Z. Phys. D 39, 225–237 (1997) ZEITSCHRIFTFUR PHYSIK Dc© Springer-Verlag 1997

Na clusters on Na-Cl surfaces – the impact of the interface potentialC. Kohl, P.-G. Reinhard

Institut fur Theoretische Physik, Universitat Erlangen, D-91058 Erlangen, Germany

Received: 25 September 1996 / Final version: 3 December 1996

Abstract. We investigate the structure and energetics of Naclusters on Na-Cl surfaces. The Na-Cl substrate is taken asinert acting on the clusters constituents essentially throughits interface potential which consists in a general surfaceattraction and is modulated by corrugation. The cluster isdescribed as a predominantly electronic system in the spiritof the ultimate jellium model. The strong surface attractionmakes planar clusters the preferred choice in any case andnon-planar stable isomers do always exist. The corrugationhas a large influence on the shapes, particularly for the largerclusters. But the total energy of the clusters is insensitiveto these details. It can be described in terms of volume,surface, and interface energy allowing a simple estimate ofthe preferred geometry in dependence of the strength of theinterface potential.

PACS: 36.40; 61.46; 68.10; 68.35.−p

1 Introduction

The structure of free metal clusters emerges from a subtlebalance between ionic and electronic shell effects. Placingthe cluster on a surface adds further competitors to this bal-ance as the impact of the interface energy, the electronicband structure of the substrate, and the surface corrugation.Consequently it is obvious to be interested in how the prop-erties of these systems are modified in the presence of asupporting surface. Clusters on surfaces are thus an fasci-nating and demanding topic which is widely studied nowa-days. There are accordingly many experimental, e.g. [1–4],and theoretical investigations, like [5–9], on the interactionof atomic clusters with solid surfaces, whereby most the-oretical approaches employ a microscopic description withexplicit ionic degrees of freedom. The vast number of pub-lications in this field can principally be divided into twomajor groups: The first one is dealing with noble or tran-sition metal clusters deposited on metal substrates, eitherfrom an experimental [10–12] or from a theoretical pointof view [13–17]. Here the problem of hybridization of the

delocalized metal valence electrons with the substrate elec-tronic states is one of the dominant features. The secondvery popular approach is to draw more attention to metal orsimple metal clusters on non-metallic substrates like graphite[18–22], MgO [23, 24], SiO [25, 26] or NaCl. Amongst theoverwhelming variety of cluster-surface combinations, themetal on insulator system Na clusters on a Na-Cl substratehas attracted particular interest [27–32]. One reason is thatthis combination has several computational advantages, e.g.both constituents have one effective valence charge and theinteraction with the remaining ions can be treated reliablywith pseudopotentials [33]. Furthermore, in this case, anyhybridization mentioned above can be omitted [29, 30]. Thecluster material Na, moreover, has a Wigner-Seitz radiusvery close to that of a free, correlated electron cloud suchthat its bulk equilibrium density is determined exclusively bythe electrons. As a consequence, ionic structure effects areminimal for free Na clusters. This allows a zero-order ap-proach in the spirit of the Ultimate-Jellium-Model (UJM) inwhich one assumes the positive background to compensateexactly the electronic charge density [34]. Despite of thewell known limitations of the UJM, this simplified modelworks surprisingly well for neutral Na clusters even whencompared with fully ionic calculations [35]. Consequently,the UJM has been applied in several investigations of thestructure of planar clusters as they are formed when a Nadrop is attached to a Na-Cl surface [32, 36] or for quan-tum dots [37]. These investigations have concentrated onthe shell structure of the planar clusters as such while ig-noring any effects from the interface. It was found in [32]that planar systems have much smaller electronic shell ef-fects than fully three dimensional clusters, in particular if itsshapes are varied without symmetry restrictions. Thus oneexpects that the effects from the interface potential or fromionic structure become equally important.

