, the two bottom layers were kept fixed at
The aim of this paper is to re-evaluate with highest the bulk separation. We used an energy cutoff for the
accuracy the properties of the surfaces which determine plane-wave basis set of 340 eV, and the electronic oc-
the stability towards reconstruction. In this section we cupation was smeared by a pseudothermal distribution
report on such calculations for the (111) surfaces of Au, of T = 0.2 eV. We performed some checks with a lower
Pt, and Cu in the density functional theory (DFT) ap- value, T = 0.1 eV, and found that the changes of the
proach. In the following subsections we describe, in or- calculated quantities were irrelevant.
der,thegeneralitiesofthenumericalcalculations,thecal- Next, we calculated the structure of clean (111) sur-
culationofsurfaceenergies,surfacestress,surfacespring faces. Therelaxationofthesurfacelayersfromtruncated
constants,andthe calculationofenergyrequiredtoslide bulk positions was foundto be rather smallfor the three
thecompletefirstlayertovariouspositionsawayfromthe metals considered. For gold, the first and second layers
most stable one. In the next section we use these results relaxlessthan0.3%oftheinterlayerdistanceinthe 111
h i
as input parameters in the (effectively one-dimensional) direction, with an energy gain of 1 meV, which is irrel-
Frenkel-Kontorova model of uniaxially compressive re- evant considering the accuracy of the calculation. The
construction. first platinum layer relaxes outwards by about 1%, with
anenergygainof2.5meV.Thisunusualexpansionofthe
interlayer distance has been observed experimentally.13
A. DFT calculations On copper, the first three layers move inwards by less
than 1%, and the energy gain is 4 meV.
We have performed first-principle calculations of the
electronic structure of (111) surfaces within the density
functionaltheoryapproach,usingthedacapo program.12 B. Surface energy
Wehaveusedtheprovidedultrasoftpseudopotentialsfor
thePerdew-Wangexchange-correlationfunctionalPW91 We obtained the surface energy per atom as the dif-
and the generalized gradient approximation (GGA). ference of the energy of the bulk and of a slab. In order
We have first made calculations for bulk metal, in or- to minimize the influence of the computational details,
dertodeterminethelatticeconstantatwhichtheenergy we calculated the energy of the bulk using a six layer
per atom is minimum, using the primitive fcc unit cell. slab (i.e., twice the abc stacking of the 111 fcc direc-
h i
The lattice constant found often differs slightly from the tion) with no vacuum layers. After that, we changedthe
experimentalvalue: forexample,ourvalueis4.00˚Acom- configurationbyintroducingfourlayersofvacuumwhich
pared with the experimental value 3.92 ˚A for Pt, 4.18 ˚A createdtwo(111)surfaces,andcalculatedtheenergy,al-
comparedwiththeexperimentalvalue4.08˚AforAu,and lowing both surfaces to relax. The surface energies γ,
3.66 ˚A compared with 3.61 ˚A for Cu. We have followed one half of the difference of the energies obtained in the
the common practice using the computed value of the two calculations, are given in the GGA column in Ta-
lattice constant in susequent calculations, which ensures ble I. These values are consistently smaller by about
that no spurious bulk stresses appear. onethirdthanthose calculatedrecently byVitos etal.14
The surfaces were described by a slab of five or more and Galanakis et al.15,16 and various experimental val-
hexagonallayersofatoms. Sincethecalculationassumes ues reportedtherein. One possible source of the discrep-
periodicityinallthreedimensions,themetallicslabswere ancy are the different pseudopotentials and exchange-
separatedby typically five layersof vacuum. In addition correlationfunctionalsused. Wethereforealsoreportthe
toidealsurfaces,wecalculatedsurfacesperturbedinvar- surface energies calculated using the LDA functional for
iouswaysinordertodeducequantitieslikesurfacestress theelectronicdensitiesobtainedintheGGAcalculations,
and surface energy, which can be used to estimate the inthecolumnlabeledasLDAinTableI. Theseelectronic
stability of the surface. More details are given in the densities are, of course, non-self-consistent with respect
respective subsections. to the LDA functional. They are always larger than the
In the case of a clean surface, the unit cell in the di- GGA results, in better agreement with Refs. 14,15,16.
