Structural and dynamical properties of Cu–Au bimetallic clustersM. J. López, P. A. Marcos, and J. A. Alonso

Citation: The Journal of Chemical Physics 104, 1056 (1996); doi: 10.1063/1.470831 View online: http://dx.doi.org/10.1063/1.470831 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/104/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Combined tightbinding and density functional molecular dynamics investigation of Si12 cluster structure J. Chem. Phys. 104, 9833 (1996); 10.1063/1.471742 Structural and vibrational properties of fullerenes and nanotubes in a nonorthogonal tightbinding scheme J. Chem. Phys. 104, 5875 (1996); 10.1063/1.471319 Tight binding molecular dynamics study of Ni clusters J. Chem. Phys. 104, 992 (1996); 10.1063/1.470823 Dynamics and structure of molten CsAu AIP Conf. Proc. 330, 129 (1995); 10.1063/1.47866 Probing bimetallic surfaces with photoelectron diffraction: Au/Cu(001) and Fe/Cu(001) J. Vac. Sci. Technol. A 8, 2494 (1990); 10.1116/1.576721

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Structural and dynamical properties of Cu–Au bimetallic clustersM. J. Lopez, P. A. Marcos, and J. A. AlonsoDepartamento de Fı´sica Teo´rica, Universidad de Valladolid, E-47011 Valladolid, Spain

~Received 3 July 1995; accepted 10 October 1995!

The effect of alloying on the structural and thermal properties of Cun2xAux (n513,14) clusters isinvestigated by constant energy Molecular Dynamics simulations. The interactions between theatoms in the clusters are mimicked by a many-body~Gupta-like! potential based on the secondmoment approximation to the tight-binding model. The minimum energy structures and thelowest-lying isomers of the pure and mixed clusters are obtained by thermal quenching. We findicosahedral-like ground state structures for the 13- and 14-atom clusters and for all theconcentrations, the only exception being Au14 which hasC6v symmetry. Mixed structures arepreferred over the segregated ones. The lowest-lying isomers of the binary clusters are thepermutational ones, i.e., isomers having the same underlying geometry as the ground state structureand different relative arrangement of the unlike atoms in the atomic positions of the geometry.However, presence of these low lying permutational isomers does not affect the gross features of themelting-like transition. The 13- and 14-atom~icosahedral-like! binary clusters melt in one and twostages, respectively, as the corresponding pure Cu clusters. In constrast the melting-like transition ofAu14 exhibits a single stage. The melting temperature is studied as a function of clusterconcentration and size. The main conclusion is that mixed Cu–Au clusters likely behave as pure Cuclusters, both from the structural and the dynamical points of view, for all concentrations. ©1996American Institute of Physics.@S0021-9606~96!01003-2#

I. INTRODUCTION

Bimetallic clusters constitute an exciting field of re-search due to the interest they generate both from the theo-retical and applied view points. Probably one of the mostinteresting features of bimetallic clusters is the structure oftheir surface, which determines their chemical activity and inparticular their catalytic properties. As has been extensivelydemonstrated, the structural and thermal properties of smallclusters change with cluster size in a discontinuous~step-wise! manner. In fact, one of the most salient features ofsmall clusters~as compared to bulk matter! is the richness ofnew structures and thermal behaviour that they exhibit as aconsequence of their finite size. On the other hand, binaryalloys1 present a number of structures and phases which aredifferent than those of the corresponding pure metals. Alloy-ing two metals, is indeed, much more than averaging them.The properties of the alloys cannot be obtained, in general,by interpolating~as a function of concentration! the behav-iour of the corresponding pure metals. Bimetallic clusterscombine the characteristics of the finite size systems togetherwith those of the alloys. One would expect then, to find newand interesting properties as a function of cluster size andconcentration for the microalloys.

Coexpansion experiments of alkali metal vapors mixedwith a small amount ('1%) of a divalent metal~Mg, Ca, Sr,Ba, Zn, Eu, Yb!2 have revealed that bimetallic clusters canbe formed even between metals which are known to be non-miscible in the bulk phase. The prominent abundancemaxima present in the experimental mass spectra2 of thesemixed clusters have been interpreted, similar to the magicnumbers of pure akali metal clusters,3 by an electronic shell

picture4–7 within the framework of a modified jellium modelin which the impurity is assumed to be placed at the clustercenter and host and impurity are characterized by a differentpositive charge background. Besides the 8 electron magicclusters, this model yields closed shell 10-electron clustersfor alkali metal clusters doped with a single, attractiveenough, divalent impurity,5 ~e.g. Na8Zn! in full agreementwith the experiments.2 Ab initio calculations8–11 for alkalimetal clusters with a single divalent impurity sitting in thecenter of the cluster support the main conclusions of theelectronic shell model. However a complete understandingof the stabilities of doped alkali metal clusters requires thesimultaneous consideration of electronic and geometricalvariables. For instance recent calculations on NanMg

12 usingthe Car–Parrinello method show that the minimum energystructures of Na6Mg and Na8Mg have the Mg impurityplaced in an off-center position which produces non-closedshell electronic structures for these two clusters. These re-sults have been confirmed by otherab initio calculations.13,14

On the other hand, density functional calculations for alkalimetal clusters with a trivalent impurity~AlLi n)

15 indicatethat the electronic structure of these heteroclusters can beviewed as changing from a localized atomic-like electronicstructure about the open shell impurity forn<5 to a per-turbed delocalized electronic shell structure forn.5 con-taining an AlLi5 core.