The studies of [32] have, in fact, employed free clustersin the UJM where the planar geometry was achieved byselecting a planar isomer. It is the aim of this paper to carrythe model one step forward and to investigate the effects ofthe interface with the Na-Cl substrate. The leading featurewill be the overall attraction extended from the substrate tothe cluster. It determines, in balance with the surface tension

226

and shell energy, the clusters preferred shape. Here, we liketo investigate whether the clusters form a monolayer or arestacked in vertical direction, depending on the strength ofthe interface energy. The next important detail is the spatialmodulation of this attraction due to the crystalline structureof the substrate, the corrugation [31]. It influences the actualposition and the shape of the area of contact and leads to anadditional energy contribution. We will consider both aspectsof the surface interaction. To that end, we start again fromthe UJM and model the surface interaction by a spatiallymodulated effective electron surface potential. This effectivepotential is quantitatively adjusted to the surface interactionas deduced in the ab initio calculations of [31] using thecode of [5, 27]. The adjustment is done for the few smallclusters which can be computed in all that detail. The modelserves then to extend the considerations to larger clusters inthe range up toNel = 24 which is enough to show the basictrends but is clearly not the limit of this model. In a finalstep, we will summarize our results in a simple macroscopicmodel which may allow further extension to even largersystems or other substrates later on.

The paper is outlined as follows: In Sect. 2, we explainquickly the necessary theoretical background, the UJM andthe energy-density functional. In Sect. 3, we present anddiscuss shapes and energies of the Na clusters on Na-Cl. InSect. 4, we discuss effects of a varying interface energy andthe transition from 3D-drops to planar clusters.

2 Theoretical background

2.1 The UJM

The valence electrons of the Na atoms are described byKohn-Sham equations derived variationally from effectiveenergy density functionals. The fully detailed models startfrom the energy density functional

E = T +Exc[ρ↑, ρ↓] + Ecoul +Es (1)

where T is the kinetic energy of the electrons,Exc theirexchange-correlation energy,Ecoul the Coulomb energy ofelectrons and ions.Es encompasses the interaction betweenthe substrate and the constituents of the cluster. The kineticenergy

T =N∑i=1

〈φi | − ~2

2m∇2 | φi〉 (2)

employs the electronic wavefunctionsφi explicitely. Theyare contained in all the other contributions via the electronicdensity as

ρ↑↓(r ) =N∑i=1

φ†i (r )σ↑↓φi(r ) , ρel = ρ↑ + ρ↓ . (3)

The Kohn-Sham equations for the electrons are obtained byvariation of the total energy with respect to the electronicwave function and yield the mean field Hamiltonian in thestandard manner [38].

The UJM is motivated by the experience that the ionicdistribution follows the electrons very closely [34, 35]. Inan extreme idealization, one assumes a perfect matching

ρion(r ) = ρel(r ) (4)

thus eliminating the ionic degrees of freedom in favor of apurely electronic description. The basic assumption (4) de-livers also a vanishing total charge distribution and this sim-plifies the treatment even more because the Coulomb energydisappears exactly. The structure of a free cluster is then de-termined completely from the interplay of kinetic energyT and exchange-correlation energyExc without requiringany further structural input. Thus the UJM is a very simpleand efficient approach for systems which are dominated byelectronic structure, as in Na clusters. It fails, of course, toreproduce quantities like the total energy (which contains alarge fraction of ionic structure energy) or the dipole barrier(which vanishes here, of course). But it can deliver a perti-nent description for the global deformation of clusters or forenergy differences, as e.g. electronic shell energies [34, 35].

Additional contributions come into play, of course, fora cluster on a surface. In accordance with the experiencefrom ab initio calculations [31], we assume the substrateto stay structurally unmodified delivering a fixed interfacepotential to the cluster. The UJM contains only electronsas variational degrees of freedom such that the effect ofthe substrate is to be summarized in an effective substrate-electron interactionEse[ρel]. Altogether, we are using theenergy-density functional

E = T +Exc[ρ↑, ρ↓] + Ese[ρel] (5)

with the kinetic energyT as given in Eq. (2). ForExc, weare using actually the spin-density functional of Gunnarsonand Lundqvist [39]. The modeling ofEse is discussed in thenext subsection.