rections along the surface plane (x y) consisted of one Anotherpossiblesourceofdiscrepancyisthesmallthick-
−
atom, and 56 k-points in the two-dimensional first Bril- nessoftheslabinourcalculation. Inthebulkcalculation
3
bulk term, as discussed earlier. The diagonal element of
TABLE I: Surface energy γ, surface stress τ, and force con-
thesurfacestresstensorτ,whichistherelevantquantity
stantsbetweennearestneighborsk0,for(111)surfacesofAu,
for the surface reconstruction, is
PtandCu. Thetwovaluesofthesurfaceenergyγ havebeen
obtained from a fully self-consistent DFT GGA calculation, ∆E
andfromapplyingtheLDAfunctionaltothesameelectronic τ = , (1)
2∆A
density,seetext. Thetreevaluesoftheforceconstantk0have
been calculated from the bulk modulus B, (k0B), and from where ∆E is the change of energy when the lattice con-
the forces obtained in DFT calculations in which the first- stantisvaried(aftersubtractingthequadraticbulkterm)
layer atoms were slightly displaced ”compressively” (k0C) or and∆Aistheassociatedchangeofthesurfacearea. The
”laterally” (k0S),see Fig. 1. numerical values are given in Table II. We have found
γ (eV/˚A2) τ (eV/˚A2) k0 (eV/˚A2) thatinthiscasethereisalmostnochangeifthenon-self-
consistent LDA values of the energy are used instead of
GGA LDA k0B k0C k0S
the GGA values.
Au 0.050 0.080 0.15 1.83 1.41 1.18
Pt 0.084 0.123 0.34 3.12 2.23 2.55
Cu 0.080 0.112 0.11 2.04 1.73 2.36 D. Surface spring constants
The simplestatomic-scalemodelofthe lattice dynam-
(i.e., 6 layers without vacuum) we used only one k-point ics of fcc metals is to assume central harmonic forces
inthez-direction,inordertokeepthesimilaritywiththe between nearest neighbor atoms. In this model there
surface calculations, and consequently the long-distance isa universalscalingofphononspectraanddifferentele-
contributions in the z-direction might not be correctly mentsoftheelasticitytensor,whichisobviouslyanover-
taken into account. This can be investigated by using simplification. Nevertheless, this model is sufficient for
elementary cells with more layers, i.e., repeating the abc the purpose of this work. The force constant k0B of the
structurethreeormoretimes,butwehavenotdonesuch nearestneighborbondinthe bulkcanbe foundfromthe
checks. Also, we used the equilibrium lattice constants bulk modulus19 B =1/3(C +2C ):
11 12
obtainedself-consistentlyinourGGAcalculationswhich
1
were larger than the experimental values and those ob- k0B = , (2)
tained by other authors from LDA calculations. 2aB
In the following section, when discussing the stability where a is the lattice constant of the conventional fcc
ofthesurfaces,weshallusetheLDAresults,whichseem unit cell.
to agree well with the best values reported in the litera- However, the effective force constant between the
ture. atomsinthefirstsurfacelayercandifferfromthevaluein
the bulk, since the lower coordination can substantially
alter electronic properties, and an ab-initio DFT calcu-
C. Surface stress lation is necessary. (Note that this force constant also
contributes to the quadratic energy term in the preced-
The surface stress can be found by considering the ing subsection, where we did not consider the possibility
changeofenergywhenthelatticeconstantinthexandy that it was modified within the first layer. However, the
direction(i.e.,inthe surfaceplane)isvaried. Inthebulk possible error which this introduces in the calculation of
calculationstheleadingcorrectiontotheenergywhenthe the surface stress is negligible.)