However, despite their interest, the studies on bimetallicclusters combining transition and/or noble metals are stillscarce.16–20 Most of them are concerned with the atomicstructure of the clusters focussing on the question of orderingand segregation effects.19 As a general trend it is determinedthat the atoms having smaller surface energy segregate to the

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surface. For instance it has been shown19 that in cuboctahe-dral and icosahedral Cu–Ni clusters with 55 and 147 atoms,the Cu atoms segregate to the surface~which is consistentwith their lower surface energy!. However a detailed andcareful analysis is needed for each bimetallic system in orderto predict its ground state structure. This is especially true incases~such as for Cu–Pd! where the tendency to form or-dered compounds compites with the segregation effect.19 Inthese studies~Cu–Ni and Cu–Pd clusters! an EmbeddedAtom potential has been used to mimic the interactions be-tween the atoms in the cluster. Larger bimetallic clusters~forsizes between 200 and 1200 atoms! of Rh–Pd, Ni–Pd, Rh–Ni, and Rh–Pt17 have also been studied using the correctedeffective medium aproach. Pd, Pd, Ni, and Pt segregate to thecluster surface, respectively. On the other hand, it has beenfound that Ag–Au clusters present phase separation for clus-ters larger than a critical size of about 270 atoms.20

Molecular Dynamics simulations and Monte Carlo tech-niques have been aplied to the study of phase changes insmall clusters. New interesting phenomena~not present inthe bulk matter! appear as a consequence of the finite size ofthe system. The first dynamical simulations in Lennard-Jonesclusters21 showed that small clusters melt over a finite rangeof temperatures. This is a common characteristic of finitesystems including simple metal, transition metal and noblemetal clusters. Sawada and Sugano22 showed that the melt-ing like transition of Ni6 clusters occurs through a ‘‘fluctu-ating state’’ in which the clusters undergo structural isomer-ization transitions which do not involve diffusive typemotion of the atoms. Transition metal and noble metal clus-ters which consist of a very stable icosahedral structure plusone ~or a few! surface atom~s! ~e.g. Ni14 and Ni20) melt intwo stages.23–27 The low temperature premelting stage hasthe character of a local melting and is associated with thelocal destabilization of the cluster due to the presence of one~or few! atom~s! on the surface of a very stable structure. Thehigh temperature stage has been identified as the global melt-ing of the cluster. Another interesting phenomenon whichwas first described for Lennard-Jones clusters28 and has alsobeen found in transition and noble metal custers29 is the‘‘surface melting.’’ The atoms of the external layer of thecluster experience diffusive type motions~liquid-like behav-iour! while the cluster core is still solid-like~the atoms os-cillate around their equilibrium positions!.

Dynamical simulations have also been performed on theevaporation of atoms from Lennard-Jones clusters.30–33Thisphenomenon corresponds to the liquid to vapor transition inbulk matter and occurs at temperatures higher than the melt-ing temperature of the cluster. A different behaviour has beenfound for instance for~molecular! ~C60)n clusters

34 where theevaporation of C60 molecules starts before the cluster melts.This phenomenon can be considered as the analog of thesublimation transition which occurs in some bulk materials.The liquid to vapor type transition exhibits itself in energizedmetal clusters as a process of fragmentation35 ~includingevaporation of atoms as a particular case!. The competitionbetween all the possible fragmentation channels is reflectedin the distribution of the fragmentation channel probabilities.

A correlation has been found between the preferred channelfor fragmentation and the smallest fragmentation energy forthat channel. However there is no systematic correlation be-tween the fragmentation channel probabilities and the frag-mentation energies for all channels.

Systematic theoretical studies on the thermal behaviourof bimetallic clusters are almost nonexistent. We can mentionthe work of Lopez and Freeman.18 These authors investigatethe structural and thermal properties of Pd6Ni7 clusters usingMonte Carlo techniques and Lennard-Jones pairwise poten-tials. Completely segregated or mixed structures are obtainedas the ground state structure depending on the relative valuesof the strengths of the interactions between like and unlikeatoms in the cluster. When the strength of the interactionbetween unlike atoms is taken as prescribed by the standardBerthelot–Lorentz combining rules36 ~i.e. as the geometricalaverage between the strengths of the corresponding pure sys-tems! the ground state has a completely segregated structure.However, if the strength of the interaction between unlikeatoms is taken higher~2% or more! than the previous value,a completly mixed ground state structure is obtained inagreement with the results of the Embedded Atom approach.The heat capacity of Pd6Ni7 as a function of the temperaturepresents peculiarities which are interpreted in terms ofisomerization transitions involving permutational isomers.

The aim of this paper is to analyze the structural andthermal properties of small bimetallic clusters—how theirproperties relate to the ones of the pure clusters and what arethe new features which appear in the bimetallic clusters dueto alloying effects. In this study we will concentrate on theCu–Au system. On one hand Cu–Au alloys present, in thebulk phase, a few fcc-based intermetallic compounds.37 Onthe other hand, it has been found~using transmission elec-tron microscopy! that copper atoms solve rapidly when de-posited onto Au clusters~with a mean size of 4 nm!, forminghomogeneously mixed Cu–Au alloy clusters.38 However,rapid spontaneous alloying becomes more difficult with in-creasing cluster size. One would expect, then, a strong mix-ing tendency between copper and gold atoms in very smallclusters~of the order of tenths of atoms!.

Another interesting feature of copper and gold is thatwhile the pure bulk metals have the same crystal structures~fcc! and approximately the same melting temperatures, purecopper and gold clusters exhibit different behaviours. Forinstance Cu and Au clusters may have the same or differentminimum energy structures depending on their size~Cu13and Au13

26,39 have both icosahedral minimum energy struc-tures while Cu14 is an icosahedrom plus one atom over oneof its triangular faces and Au14

40 is a centered hexagonalantiprism with one atom over one of its bases!. FuthermoreAu clusters experience a much more drastic reduction intheir melting temperature~with respect to the bulk meltingtemperature! than Cu clusters. The melting mechanism isalso different for Cu14 and Au14 clusters. The melting-liketransition of Cu14 clusters exhibits a premelting stage which

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is absent in Au14.40 It will be of interest, then, to study how

the structural and thermal properties of 13- and 14-atomCu–Au bimetallic clusters evolve, as a function of concen-tration, between the limiting cases corresponding to the pureCu and Au clusters. We have performed constant energy Mo-lecular Dynamics simulations on Cu–Au bimetallic clustersto investigate their minimum energy structures, isomer hier-archy as well as the melting-like transition.