2.2 The interface potential

In a detailed ab initio calculation, the ions and electrons ofthe substrate are treated on the same footing as the con-stituents of the cluster, i.e. by an energy-density functionaland pseudopotentials for the interaction of ions and valenceelectrons [5, 30]. Experience for the system of Na on Na-Clshows that the substrate remains rather inert and essentiallyinteracts with the cluster as a surface interaction potential[31]. The surface of the substrate polarizes the cluster as awhole which leads to an attractive polarization force. Polar-ization effects between the cluster ions can be neglected andthus the polarizability is proportional to the particle num-ber N which allows a description in terms of a potentialacting on each single atom separately. Such a potential hadbeen adjusted as a local surface-atom potentialVsa(r ) in [41]by fitting an analytical function to the interface energy ofa sodium atom on top of the surface in a MD-simulation(see [31]). This potential was modeled in a separable formV⊥(z)V‖(x, y) with a Van-der-Waals like surface attraction

V⊥ ∝ (σ/z)6 − (σ/z)4

and ax-y-periodic potentialV‖ to account for the surfacecorrugation when moving from a (more attractive) Cl− siteto a (less attractive) Na+ site. The perpendicular directionsees a van-der-Waals potential where the power 4 of the

227

attractive part ofV⊥ emerges from superposing the usual1/z6 from each surface atom of the substrate (this is strictlytrue only for a thin film - an integration over an infinitelythick material would lead to a power 3 dependence). Theminimum of V⊥(z) lies approximately 5.7 a0 away fromthe surface as produced by the MD data for clusters [31].The strength was chosen to reproduce the MD-value for thepotential-depth of≈ −0.148eV at the minimum.

The potential of [31] is an effective surface-atom poten-tial in the same spirit as one often uses effective atom-atompotentials for MD simulations of ionic motion [40]. TheUJM, however, has eliminated the localized ionic cores andreplaced it by a smooth distribution according to Eq. (4). Wethus need an effective surface-electron potential. This poten-tial should be similar to that of [31] describing the long-rangevan-der-Waals attraction with change to repulsion at shorterdistance to the surface. But the hard repulsive core,∝ 1/z6,is inappropriate because, unlike point ions, the smooth elec-tron distribution comes arbitrarily close to the boundaries ofthe interface. Therefore, we cut off the hardcore ofV⊥(z) ata limiting potential of 0.5 eV which equals the bandgap inNaCl (mind that the electron states of the cluster lie in thisbandgap). Furthermore, we symmetrize the potential aroundthe minimum which we place, for numerical convenience,at z = 0. Altogether, we use the effective surface-electronpotential

Vse(x, y, z) = Vse,⊥(z)V‖(x, y) (6)

Vse,⊥(z) = min(0.5 eV, 1.38 eV

[Z6 − Z4

]),

Z =σ

|z| + z0(7)

V‖(x, y) = 1.15714 + 0.62857 sin(kxx) sin(kyy)

(8)

with σ = 4.688ao. The strength of the surface-electron po-tential has now been determined by an adjustment of theUJM results for the surface-cluster binding energy to theMD results [42], similar as it was done formerly for thesurface-atom potential. For details, see subsection 2.3.

The corrugational partV‖(x, y) has been taken over un-changed from [31]. Its parameters were chosen to yield acorrect corrugation of an atom on top of a sodium ion(≈ 0.19eV) or on top of a hollow site (≈ 0.09eV). The over-lap with the substrate electronic structure and electron trans-fer are neglected. That is similar to a local pseudopotentialapproach where the effect of the core electrons is also medi-ated exclusively through an effective potential. Such an ap-proach is justified because the underlying electron structureis rather rigid, as shown in [31, 30], and the large bandgapof 0.5 eV in Na-Cl supports this point of view.