lattice constant is varied around the equillibrium value Wemadeslabcalculations(withrelaxationturnedoff)
is quadratic by construction, since we used the lattice in which rows of atoms in the first layer were displaced
constant corresponding to the energy minimum. Owing by about 0.05 ˚A either in the direction of the rows or
to the reduced coordination, the optimum interatomic perpendicularly to it, as shown in Fig. 1. Unlike other
distance in the surface layeris smaller than the bulk lat- calculationsin this paper, here we were not interestedin
tice constant, as already mentioned, causing an intrinsic the total energy, but rather in the restoring forces ap-
tensilesurfacestress.17,18 Therefore,thesurfacecontibu- pearing owing to the nonequlibrium configurations. We
tion to the energy has a leading linear term if the lattice analyzed the results assuming central harmonic interac-
constant in the x and y direction is varied. tions (“springs”) between nearest neighbors, neglecting
In order to find the intrinsic surface stress, we cal- thecouplingtothesecondlayer. Takingintoaccountthe
culated the energy of slabs consisting of six layers, as in numberofnearestneighborbondsandtheangles,wecal-
theprecedingsubsection,bothunperturbedandwiththe culatedthe force constants k0S. The results for both the
unit cell compressed or expanded along the coordinates bulk values obtained from Eq. (2) and the surface DFT
x and y (i.e., in the directions in the surface plane) for calculations are shown in Table I. We note once again
around 0.1%, not allowing any additional relaxation in that this approach does not attempt to give a complete
thez direction. Wealsocalculatedsimilarconfigurations descriptionofthelatticedynamicalpropertiesofthefirst
withoutvacuumlayers,inordertosubtractthequadratic surface layer. In general, even if the interactions with
4
TABLE II: The parameters of theFrenkel-Kontorovamodel of uniaxial compressive reconstruction. A is thearea per surface
atom, a is the distance between thefcc and bcchollow site on the (111) surface. The other parameters were determined from
theDFTcalculations as described in thetext. These are thesurface stress τ,thenon-self-consistent LDA valueof thesurface
energy γLDA, the force constant µ = 3/2k0C, and the average amplitude of the corrugated potential of the second layer W.
The quantitiesα and β defined in Eq.(4) determine thestability of the surface in theFrenkel-Kontorovamodel.
A (˚A2) a (˚A) τ (eV/˚A2) γLDA (eV/˚A2) µ (eV/˚A2) W (eV) α β Reconstruction
Au 7.59 3.41 0.15 0.080 2.115 0.038 22.90 −23.97 Compressive
Pt 6.94 3.27 0.34 0.123 3.345 0.061 21.80 −37.58 Compressive
Cu 5.79 2.99 0.11 0.112 2.595 0.056 18.32 −3.58 No
f
b
h
ht
b t
(a) (b)
ft
f
FIG. 1: The calculation of the force constants between the
atoms in the surface layer of (111) surface. The atoms in
the top layer of the slab were displaced by around 0.1% of
theinteratomicdistance, andtherestoringforces werecalcu-
lated. Assuming that only central nearest-neighbor restoring
forces exist, the force constants are k0C = (1/6)(F/δy) for
”compressive” displacements (a), and k0S =(1/2)(F/δx) for
”lateral” displacements (b).
FIG. 2: Various positions of the first-layer atoms (small cir-
cles) on a (111) surface, relative to the second layer (large
circles) and the third layer (grey). In the unreconstructed
subsurfaceatomsarenottakenintoaccount,asetofforce
phase,all first-layeratoms are in thefcc positions (f). Upon
constantsbetweensecondandfurtherneighboursand/or
reconstruction,theatomsarefoundinvariouspositionsalong
angularforceconstantswouldbenecessaryinordertore- thepath f −b−h−b−f.
produceaccuratelythelatticedynamics,i.e.,thephonon
spectra. The present approach is a simplification to be
used only in an estimation of the stability of the surface
slightly. A good estimate of the energy involved can be
layer. One may further object that the reconstruction
obtained by considering structures in which all atoms in
has a large wavelength, while the displacements shown
the first layer are simultaneously displaced by the same
in Fig. 1 correspond mainly to short-wavelength modes
vector. We performed DFT calculations of such configu-
aroundthe edgeofthe Brillouinzone,andthe estimated
rations,keepingthedisplacementinthex y planefixed
kS need not be the same. In our opinion, the differences −
and allowing the atoms to relax in the z direction.
are small.