A Gupta-like potential41,42 ~based on the second momentapproximation to the tight-binding model!, has been used tomimic the interactions between the atoms in the cluster. Theinteresting feature of this semiempirical potential is itsmany-body character~let us stress here that pairwise poten-tials are unable to describe metallic cohesion!. This potentialhas been extensively used in the literature for studying struc-tural and thermal properties of bulk metals,43–45defects, sur-face reconstruction,46 alloys47 as well as in clusterstudies.22–24,26,35,39,40,48–50The Gupta-like potential providesa similar approximation to the interaction energy of the metalatoms in the cluster~even though they are based on differentphysical grounds!, as the Embedded Atom method~EAM!51–53 and the Effective Medium approach.54,55 Theselast two semiempirical potentials have also been applied instudies of transition and noble metalclusters.16,17,19,25–27,29,56–59

The outline of the paper is as follows. In section II wepresent the semiempirical potential used to mimic the inter-actions between the atoms in the clusters~Gupta-like poten-tial! and the dynamical quantities which will be used to char-acterize the thermal behaviour of the clusters. Section IIIshows the results and their discussion. A brief summary isgiven at the end of section IV.

II. THEORETICAL BACKGROUND ANDCOMPUTATIONAL PROCEDURE

In this work we use a semiempirical many-body~Gupta-like41,42! potential to mimic the interactions betweenthe atoms in the cluster. This potential was introduced origi-nally to represent the cohesive energy of the fcc-lattice fortransition metals.41,60 The attractive part of the potential isgiven by the band energy calculated in the second momentapproximation to the tight-binding model. This term incor-porates the many-body effects. The repulsive part of the po-tential, which is introduced to guarantee the stability of thelattice, is represented by a pairwise interaction in the form ofa sum of Born-Mayer terms.

The Gupta-like potential has been extensively used tostudy structural and thermal properties of transition metalssuch as defects, surface reconstruction, diffusion coefficients,as well as for noble ands–p metals and alloys.44 This po-tential has also been applied to dynamical simulations ofmetal clusters.

For bimetallic systems the Gupta-like potential can bewritten as

V51

2(a51

2

(ia51

na F (b51

2

(jb51~ jbÞ ia!

nb

UabAab

3exp2pabS r i jr 0ab

21D 2S (b51

2

(jb51~ jbÞ ia!

nb

Uab2

3exp22qabS r i jr 0ab

21D D12G , ~1!

wherea(b)51,2 indicates the type of atom,na(nb) is thenumber of atoms of typea(b) in the cluster, andr i j is thedistance between atomsi and j . Uab , Aab , pab , qab , andr 0ab

(a,b51,2 anda,b) are the adjustable parameters ofthe potential. Notice that for each pairab there are only fourindependent parameters. It is customary then to fixr 0ab

tothe equilibrium distance of the lattice of the pure metals~fora5b) and to the equilibrium nearest neighbors distance be-tween unlike atoms of a chosen intermetallic compound~foraÞb).

Garzon and Jellinek23,39 have interpreted the parametersU and r 0 ~for pure systems! as the units of energy andlength, respectively, and consider them not only as materialbut also as cluster-size dependent parameters.

In this work we use the values of the parameters givenby Cleri and Rosato.44 For a5b the parameters were ad-justed to the cohesive energy, lattice constant, and elasticconstants of the fcc Cu and Au lattices separately. The fourremaining parameters~for a Þ b) were optimized~keepingfixed the parameters for the like atoms! to the cohesive en-ergy, lattice constant, and elastic constants of the fcc-basedintermetallic compound Cu3Au.

The structural and thermal properties of Cun2xAux(n513,14) clusters are investigated by constant energy mo-lecular dynamics simulations. The time evolution of the sys-tem is obtained by solving Newton’s equations of motion

mi

d2r idt2

52]V

]r i, ~2!

using the velocity version of the Verlet algorithm.61 The timestepDt51 fs used in the numerical integration of the equa-tions of motion guarantees energy conservation within0.01%. We will consider only non-translating and non-rotating clusters, i.e., clusters with total linear and angularmomenta equal to zero. The lower lying isomers of the clus-ters~local minima of the potential energy surface~PES!! andthe ground state structures~absolute minima of the PESs! areobtained by the thermal quenching method which consists ofcooling down the clusters from a high energy configurationinto a rigid structure at the bottom of a well of the PES.

To characterize the thermal behaviour of the clusters, asa function of their internal energy, and the melting-like tran-sition we will use two different types of magnitudes. 1! Glo-bal quantities such as the temperature~average kinetic en-ergy per degree of freedom!, the root mean square bondlength fluctuation (d), and the specific heat. These quantities

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are calculated as time averages over a whole trajectory at agiven energy. 2! Time dependent quantities such as the meansquare displacement and the velocity autocorrelation func-tion. These quantities can be calculated as averages over tra-jectories~all of them corresponding to the same energy! foreach time, or equivalently, as averages over well separatedtime origins defined along a single trajectory.

The caloric curve or microcanonical equation of state ofclusters ~temperature versus energy! is a smooth functionwith a constant slope for solid-like~liquid-like! clusters. Achange in the slope~which occurs over a finite range ofenergies! connecting the solid-like behaviour to the liquid-like one is a signature of the melting-like transition.

The cluster temperature is given by

kBT52

3n26^Ekin& t , ~3!

where n is the number of atoms in the cluster,kB is theBoltzman constant,Ekin is the total kinetic energy of thecluster, and & t represents the time average along the wholetrajectory ~each trajectory has been propagated over2.53105 time steps which corresponds to 250 ps.!.