The surface-electron potential (6) acts as an external lo-cal potential on the cluster electrons. It can easily be inte-grated to the surface-electron termEse in the energy-densityfunctional (5) yielding

Ese[ρel] =∫

d3r ρel(r )Vse(r ) . (9)

This completes the model Hamiltonian. The calculations arebased on the code of [43], modified to run the UJM forclusters on a substrate, extended to spin-density formalism,extended to remove any symmetry restrictions, and comple-

Fig. 1. Energy differences between clusters in the vacuum and clusters onan insulating surface versus clustersize. Thefull line denotes the energy gainof planar clusters when deposited on the surface, thedashed lineindicatesthe additional interface energy for the corresponding three-dimensional sys-tems. The data was obtained by fitting the depth of the surface potential tothe MD-results of [31] which are labelled by stars

mented by annealing methods to stabilize the iteration inregions of abundant isomers.

2.3 Test of the effective surface-electron potential

Outgoing from the successful surface-atom potential of [41],we have constructed a similar surface-electron potential(7) by applying only minimal modifications. This requires,nonetheless, a new fine-tuning of the potential parameters.The height is fixed at the bandgap of Na-Cl. The remainingdepth of the attractive well is adjusted such that the inter-face energy of clusters from the present model coincidesoptimally with the same observable as obtained from thefull MD calculations [29, 31]. The interface energy is de-fined as the additional energy gained from the binding ofthe cluster to the substrate, i.e.

∆Einterface=|Ebound(N )| − |Efree(N )|

N(10)

whereEbound is the energy of the cluster on the surface andEfree the energy of the corresponding free cluster. This is anenergy difference where all bulk effects cancel out. Thus theUJM is applicable as a first approach for discussing interfaceeffects.

In Fig. 1 we compare the interface energies of the presentUJM (full line for flat clusters and dashed line for 3D-shapes)with the results of the MD simulations (labelled as stars)for all available systems. We find a good agreement overthe whole range of clusters, giving confidence to the UJMapproach for the present purposes of studying Na clusterson a Na-Cl substrate.

At a second glance on Fig. 1, we see two different en-ergies for the clusters withN = 7 and 8. In these cases,there appear two solutions, a planar cluster which coversone layer on the substrate and a “three dimensional” clus-ter which resembles a soft-landed drop. Both configurations,the planar as well as the three dimensional one, are stablein their respective vicinity and there is a strong barrier be-tween the different shapes such that the three dimensionalcluster iterates to a stable minimum configuration once it hasbeen caught in the corresponding attractor region. Thus there

228

is a clear shape isomerism for these two largest clusters inFig. 1, and we will see more examples of isomerism in thesubsequent applications for larger clusters. It will be shownthat the planar shape comes out to be energetically preferredwhich is a consequence of the higher gain in interface en-ergy for the planar clusters with respect to the corresponding”3D”-systems (here about 0.5−1 eV). This advantage over-compensates the energetic preference of 3D-clusters in thevacuum as will be discussed later. More important in thepresent context is the result that the UJM reproduces nicelythe isomerism as found in the full MD simulations up tothe detailed values for the interface energies. This strength-ens the confidence in the UJM approach and it hints thatthe isomerism is predominantly an effect of electronic struc-ture. This motivates us to proceed and to apply the effectivesurface-electron potential to larger systems where the sim-plicity of the UJM can unfold its strengths.

3 Shapes and energetics

3.1 Shapes of planar configurations

We have investigated systems up toNel = 24. The planarconfiguration turned out to be the groundstate geometry forall investigated clusters. To give a first impression of theemerging shapes, we show in Fig. 2 for a variety of systemsthe density of the planar configurations at fixedz = 0 (i.e.at the minimum of the interface potential) as contour plotsin the x-y plane. The structure of the underlying substrateis visualized by circles and stars, where circles denote theposition of a Cl− ion and the stars a Na+ ion. For the smallersystems up toNel = 8 results from full MD simulations areavailable in [29, 31]. And the densities, as computed withthe present model, are very similar to those of the MD calcu-lations. This complements the agreement found already forthe energies and demonstrates that we can also rely on thedensities as predicted by the model.