We first calculated the energy of the regular fcc con-
figuration, denoted by f in figures and tables, which is
energetically the most favorable. Next, we considered
E. Potential between the top layer and the bulk the configuration with the first layer atoms in the hcp
hollows of the second layer (h), which is also a local en-
Whenreconstructionoccurs,somefirst-layeratomsare ergy minimum. We furthermore calculated the energy
displacedtoenergeticallylessfavorablepositionswithre- of some other configurations which are not local energy
spect to the second layer. The last quantity which we minima, keeping the x and y coordinates fixed, so that
need for anestimate of the stability is the amountof en- the algorithm for the atomic relaxation ignored the lat-
ergylostbytheatomswhendisplacedtovariousnonopti- eralforcesactingonthe first-layeratoms. Theseare(see
malpositions. Sincetheperiodicityofthereconstruction Fig. 2) the “bridge” position (b) in which the first-layer
pattern is large compared with the substrate periodicity atoms are halfway between fcc and hcp hollows, the on-
(e.g., by a factor of around 22 on Au(111)), the posi- top configuration (t), the configurations (ft) and (ht)
tion of each subsequent atom along the reconstruction halfway between (f) and (t), and (h) and (t), respec-
direction with respect to the second layer changes only tively, etc. (Not all calculations were performed for all
5
t tf f fb b bh h ht t
In Fig. 3 we show the calculated energies, measured
withrespecttotheenergyofthefccconfiguration,which
was the lowest for all three metals. The “bridge” posi-
5 layers
0.2 tion, midway between fcc and hcp hollows, is approxi-
6 layers matelyasaddlepoint. Allpoints werecalculatedforfive
Au
7 layers layers (first one fixed in x and y, two free, two fixed),
and some points were also calculated for six and seven
0.1 layers. It was found that the energies changed by sev-
eralmeV, but the qualitative behaviorremainedsimilar,
and the estimates of stability towards reconstruction in
the following section are not affected. Generally, the en-
0 ergies of symmetric configurations (h,b,t) increase with
increasingnumberoflayers,whilethoseofnonsymmetric
) configurations,like(fb)or(ht),decrease. Thiscanbeat-
V 0.1 Pt
tributedtothefactthatinthelattercasetherearemore
e
( atomic layers which are free to relax laterally, leading to

E
a more complete relaxation.
∆
The first-layer atoms in positions other than the reg-
0
ular fcc one (f) are also protruding further out. For all
threemetals,thevaluesaresimilar,namely,around0.1˚A
forthe bridge(b),0.02˚Aforthe hcp(h),and0.3 0.4˚A
−
for the on-top position (t).
0.2 Cu
The relevant quantity for the stability estimate is the
average energy W along the path f b h b f, in
− − − −
the approximation of making a Fourier expansion and
keeping only the two lowest terms:
0.1
W =∆E(b)+∆E(h)/2. (3)
0
III. STABILITY AND RECONSTRUCTION
t tf f fb b bh h ht t
In this section we use the calculated quantities to dis-
FIG. 3: The energy needed in order to move the complete
cuss the stability of the surfaces of platinum, gold and
first atomic layer away from the optimum fcc into other con-
copper. We first introduce the one-dimensional Frenkel-
figurations. The variouspositions are depicted in Fig. 2, and
Kontorovamodel, andthen discuss, in order,eachof the
theenergies are expressed in eV/atom.
consideredsurfaces,calculatingthestabilitycriteria,and
comparingtheresultswiththeknownexperimentalfind-
ings.