The relative root mean square bond length fluctuation(d) is defined by

d52

n~n21!(i. j

~^r i j2 & t2^r i j & t

2!1/2

^r i j & t. ~4!

This magnitude experiences an abrupt~but continuous! in-crease associated with the melting-like transition of the clus-ter. As an estimation of the melting temperatureTm we willconsider the value ofT in approximately the middle of thetransition region (d50.18).

For bimetallic systems we introduce also partiald8s(daa and dab with a(b)51,2 anda,b) calculated fromthe like-atom distances and the unlike-atom distances respec-tively. Differences between the partiald8s and the totaldwill indicate different mobilities of the two types of atoms asa function of energy. The analysis of the partiald8s willprovide, then, further insight into the mechanism of themelting-like transition for bimetallic clusters.

Another indicator of the melting-like transition is thespecific heat defined by

Cv5Fn2nS 122

3n26D ^Ekin& t^Ekin21& tG21

, ~5!

whereCv is given in units ofkB . This quantity is related tothe fluctuations in the kinetic energy.

The mobility of the atoms in the cluster can be analyzedin terms of the mean square displacement

^r2~ t !&51

nnt(j51

nt

(i51

n

@r i~ t0 j1t !2r i~ t0 j !#2, ~6!

wherent is the number of time origins (t0 j) considered alonga trajectory. When the atoms have an oscillatory motionaround their equilibrium positions this magnitude is a con-stant as a function of time~besides the initial increase fromzero to the constant value!. As the energy of the cluster in-

creases the atoms begin to move in a diffusive way and then,the mean square displacement increases linearly with time.

The velocity autocorrelation function given by

C~ t !5^v~ t01t !•v~ t0!&

^v2~ t0!&5

( j51nt ( i51

n vi~ t0 j1t !•vi~ t0 j !

( j51nt ( i51

n vi2~ t0 j !

,

~7!

provides complementary information on the motion of theatoms in the cluster. For low energies, the motion of theatoms is highly correlated andC(t) exhibits oscillations as afunction of time. This is the typical behaviour of a solid-likecluster. For high energies~corresponding to liquid-like clus-ters!, the oscillations inC(t) are completely lost~after thefirst minimum! which indicates an uncorrelated motion ofthe atoms in the cluster.

III. RESULTS AND DISCUSSION

The structural and thermal properties of Cu–Au bimetal-lic clusters are studied as a function of concentration, for thesizesn513 and 14. Interesting questions are how the prop-erties of the mixed clusters evolve between the ones of thecorresponding pure copper and gold clusters and what are thenew and specific features of the alloys. We have chosen tostudy 13- and 14-atom clusters, because of the particular anddistinctive structural and thermal properties of the pure Cuand Au clusters of these sizes.

As has been described before, pure transition metal andnoble metal clusters tend to have structures based on icosa-hedral packing.57,59 In agreement with this general trend wedetermine that the ground state structures of Cu13 and Au13are both perfect icosahedra, and the configuration of Cu14 isan icosahedron plus one atom decorating one of its triangularfaces. Au14, however, does not follow the icoshedral growthsequence. Its ground state structure is a centered hexagonalantiprism with one atom over one of its bases (C6vsymmetry!.40

The qualitative features of the melting-like transition forCu13 and Au13 are quite similar. In both cases the transitiontakes place over a finite range of temperatures~this is a gen-eral feature of small systems!, and it can be described as a‘‘one step’’ transition. We find a similar behaviour forAu14.

40 The melting behaviour of Cu14 is, however, quitedifferent. The transition takes place in two steps. The lowertemperature step has been identified with a premelting stage~local melting of the cluster; see Sec. III B below!, and thehigh temperature step can be identified as the ‘‘true’’~global!melting of the cluster.

The melting-like transition can be characterized quanti-tatively by the melting temperature~defined as the tempera-ture corresponding to the middle of the transition region!.Small clusters are expected to have lower melting tempera-tures than the corresponding bulk materials. The meltingtemperature of Cu13 (Tm5986 K! is about 27% lower thanthe melting temperature of the bulk-Cu (Tm51356 K!.Au13, however experiences a much more substantial reduc-tion ~of about 67%! in its melting temperature (Tm5441 K!with respect to the bulk-Au (Tm51336 K!. Cu14 and Au14

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have approximately~within the accuracy of the calculation!the same melting temperatures as Cu13 and Au13, respec-tively.

In conclusion, we can say that pure copper and gold13-atom clusters have similar structural and thermal features,while the corresponding 14-atom clusters exhibit very differ-ent behaviours. This means that these two sizes~13 and 14atoms! are particularly interesting for the study of alloyingeffects.

A. Structure

First we concentrate on the structural features~groundstate structures and isomers! of the Cu–Au bimetallic clus-ters. The minimum energy structures and the isomer hierar-chy are, in fact, important characteristics of the potential en-ergy surface~PES!. We have performed an extensive searchof local minima on the PESs corresponding to the differentbimetallic and pure clusters. For each cluster size and con-centration we have optimized at least 400 initial configura-tions using the thermal quenching procedure. Those initialconfigurations were recorded along trajectories at several to-tal energies~corresponding to solid-like clusters, to themelting-like transition region, and to liquid-like clusters!.