The global deformation properties of the shapes of thesmaller clusters up toNel = 12 remain very close to thegroundstate configurations found for planar clusters withoutsurface potential, and in particular without corrugation [32]which hints at the still leading role of electronic shell struc-ture. This is further supported by the fact that electronicshell structure yields the systemsNel = 6 , and 12 as magicelectron shells in planar geometry. This feature persists inspite of the visible small modifications in the details of thedensities through the corrugation which adds some “edgi-ness” to the elsewise well rounded shapes. Thus we see thatelectronic shell effects still dominate for small clusters.

The situation changes for the larger clusters,Nel > 12.The shapes differ significantly from the planar clusters with-out corrugation, as shown in [32]. The corrugation gener-ally drives towards more asymmetric shapes. That can beunderstood from a counterplay of electronic structure, pre-ferring smooth distributions, and the fixed atomic positionsin the substrate, pulling the electrons to optimum overlapwith the chlorine positions. The electronic shell effects de-crease with increasing system sizes and the electron cloudas such becomes much softer against perturbation. Thus cor-rugation can take a larger share in determining the clus-

ters shape. Subsequently much different and less symmetricshapes emerge.

Increasing structural differences are expected with fur-ther increasing systems size. In Fig. 3, we show the shapesfor Nel = 16, 20, and 24 and compare it with results froma calculation without corrugation in the interface potential,i.e. settingV‖=1 in the potential (8). First we note, that theshapes with interface potential but without corrugation, asshown on the right column of Fig. 3, match perfectly the re-sults of planar clusters without any confining potential [32]which shows that a smooth interface potential has no effecton the planar distribution. The changes from corrugation,however, are large. In particular, the ”magic”Nel = 24 hasperfect axial symmetry (associated with an axial shell closure[32]) in planar geometry whereas corrugation breaks thissymmetry, breaks even theD4-symmetry of the underlyinglattice, and yields a shape which looks dramatically different.Less symmetry is provided in mere planar geometry for thetwo other systemsNel = 16 and 20. Nonetheless, the corru-gation modifies the shapes further. The reflection symmetryis broken forNel = 20. But the corrugation enhances thesymmetry forNel =16 which was rather asymmetric before.It is visible how the underlying lattice symmetry arrangesthis particularly soft system. Altogether, corrugation playsa dominant role in determining the shape of these systemsin this regime of medium large clusters which was found tohave negligible electronic shell effects in an earlier investi-gation [32]. Even a small perturbation is sufficient to changethe shape substantially. The influence of the corrugation onthe energetics will be discussed later on.

Finally a sideremark: we see from Figs. 2 and 3 thatthe most magic and best bound subsystem, the dimer withNel = 2, appears as a major building block in most config-urations, particularly the less symmetric ones. This featurewas also observed in the UJM for free three dimensionalclusters and it was shown that a simple description in termsof a tight-binding model of dimers is possible [44]. Withsome precaution this could also be an option for the planarclusters here.

3.2 Isomers

As seen in Fig. 1, the high interface energy favors clus-ters which have a large area of contact with the substrate.Thus the planar clusters represent always the ground stateconfigurations in the system Na on Na-Cl. These planar con-figurations have been discussed in the previous subsection.But the Fig. 1 hints that there are also “three-dimensional”isomers, i.e. clusters which stack the electrons in more thanone layer. We will now have a closer look at the isomers inthis subsection.