three metals.) For the quasi-one-dimensional compres-
sive reconstruction,the relevantconfigurationsare along
the path f fb b bh h and back to b and f. (The
− − − − A. Reconstruction of (111) surfaces
on-top configuration, which is not occupied in the re-
construction, was calculated with the aim to obtain a
better insight into the difference between various met- As already said, the electronic structure of closed-
als.) Inpractice,the calculationswereperformedsothat packed metal surfaces is very different from that in the
initially, all layers except the top one were kept fixed at bulk, with an abrupt decrease in density of conduction
the bulk configuration, the top layer had fixed x and y electrons. The consequences may be a change of the
coordinates, and the z coordinate was allowed to relax. length of the bonds between surface atoms, occasionally
In a second step, the bottom two layers were kept fixed, accompaniedbyareconstructionwhichinvolvesachange
the intermediate layers were allowed to relax in all di- of the number of atoms in the first surface layer.
rections, and the top layer was allowed to relax in the The reconstruction of the fcc (111) surface involves a
z direction only. The second step produced only minor large number of atoms in inequivalent positions with re-
changes for the symmetrical configurations (f), (h), (t), spect to the underlying layer of atoms. In scanning tun-
and (b), but the nonsymmetric configurations changed nelingmicroscopyexperiments,the reconstructionofthe
significantly, as the atoms in the second layer (and to a Au(111)surfacecanbeclearlyseenasbrightstripes,1,3,20
lesserdegreeindeeperlayers)relaxedlaterallyunderthe owing to the increased height of atoms which are out of
force exerted by the first-layer atoms. registry with the second layer. On a larger scale, the
6
stripes form a herringbone pattern, as the quasi-one- closely related to) those calculated in the preceding sec-
dimensional reconstruction proceeds along three equiv- tion for real metal surfaces. There is no reconstruction
alent directions on the surface. A typical size of the re- for P <1,thereconstructioniscompressiveforP < 1
constructed Au(111) supercell is 70˚A 280˚A. The re- and|ex|pansive for P >1. The quasicontinuum approa−ch
×
constructionofthe Pt(111)surface is similar,but occurs is valid if the magnitudes ofthe dimensionless quantities
only at high temperatures4,5 or in the presence of sat- α and β defined in Eq. (4) are large, say larger than 5.
urated Pt vapour.6,7 STM micrographs show that the In the following subsections we apply this analysis to
stripes do not form continuous patterns, but are instead the (111) surfaces of gold, platinum, and copper. The
well separated, with various types of intersections.7 The relevant quantities derived in Section II are summarized
stripes in both systems consist of quasi-one-dimensional in Table II, and the resulting α and β are given.
compressive reconstruction1,2 and can be treated as
solitonic solutions of the Frenkel-Kontorova model.21,22
The large-scale two-dimensional structure of the recon- C. Au(111)
structed phase depends upon details of the interactions,
and will not be discussed here.
In our calculations the first layer of an ideal Au(111)
surface relaxes outwards by about 0.005 ˚A, less than
0.3%,whichisatthelimitofaccuracyofthecalculations.