Figure 1 shows the ground state structures of the 13-atom Cu–Au bimetallic clusters for several concentrations.As we have already mentioned, both Cu13 and Au13 haveicosahedral ground state structures. Mixing Cu and Au doesnot destroy the geometry of the minimum energy structure,which is icosahedral for all concentrations~slight distortionsfrom a perfect icosahedron appear for the mixed clusters dueto the size difference between Cu and Au!. The central atomtends to be of the type of the minority atoms in the cluster.For compositionsx5124, the central atom is gold, whilex55212 has copper as the central atom~see Fig. 1!. Thisseems to indicate the preference of the central atom to besurrounded by atoms of a different type. A competing effectis the slightly stronger tendency of the Cu atoms to occupythe central position in the cluster. The degree of mixing onthe cluster surface can be measured~following Freeman18!by the numberNm of unlike-atom nearest neighbor interac-tions ~mixing number!. Nm characterizes the different rela-tive arrangements of the unlike atoms on a certain geometri-cal structure~one should notice, however, that two different

arrangements may correspond to the same value ofNm). Forinstance for 6 Cu atoms and 6 Au atoms on an icoshedrallayer, the mixing number equal to 10 corresponds to the per-fectly phase separated system, andNm520 ~which is themaximum possible value of the mixing number!, corre-sponds to the perfectly mixed system. From Fig. 1 a verystrong mixing tendency is aparent in the 13-atom Cu-Aubimetallic clusters, even though the ground state structuresdo not maximize, in general, the mixing number~see forinstance Cu7Au6 in Fig. 1 for which the surface mixing num-ber is 18!.

Using ~pairwise! Lennard-Jones potentials to mimic theinteractions in the clusters, Freeman18 obtains for Pd6Ni7 ei-ther completely phase separated or completely mixed struc-tures depending on the strength (a) of the interaction be-tween unlike atoms, with the exception of a very narrowrange ofa values for which partially mixed structures wereobtained. Embedded Atom calculations show that the mini-mum energy structure of Pd6Ni7 is completely mixed withtotal mixing numberNm526. The completely phase sepa-rated and completely mixed structures can be explained byarguments based on the relative strength and number of thenearest neighbor interactions between like and unlike atoms.However, these simple arguments fail in predicting theground state structures that we obtain for Cu–Au bimetallicclusters due to the many-body character of the interatomicpotential here considered~let us stress that the many-bodycharacter of the potential is essential for the appropriate de-scription of metal clusters!. The mixing tendency that wefind in small Cu–Au clusters is in agreement with the rapidspontaneous alloying which has been observed when copperis deposited onto gold clusters of approximately 4 nm in themean size.38

For bimetallic clusters two different types of isomers~local minima of the PES! can be described. 1! Isomerswhich involve topological changes on the geometrical struc-ture ~from now on we will refer to this type of isomer as‘‘topological’’ isomers!. 2! Isomers which only involve rear-rangements in the relative positions of the unlike atoms on afixed geometry~we will refer this type of isomer as ‘‘permu-tational’’ isomers!.

Clearly, pure clusters have only topological isomers. Thelowest-lying ~topological! isomers of Cu13 are based on anicosahedral packing~see Fig. 2!. One of the atoms of theicosahedron is promoted, either over one of its triangularfaces~isomers 2,3 and 4 of Fig. 2; notice that there are 3inequivalent triangular faces in an icosahedron minus oneatom!, or over one edge~isomers 5 and 6 of Fig. 2!. Au13,however, presents low lying topological isomers which breakthe icosahedral-like structure.

The number of permutational isomers of the ground stategeometry of bimetallic clusters depends on the concentra-tion. For n513 andx ~or n2x)51,2,3,4,5, and 6 there are2,4,8,15,21, and 28 possible permutational isomers of theicosahedral ground state structure, respectively. Even thoughnot all possible isomers of the ground state structure havebeen identified in our simulations, due to the computational

FIG. 1. Minimum energy structures of Cu132xAux clusters for x51-7 and12. The dark and clear shaded atoms represent copper and gold respectively.

1060 Lopez, Marcos, and Alonso: Cu–Au bimetallic clusters

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limitations, we have strong evidence that all of them corre-spond to local minima of the PES. The first few lowest-lyingisomers of Cu132xAux are permutational isomers of theground state geometry for all concentrations, i.e. all of themhave icosahedral structures, the only exception beingCu1Au12 for which the only possible permutational isomer ofthe minimum energy structure is the 12th isomer~orderingthe isomers by increasing energy and calling first isomer tothe minimum energy structure!. Among the low lying permu-tational isomers the lowest ones~for 3<x ~or n2x)<6)have the same type of central atom as the correspondingminimum energy structures. The topological isomers of the13-atom Cu–Au bimetallic clusters have the same geom-etries as the isomers of Cu13. Each of the topologically dif-ferent structures gives rise to a~concentration dependent!number of permutational isomers. As an illustration of thepermutational and topological isomers that we find for bime-tallic clusters, in Fig. 3 we present the lowest-lying isomersfor Cu6Au7 . The first 15 isomers~including the ground state!are permutational isomers of the ground state structure. Fromisomers 1 to 9, the clusters have copper occupying the cen-tral site, and from isomers 10 to 15, the central atom is gold.Isomers 16 to 20 are topological isomers of the ground statestructure.

The interesting features to extract in conclusion are that,for bimetallic clusters many more isomers are found than forthe pure clusters, and that a number of permutational isomersare lower in energy than the lowest-lying topological isomer.We will address later in this section how these features of the

PESs of bimetallic clusters affect their thermal behaviour~i.e. their melting-like transition!.

The case of 14-atom clusters is especially interesting. Asmentioned above, Cu14 and Au14 have different ground statestructures. Naively, one could think that Cu–Au 14-atom bi-metallic clusters would have, either Cu-like or Au-likeground state structures for low copper and low gold concen-trations, respectively, and that for a certain concentration~orconcentration range!, the transition from one to anotherwould take place. The striking result is that Cu–Au bimetal-lic clusters of all concentrations have Cu-like ground statestructure~see Fig. 4!. In other words, it is enough to substi-tute in Au14 one gold atom by a copper atom to change thetopology of its ground state structure.

Two competing effects determine~as for the 13-atomclusters! the type of central atom. On one hand, the mixing

FIG. 2. Lowest-lying isomers of Cu13 clusters. The numbers indicate order-ing by increasing energy. The corresponding energies measured from theminimum energy structure~isomer 1! are: ~2! 0.940 eV,~3! 0.941 eV,~4!0.958 eV,~5! 0.993 eV, and~6! 1.088 eV.