The Figs. 4 and 5 show contour plots of the electronicdensity in the x-z and the y-z plane for the clustersNel =8,18, 20, and 24. In each case, one finds the planar ground stateconfiguration in the left column, and the two energeticallynext isomers in the right column, with the second isomericstate in the upper panel and the third isomer in the lowerpanel. All isomers have been checked to represent stablelocal minima well separated from each other and from theground state. It is, furthermore, to be noted that the shape

229

N = 4 N = 5 N = 6

N = 7 N = 8 N = 12

N = 13 N = 14 N = 15

N = 16 N = 17 N = 18

Fig. 2. Contour plots of the electronic groundstate densities for various planar clusters in the x-y plane. Thestars and circles indicate the surface of anNaCl(100)solid wherecircles denote the position of a Cl− ion and thestarsa Na+ ion

of the 3D-isomers of the cluster withNel = 8 is identical tothe same isomer in the MD-simulations [30, 31].

One sees clearly that the isomers are more extended invertical direction. The effect is most pronounced for the twosystemsNel =8 and 20 which are magic as free clusters. Themagic electron shell persists as a strong counteracting forceagainst the interface attraction such that the former magic

shells survive in the isomers. The shapes look, quite ex-pectedly, like the free spherical drop, pear-shaped deformedthrough the attachment to the surface. The “magic” attractorregion is so strong that it rules at least two isomers whichlook very similar in their global pattern and are distinguishedonly by different perturbations from the interface, mean-ing that the energetically closest isomer is centered above

230

N = 16

N = 20

N = 24

WITH CORRUGATION WITHOUT CORRUGATION

Fig. 3. Contour plots of the electronic densities of some chosen planar clusters in the x-y plane. The shapes in theright column where calculated byneglecting any corrugation of the surface potential. In theleft columnthis corrugation was explicitely taken into account

231

N = 8

N = 18

PLANAR GROUNDSTATES 3D-ISOMERS

Fig. 4. Contour plots of the electronic densities of the planar groundstate(left column) and the next two 3D-isomers (right column) in the x-z plane(left plots) and the y-z plane (right plots in either column) forNel = 8 and18. Theupper plotson the right panel show the isomer next to the groundstate and thelower plotsshow the third isomer for each clustersize. In allsub-plotsthestarsandcircles indicate the first monolayer of the NaCl(100)solid

a sodium ion and the next isomer above a chlorine ion. Inthe case of the 20-electron cluster, the energetic differenceto the planar ground states is rather large (see Fig. 6). Butthe collapse to the favored planar configuration is associatedwith a huge structural change such that the two states are sep-

arated by a huge energy barrier. Thus the three-dimensionalmagic clusters might be maintained applying very soft land-ing techniques and low temperatures.

The situation is different for the isomers of non-magicclusters, such asNel =10, 12, 16, 18, 22, or 24. These try to

232

N = 20

N = 24

PLANAR GROUNDSTATES 3D-ISOMERS

Fig. 5. Contour plots of the electronic densities of the planar groundstate(left column) and the next two 3D-isomers (right column) in the x-z plane(left plots) and the y-z plane (right plots in either column) forNel =20 and24. Theupper plotson theright panelshow the isomer next to the groundstate and thelower plotsshow the third isomer for each clustersize. In allsub-plotsthestarsandcircles indicate the first monolayer of the NaCl(100)solid

keep closer contact to the surface and produce less verticalextension. This effect is particularly pronounced forNel =18(see upper right panel in Fig. 4) which is not so surprisingwhen reminding that the corresponding free cluster is alreadyoblate. Even the next higher isomer (lower right panel) hes-

itates to lift off into vertical direction. For the system withNel = 24 (see Fig. 5) the move into the third dimension isobviously easier. The lowest isomer (upper right panel) hasa very peculiar shape, looking like the flat isomer ofNel =18with the four extra electrons stacked into the second layer.

233

Fig. 6.Binding energy difference per particle between the energetically mostfavorable planar cluster and the three dimensional configurations versusclustersize. The best 3D-isomers indicated bycircles correspond with theupper density plotsof the right columnof Fig. 4 and Fig. 5, thestarsshowthe energy difference to the second best 3D-configurations drawn in thelower plots

The non-magic systems are similar to the magic ones inthat at least the higher isomers are well separated from thegroundstate by an energy barrier. Therefore one can con-clude that both planar and 3D-configurations are expectedto be stable on the surface, once they have been producedsuccessfully.