B. The Frenkel-Kontorova model The energy per surface atom of the hcp configuration is
higher than that of the ideal (fcc) structure by around
The one-dimensional Frenkel-Kontorova model8,9,10 10 meV per atom, and that of the bridge by 33 meV
consistsofalinearchainofatoms,subjecttotwocompet- (Fig. 3). This is in excellent agreementwith the calcula-
inginteractionswithdifferentintrinsicperiodicities. The tions of Takeuchi et al.21 and Galanakis et al.,15,16 who
atoms interactvia a nearest-neighborharmonic coupling used a density functional approach with a mixed basis
with a preferred lattice constant b. In addition, there is set. These energies are the smallest of all metals consid-
an external periodic potential (i.e., the potential of the eredhere. The surfaceenergyis alsosmall,andowingto
secondatomiclayer)ofaperiodicitya. Dependingonthe a moderately large tensile surface stress we obtain that
strength of the external potential and the magnitude of β is slightly larger than α, indicating that the surface
the spring constant µ, various stable solutions are possi- reconstructs. Experimentally, the surface appears par-
ble. Atsmall“pressures”(i.e.,smalldifferenceofaandb, ticularly prone to reconstruction, and a dense pattern of
smallµ)theexternalperiodicpotentialdominatesandall stripes of 23 √3 reconstruction is observed at room
atoms are in potential minima, forming a commensurate temperatures×with STM.1,3,20
phase. As the “pressure”increases,the naturalperiodic-
ity of the atomic chainbecomes more important. If, say,
b < a, solitons appear in which the atoms are closer to D. Pt(111)
eachother,thus gainingenergyfromthe harmonicinter-
action but paying the cost of increased potential energy Our calculations show that the first atomic layer on
in the external potential. Finally, when the external po- Pt(111) relaxes outwards by 0.023 ˚A or 1%. This some-
tential is weak compared with the elastic energy needed what exceptional behavior has also been found in other
to stretchthe atomicchain, the atoms followthe period- calculations andconfirmed experimentally.13 It has been
icity b, forming an incommensurate phase. attributed to an outward pointing electrostatic force on
Mansfield and Needs23 have applied the Frenkel- the positively charged atoms of the first layer owing to
Kontorova model to the reconstruction of (111) surfaces thekindofspill-outofelectronsawayfromthesurface.24
of the fcc metals. They have found that the relevant Theenergyofthebridgeconfigurationisaround46meV
quantity is larger than that of the regular fcc surface. The energy
of the hcp configurationis only slightly smaller than the
A(γ 4τ/3) Wβ bridge (even less so in calculations with 6 and 7 layers),
P = − = , (4)
2p2µWa2 Wα whichisdifferentfromtheothersurfacesconsidered. The
π
force constant k0C calculated assuming a “longitudinal”
where the parameters of the Frenkel-Kontorova model displacement of rows of first-layer atoms is reduced by
have already been expressed in terms of the physical about 30% compared with that derived from the bulk
properties of real (111) surfaces. Thus, γ is the surface compressibility. The intrinsic tensile surface stress is
energy,τ isthesurfacestress,µis3/2ofthesurfaceforce large,more than twice largerthan in the other two met-
constant, and W is the average potential energy. (The als considered. Although the surface energy is also quite
factors 4/3 in the stress term and 3/2 in the definition large,the quantity β is much largerthan α, predicting a
of µ appear because the path f h f is not straight.) strong tendency to reconstruct.
− −
a is the distance from the fcc hollow site to the nearest Experimentally, the reconstruction of Pt(111) is ob-
hcp hollow site on the (111) surface, and A is the sur- served only at high temperatures or in the presence of
face area per atom. These quantities are the same (or saturatedplatinumvapor.4,5,6,7 Thenature ofthe recon-
7
struction is similar to that of Au(111), but the stripes the force–relaxationratiois consistentwiththe modelof
of compressive “solitons” do not form a dense pattern. elastic constants calculated in Section II. Surprisingly,
Instead, they are sparse, and occasionally intersect in in our calculations (one adatom per four surface atoms)
several distinct ways.7,25 the adsorption into a hcp hollow site was slightly more
favorable than into a fcc hollow site for all three metals
considered. We have not found any conclusive experi-
E. Cu(111) mental data about this property.
Theapproachcanalsobeappliedtorelaxationaround
Unlike the other two metals considered, we find that defects and chemisorbates of different species, energetics
the first layer on an ideal (fcc) Cu(111) surface relaxes ofself-andheterodiffusion,andotherpropertiesof(111)
inwardsby0.014˚A.Thecontractionoftheinterlayerdis- surfaces.
tance is usual for close packed surfaces of many transi-
tionmetals. Thehcpconfigurationisaclearlocalenergy
minimum, and the energy of the on-top configuration is
IV. CONCLUSIONS
found to be exceptionally large at about 250 meV. The
intrinsic surface stress is again tensile, but rather small.