FIG. 3. Lowest-lying isomers of Cu6Au7 clusters. The isomers have beenordered by increasing energy. The first 15 isomers are permutational isomersof the minimum energy structure~isomer 1!. The energy of isomer 2 is 0.02eV, the one of isomer 10~the first isomer having Au as the central atom! is0.54 eV, and the one of isomer 16~the first topological isomer! is 0.69 eV;these energies are measured from the ground state. The coding of the type ofatoms is the same as in Fig. 1.

FIG. 4. Minimum energy structures of Cu142xAux clusters for x50-4,7, and13-14. The pure Cu14 and Au14 clusters~corresponding to x50 and x514respectively! have been included for comparison. The coding of the type ofatoms is the same as in Fig. 1.

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tendency favours the minority atoms occupying the centralsite, and on the other hand, the copper atoms have a strongerpreference for the central site. As a result, for concentrationsx>4 copper sits at the center of the cluster. The arrangementof the atoms on the surface of the cluster also exhibits astrong tendency for mixing the two types of atoms, eventhough the ground state structures are not completely mixed~following Freeman criterion18! ~see Fig. 4!.

The isomer hierarchy of 14-atom clusters is character-ized by an increase in the number of isomers~with respect tothe 13-atom clusters!, and by the presence of low-lying to-pological isomers. Figure 5 shows the lowest-lying isomersof Cu14. The second isomer is only 0.13 eV higher in energythan the minimum energy structure~notice that the secondisomer of Cu13 is 0.94 eV higher than the ground state!.Together with icosahedral-like isomers we find~in the samerange of energies! isomers which are not based on icosahe-dral packing~i.e., isomer 5 hasC6v symmetry!. The lowest-lying isomers of Cu142xAux ~except forx and 14-x51! arepermutational isomers of the ground state structure whichpreserve the type of the central atom. Isomers having thesame geometry as the second isomer of Cu14 and with thesame type of central atom as the corresponding ground stateare lower in energy than the permutational isomers of theminimum energy structure which change the type of the cen-tral atom. These low energy topological isomers will giverise to particular features~see below! of the melting-liketransition of this cluster size~as has previously been de-scribed for Ni14). Other low energy topological isomers ap-pear also for the bimetallic clusters.

B. Thermal effects

To investigate the thermal features of the Cu–Au bime-tallic clusters, we generate the caloric curves~equation ofstate of the clusters! starting from the corresponding mini-mum energy structures. The kinetic energy of the clusters isincreased from point to point of the caloric curves under theconstraint of total linear and angular momenta equal to zero.Figure 6 shows the caloric curves forn513 and 14 for sev-

eral concentrations~including pure copper and gold clusters!.A change in the slope in the caloric curve correlates with analmost stepwise increase in the relative root mean squarebond length fluctuationd ~Fig. 7!. A similar behaviour isobserved for the caloric curves corresponding to bimetallicclusters of different concentrations~and pure copper clus-ters!, for a fixed cluster size and the same behaviour is no-ticed ford. This indicates a similar behaviour of the meltingtransition. Pure gold clusters on the other hand, exhibit dif-ferent features.

As the cluster energy increases, mixed 13-atom clustersexperience, similar to Cu13, a transition from a solid-likeform, in which the motion of the atoms is restricted to oscil-lations around the equilibrium configuration, to a liquid-likeform characterized by the uncorrelated motion of the atoms.The melting temperature is almost independent of concentra-tion and is close to the melting temperature of Cu13 ~see Fig.

FIG. 5. Lowest-lying isomers of Cu14 clusters. The isomers are ordered byincreasing energy. The corresponding energies measured from the minimumenergy structure~isomer 1! are: ~2! 0.13 eV, ~3! 0.56 eV, ~4! 0.59 eV, ~5!0.61 eV,~6! 0.79 eV,~7! 0.81 eV, and~8! 0.88 eV.

FIG. 6. Caloric curves~averaged kinetic energy per atom versus total energyper atom! of 13- ~upper panel! and 14-atom clusters~lower panel!.

FIG. 7. Relative root mean square bond length fluctuation (d) versus thetemperature of the cluster. Each panel shows the comparison ofd for the 13-and 14-atom clusters of the same concentration.

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8!. However, the melting-like transition of mixed 14-atomclusters, as for Cu14, takes place in two steps.~see Fig. 7!.The high temperature stage of the transition, corresponds tothe global melting of the cluster, and takes place at the sametemperature as the melting transition of the corresponding13-atom clusters. The low temperature premelting stage con-nects a solid-like cluster with a cluster in which the solid-likeand the liquid-like phases coexist. The 14th atom~the onesitting on the surface of the cluster! modifies the structure ofthe underlying icosahedron in its close neighborhood andeventually inserts itself into the surface of the cluster, push-ing out another atom onto the surface. This last atom thenstarts to play the role of the 14th atom. As a consequence, thevicinity of the 14th atom behaves as a liquid-like subsystemof the cluster. On the other hand, the rest of the atoms in thecluster remain oscillating around their respective equilibriumpositions, i.e. they behave as a solid-like subsystem of thecluster. The premelting stage~or local melting! was first de-scribed for Ni14

23,25 and it has been corroborated for othertransition and noble metals.26,27 This phase transition takesplace over a temperature range which depends on concentra-tion. For pure copper clusters and forx51, d presents analmost flat plateau corresponding to the premelted cluster.For concentrationsx57 and 12, there is not a plateau ind.However two, well separated, steps of the transition can beclearly identified by comparison with thed of the 13-atomclusters for the samex. The temperature of the premeltingtransition is also shown in Fig. 8.d8s for Au13 and Au14 lieon top of each other. This shows clearly that there is notpremelting stage for Au14.