The pictures of the shapes need to be complemented byinformation on the relative energies of the isomers. Figure 6shows the energy difference between the planar ground stateand the two closest isomeric configurations. The positive val-ues indicate that the planar configuration is energetically fa-vored. A general trend is that the energy difference rises withcluster size due to the high attraction exerted by the interfaceenergy. It shows a clear odd-even staggering. In spite of thehigh energy differences, the isomers are found to be stable.As already mentioned, this is due to the fact that the largersystems require also much larger spatial rearrangement andthus have larger barriers between the configurations. Over-layed to the general trend, there are huge fluctuations whenstepping from one to the next electron number. The caseNel = 8 has a particularly low energy difference which isdue to the particularly large electronic shell energy of themagic number 8. The actual value of≈ 0.057eV for the en-ergy difference is practically identical to the MD-value from[28]. Moreover, the two isomers of the 8-electron cluster areessentially degenerate which corroborates the interpretationas two slightly perturbed spherical clusters with somewhatdifferent perturbations. The same degeneracy is found againfor the next free magic shell,Nel = 20. Here, however, wehave a generally larger energy difference because the shelleffect is not as pronounced and because the interface en-ergy has gained more weight. The systemNel =18, althoughlarge, has a particularly low energy difference for the bestisomer. This can be understood from a look back on Fig. 4where we see that this isomer remains still very flat thus ex-ploiting the large interface energy together with the anyway

oblate electronic shell structure. The systemNel =24, on theother hand, has no particularly high or low energy value.The energy gain of the planar bottom is counterweighted bythe cost to stack some electrons in a second layer. It is ageneral feature for larger clusters that a relatively small en-ergy difference of the best isomer to the groundstate oftengoes along with a large difference to the 2nd best isomer(see e.g.Nel = 16, 18, 24) and vice versa (e.g.Nel = 14, 20).This fits nicely to the shape considerations from Figs. 4 and5.

3.3 Energetics and corrugation

As a complement to the shapes of the ground states, dis-cussed in subsection 3.1, we are now going to look at thecorresponding energies. Remind, that the present UJM ap-proach confines the discussion to trends and energy differ-ences. The lower part of Fig. 7 shows the binding energyper particle for the ground states with (circles) and with-out (squares) corrugation. Only even systems are producedto suppress the even-odd staggering and to pronounce thesmooth overall trends. The smallest systems are least boundand the binding energy increases with systems size until itlevels off towards a constant binding at aboutNel≈12. Thetwo curves, with and without corrugation, proceed much inparallel with an energy difference of 0.095 eV per particle.The corrugation thus averages very well over the clusterssize amounting just to a constant downshift in energy. Infact, the total interface potential averages in that manner.This is demonstrated by the comparison with the energy ofplanar clusters without surface, as they had been computedearlier in [32]. We have taken these values of free planarcluster and shifted the curve by a constant interface energyper particle of

δεint,cor = 0.571 eV, δεint,sm = 0.476 eV (11)

where δεint,cor stands for the case with corrugation andδεint,sm for the smooth interface potential without corruga-tion. The agreement is astonishing. Except for the smallestclusters, the energetic effect of the surface amounts just toa constant shift. This is similar to the ionic structure energywhich can be derived from perturbative electron responseto ionic structure [45] and which can very well be param-eterized as a global energy correction to be used in simplemodels for electrons [46]. This suggests that the structureand energetics of clusters on surfaces could be estimated insimple models accounting for the balance between volumeenergy, surface tension, and the average interface energy.And first estimates of these separate energy contributionsmay be drawn already from fairly small systems, just ac-cessible to detailed ab initio calculations. This idea will betaken up and worked out in the next subsection. The enor-mous similarity in energy stays in sharp contrast to the largedifferences in shape between systems with and without cor-rugation, as discussed in subsection 3.1. It hints that verylittle energy is involved in this spatial reshuffling. Even thehigh impact of the corrugation on the groundstate shapes forclusters with 16, 20, and, 24 electrons (see Fig. 3) is not dis-tinctly reflected in this energetic inspection. This means that