Using the density functional theory approach we have
Thecalculatedβisclearlysmallerthanα,indicatingthat
calculatedkeypropertiesof(111)surfacesofseveralmet-
the surface does not reconstruct. Indeed, no reconstruc-
als, and evaluated the stability towards reconstruction
truction has been observed experimentally.
into a uniaxially compressed reconstructed layer, using
a one-dimensional quasicontinuum Frenkel-Kontorova
model. The intrinsic surface stress is tensile for all sur-
F. Discussion
faces, i.e., the atoms in the first layer prefer a denser
packing than dictated by the potential of the second
The use of the stability condition of (111)surfaces de-
layer,whichreflectsthebulkperiodicity. Ontheopposite
rived in the one-dimensional Frenkel-Kontorova model,
sideoftheenergybalanceisthe energyrequiredtobring
Eq. (4), gives correct qualitative answers, namely, that
an extra atom into the surface layer,and the energy lost
the surfaces of gold and platinum reconstruct compres-
owing to the fact that surface atoms in a reconstructed
sively and the surface of copper does not. From a quan-
phaseareno longerin the minima ofthe potentialofthe
titativepointofview,theagreementbetweentheoryand
second layer. We have found that Pt(111) reconstructs
experiment is less satisfactory. We find that the condi-
owingtoalargeintrinsictensilesurfacestress. Thestress
tionforreconstruction, β >α,isbarelysatisfiedforthe
| | on Au(111) is considerably smaller, but the surface en-
Au(111)surfaces,whileinexperimentsadensepatternof
ergy is also smaller and the reconstruction criterion is
reconstructionstripesisalwaysfoundonthissurface. On
satisfied. The intrinsic surfacestressinCu(111)is some-
the other hand, we obtain that the condition of Eq. (4)
what smaller than that in Au(111), but other quantities
is amply fulfilled for the Pt(111) surface, while experi-
are unfavorable,and the surface does not reconstruct,in
mentally,reconstructionisobservedonlyunderfavorable
agreementwith experimental observations. Experiments
thermodynamic conditions, at high temperatures, or in
show that Au(111) has the largest tendency to recon-
thepresenceofPtvapor. Ithasalreadybeennoticedthat
struct, and dense reconstruction patterns are observed
calculations predict only a marginal tendency for recon-
using STM, while Pt(111) reconstructs only under fa-
structiononAu(111),atvariancewithexperiment. Ithas
vorable conditions, and only a small fraction of the sur-
been suggested that the ordering of the reconstruction
faceisreconstructed. Thisisthe oppositefromwhatour
stripes into a herringbone pattern is indeed an essential
reconstruction criteria suggest, where Pt(111) is much
contributing factor to the stability of the reconstructed
deeper in the reconstruction regime. The reason may
phase.20,22 The theoretical overestimate of the tendency
be further stabilization of the reconstruction of the gold
to reconstruct in the case of Pt(111) may be due to the
surface by the formation of a two-dimensional herring-
large value of the surface energy, which makes it unfa- bone structure,20,22,26 which is not included in the one-
vorabletobringadditionalatomstothesurfaceoncethe
dimensional model used in the derivation.
surfacestresshasbeenreducedbytheformationofafew
sparsereconstructionstripes. Foramorecompletetreat-
ment, a two-dimensional Frenkel-Kontorova model with
a more realistic potential, calculated in a region around Acknowledgments
the minima, should be used.
The approachusedinthis papercanalsogiverelevant This work was supported by the Ministry of Science
parameters for other physical properties. We have cal- andTechnologyoftheRepublicofCroatiaundercontract
culated relaxations around adatoms of the same species No. 0098001. One of us (R. B.) acknowledges helpful
on the surfaces considered in this paper, and found that discussion with Dr. S. Narasimhan.
8
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