The partiald8s, dCu2Cu anddAu2Au anddCu2Au , evalu-ated from the distances between the like atoms and the un-like atoms in the cluster, respectively, exhibit the same typeof behaviour as the totald. ThedCu2Cu anddAu2Au lie on topof the totald. This indicates that there is no difference in themobility of the copper and gold atoms in a given mixedcluster. Some differences, however, can be observed betweendCu2Au and the totald for mixed clusters with concentrationsof x51 and n2x51 ~Fig. 9!. For Cu12Au

~CuAu12), the abrupt increase ind ~corresponding to themelting-like transition! begins at an energy a little bit lowerthan the step indCu2Au . From the middle of the transitionregion on, dCu2Au , lies on top of the totald. Takinginto account that the ground state structure of Cu12Au~CuAu12), is an icosahedron with gold~copper! as the centralatom, it is clear that the differences between the totald anddCu2Au represent differences in the mobility of the surfaceatoms and the central atom. The surface atoms undergo dif-fusive motions at a lower energy than the central atom. Thatis, the melting-like transition begins by the surface atomsand at a little bit higher energy also involves the centralatom. This is in agreement with the surface melting phenom-enon which has been observed for Ni55.

29 The premeltingstage masks this feature in Cu13Au and CuAu13 clusters.However, it is interesting to observe in Fig. 9 that the valueof dCu2Au in the premelted cluster is much lower than thecorresponding value of the totald. The comparison betweenthe total d and dCu2Au reflects, as in the case of 13-atomclusters withx ~or n2x)51, differences in the mobility ofthe surface atoms, some of which exhibit a liquid-like behav-iour, and the central atom which does not participate in thepremelting of the cluster.

The melting-like transition can be also identified throughthe specific heatCv . Figure 10 showsCv for Cu6Au7 . Thepeak inCv coincides with the end of the transition region. InFig. 10 we also present the results for Cu7Au7 . A small peakin Cv , on the left of the main peak, corresponds to the pre-melting stage.

The mean square displacements inform about the mobil-ity of the atoms in the cluster. For mixed clusters we recoverthe same type of behaviour which has been observed for pureclusters. Figure 11 showsr2(t)& for Cu6Au7 and for threedifferent energies corresponding to, a! the solid-like cluster,b! the transition region, and c! the liquid-like cluster. For asolid cluster^r2(t)& has zero slope for long times, whichreflects the oscillatory motion of the atoms. In the middle of

FIG. 8. Melting temperatureTm as a function of concentration for 13-atomclusters, and premelting temperatureTpm for 14-atom clusters.

FIG. 9. Partial relative root mean square bond length fluctuation for unlikeatoms (dCu2Au) compared with the totald. The upper panel corresponds toCu12Au and the lower panel to Cu13Au.

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the transition region, the diffusive motion of the atoms in thecluster begins and the slope in^r2(t)& starts to increase. Forliquid-like clusters,^r2(t)& has a constant slope, related tothe diffusion coefficient in the cluster.

In Fig. 12 we present the velocity autocorrelation func-tion for Cu7Au6 for energies corresponding to a! solid-like,b! transition region, and c! liquid-like clusters. The persistentoscillations inC(t) for a solid-like cluster indicate that themotion of the atoms is highly correlated~oscillatory type!.For energies corresponding to the transition region, the os-cillations inC(t) disappear after a few vibrations. The sys-tem has short-time memory, that is, diffusive motions of theatoms in the cluster begin to be present. For liquid-like clus-

ters, the correlation in the motion of the atoms is completelylost ~diffusive motion!, andC(t) does not exhibit oscillations~after the first minimum!.

An interesting question is what is~if any! the relation-ship between the isomer hierarchy and the melting behaviourof the clusters. The situation for pure copper clusters is simi-lar to the one which has been previously described for nickelclusters. On one hand the high melting temperature~close tothe bulk one! of Cu13 can be related to the structural stabilityof its ground state geometry. As we have already discussed,the second isomer is rather high in energy~0.940 eV! abovethe minimum energy structure. On the other hand the lowtemperature premelting stage of Cu14 can be associated withthe local destabilization of the cluster caused by the 14thatom over the icosahedral surface of the 13-atom cluster. Thetwo lowest-lying isomers of Cu14 ~icosahedron plus one atomover a triangular face and icosahedron plus one atom overone edge!, preserve the icosahedral structure and are veryclose in energy~0.13 eV!, and isomerization transitions be-tween the two take place at rather low temperatures. How-ever, the isomer hierarchy of Au14 does not present isomersvery close in energy to the minimum energy structure~ge-ometry withC6v symmetry! and consequently its melting-like transition does not exhibit a premelting stage. Then, acorrelation seems to exist between the difference in energybetween the lowest-lying isomers and the melting~or pre-melting! temperature. On this basis one would expect to findlow temperature~below melting! isomerization transitionsbetween the low-lying permutational isomers of the bimetal-lic clusters~which are very close in energy!. However, notrace of those transitions appears either in the caloric curves,or in thed8s, or in the specific heat~these curves exhibit thesame features as the ones of the corresponding pure copper

FIG. 10. Specific heat as a function of temperature for Cu6Au7 andCu7Au7 . The arrows indicate the melting and premelting temperatures asdetermined from the conditiond50.18.

FIG. 11. Mean square displacement of Cu6Au7 evaluated at the tempera-tures: a! 0.054 eV, b! 0.081 eV, and c! 0.153 eV which correspond to asolid-like cluster, to the melting-like transition region and to a liquid-likecluster, respectively.

FIG. 12. Velocity autocorrelation function of Cu6Au7 . Each panel corre-sponds to a different temperature of the cluster: a! 0.0045 eV~solid-likecluster!, b! 0.080 eV~melting transition region!, and c! 0.113 eV~liquid-likecluster!.

1064 Lopez, Marcos, and Alonso: Cu–Au bimetallic clusters

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clusters!. This result is in contrast with the isothermal MCsimulations of Pd6Ni7 in Ref. 18. The heat capacity of thiscluster exhibits a low temperature anomaly~maximum!which has been identified as an isomerization transition in-volving permutational isomers of the minimun energy struc-ture.

In order to rule out unambiguously the possibility ofthese low energy isomerization transitions~at least in con-stant energy simulations!, we have carefully examined whichisomers are accessible in the solid-like state of the cluster,when the energy is increased from the ground state structureup to the energy corresponding to the beginning of the tran-sition region, either melting or premelting. The thermalquenching analysis reveals that only the minimum energystructure is present in the solid-like state. That is, just as pureclusters do, solid-like bimetallic clusters oscillate around theequilibrium configuration in their corresponding ground statepotential energy wells. Isomerization transitions do not occurat these energies. The permutational isomers, together withthe topological isomers, start to appear at energies corre-sponding to the transition region. This means that one needsto melt the cluster to be able to access permutational isomers,even though they are very close in energy to the ground state.Clearly, what determines the possibility of accessing differ-ent isomers is not the depth of the wells on the PES but theheight of the saddles connecting them. Our result, then, sug-gests that the saddles of the PES which connect permuta-tional isomers are of a height comparable to the one of thesaddles connecting topological isomers. This is not a surpris-ing result if one considers that the motions of the atomsinvolved in the exchange of two atomic positions in the clus-ter ~i.e. the motions involved in an isomerization transitionbetween two permutational isomers! are of the same type asthe ones involved in a topological change of the cluster. Inconclusion, we can say that the presence of low-lying per-mutational isomers does not change the main features of themelting-like transition of bimetallic clusters. The meltingmechanism of bimetallic clusters~likely as for pure clusters!consists on isomerization transitions which involve topologi-cally different isomers.

IV. SUMMARY

We have studied the structural and thermal properties ofCu–Au 13- and 14-atom bimetallic clusters. A many-bodyGupta-like potential41,42 has been used to mimic the interac-tions between the atoms in the clusters. The many-body char-acter of the potential is an essential ingredient for the appro-priate description of this type of clusters.

The ground state structures of the mixed clusters andtheir isomer hierarchy have been investigated using the ther-mal quenching procedure, starting from configuration pointsgenerated along trajectories at several total energies~corre-sponding to solid-like clusters, to the melting-like transitionregion and to liquid-like clusters!. Just as with pure Cu13 andAu13 clusters, Cu132xAux clusters have icosahedral groundstate structures for all concentrations. That is, mixing Cu andAu does not change the minimum energy structure of the

13-atom clusters. On the other hand, the ground state struc-ture of Cu142xAux clusters does not experience a progressivechange between the one of Cu14 ~icosahedron plus one atomover one of its triangular faces! and the one of Au14 ~centeredhexagonal antiprism plus one atom over one basis! asx in-creases. Fromx51 on, all the 14-atom clusters prefer a mini-mum energy structure like that of Cu14. The central atom inthe mixed clusters tends to be one of the minority atoms,even though we have observed a slightly stronger preferenceof the Cu atoms for the central site. There is also a strongmixing tendency of the unlike atoms on the surface of thecluster which is in agreement with the spontaneous alloyingeffect observed in small~nm sized! Cu-Au clusters.38 How-ever, in contrast to Pd6Ni7

18 whose minimum energy struc-ture is completely mixed~total Nm526!, the ground stateconfiguration of Cu–Au clusters does not obey the maximumNm rule.

The isomer hierarchy of bimetallic clusters is richer thanthe one of pure clusters due to the presence of permutationalisomers~minima of the PES which correspond to the samegeometry and to different arrangements of the unlike atomsin the atomic positions of the geometrical structure!. Thebimetallic clusters ~except Cu1Au12, Cu13Au1 , andCu1Au13) present a number of low-lying permutational iso-mers of the minimum energy structure which are lower inenergy than any topological isomer. The interesting featureof the 14-atom bimetallic clusters is that, similar to Cu14,present topological isomers~having the same geometry asthe second isomer of Cu14) which are very low in energy ascompared with the lowest-lying topological isomer of thecorresponding 13-atom cluster.

The thermal behaviour of the clusters is examined byconstant energy molecular dynamics. As the energy of thecluster is increased~starting from the ground state structure!,the 13-atom clusters exhibit a one stage melting-like transi-tion, while the 14-atom clusters~except Au14) melt in twostages. The low temperature~premelting! stage of the 14-atom clusters has the character of a local melting and in-volves the two lowest-lying topological isomers of theseclusters~i.e. icosahedron plus one atom over one of its trian-gular faces and over one edge, respectively!. The high tem-perature stage corresponds to the global melting of the clus-ter. We find that the melting temperature of the bimetallicclusters is close to the melting temperature of copper clus-ters.

We have not found differences between the mobility ofthe copper and gold atoms of a given cluster at a given en-ergy. However, there is a difference between the mobility ofthe central atom and the surface atoms. The surface atomsbegin diffusive-type motions at a slightly lower energy thanthe central atom. This is the analog of the surface meltingphenomenon which has been found for larger cluster sizes. Itis interesting to notice that the presence of the low-lyingpermutational isomers in the bimetallic clusters does notchange the main features of their melting-like transition ascompared to the melting-like transition of pure copper clus-ters of the same sizes. This is a consequence of the absenceof low energy isomerization transitions between permuta-

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tional isomers~even when they are very close in energy!.In conclusion, we have studied the effect of alloying two

metals~copper and gold! in small 13-atom and 14-atom clus-ters. In contrast to the bulk matter, Cu–Au bimetallic clustersdo not experience a smooth transition between the propertiesof pure copper and pure gold clusters as the concentrationchanges. As we have shown, the main structural and dynami-cal features of Cu–Au bimetallic clusters are mainly deter-mined ~besides the size! by the presence of copper atoms inthe cluster.

ACKNOWLEDGMENT

This work has been supported by DGICYT~Grant No.PB92-0645-C03-01!.

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1066 Lopez, Marcos, and Alonso: Cu–Au bimetallic clusters

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