234

Fig. 7. Upper plot: Second differences of planar clusters in the vacuum (fullline) and of the corresponding clusters on the surface (dashed line) in thesize range 2≤ Nel ≤ 24. Middle plot: Second differences of free three-dimensional clusters in the vacuum.Lower plot: Binding energy per particleversus clustersize for even clusters. The circles denote planar clusters onthe surface using the full effective interface potential. The squares indi-cate results for planar clusters using an effective interface potential withoutcorrugation. The binding energies for planar clusters without any interfacepotential are shown for comparison in two manners: the stars representthese energies shifted by a constant amount of 0.571 eV which stands forthe full interface energy per surface particle, and the triangles come froma shift by 0.476 eV standing for the corrugationless interface energy perparticle

simple models will work for the energetics but may turn outinsufficient for computing detailed shapes.

The energies as such, as shown in the lower part of Fig. 7tend to hide the subtle shell fluctuations. These are madebetter visible by displaying the second differences [47]

∆E2(N ) = E(N + 1)− 2E(N ) +E(N − 1) . (12)

The upper part of Fig. 7 shows the second differences for theplanar clusters, with (dashed line) and without (solid line)interface potential. The difference between these two optionsis minimal, even when looked at with the amplifying glassof the ∆E2. This justifies once more the constant shift byan interface energy. Unlike the binding energies as such, the∆E2 are produced for even as well as for odd clusters. The

results show a pronounced even-odd staggering because oddsystem are generally less bound than even systems.

Magic shells are distinguished by pronounced peaks inthe ∆E2 plot. This is demonstrated in the middle part ofFig. 7 which displays the second energy difference for freeNa clusters in the UJM. We have to concentrate on the evenclusters (the upper spikes of the staggering) and to look forpeaks sticking clearly out of the average. This is obviouslythe case for the magic free clustersNel =8 andNel =20 witha shell energy (i.e. enhancement above average) of about0.5 eV. No effect of this sort or size can be seen for planarclusters as displayed in the upper part of Fig. 7. As arguedearlier [32], the shell effects are generally smaller in two-dimensional systems and they are completely washed outby the energy gain through free shape adjustment of theclusters on surfaces without any symmetry restriction. In thissense the already weak ”magic” number 24 from free planarclusters [32] has been suppressed completely associated withthe loss of the perfect axial symmetry through corrugation,as can be seen in Fig. 3. The 10- and 20-electron clusters,on the other hand, gain weight through corrugation whichin this case drives closer to axial shapes. Altogether, thecorrugation smoothens the energy differences even furtherand wipes out completely any trace of magic numbers forplanar clusters in the size range studied here.

4 Interplay between interface and surface energy

4.1 Variation of the interface potential

All above studies have been performed for the fixed Na onNa-Cl system and our results hint on a rather strong inter-face potential rendering all ground states to assume planarshapes tightly bound to the surface of the substrate. It isinteresting to see to what extend these result do depend onthe actual strength of the interface potential and how theychange if that strength is varied. This serves as a model forother insulating substrates with a similar surface geometrybut a different polarization effect on a sodium cluster. Tothat end we have performed a series of calculations withvaried strength of the interface potential (6). We reoptimizethe shapes of the lowest planar state and the lowest three-dimensional configurations for each new strength. We findthat the shapes for a given cluster do change very little withthe strength. Mainly the relative energies undergo substan-tial and systematic changes. The results for the energies areshown in Fig. 8 where we plot the energy difference betweenthe lowest planar and the lowest three-dimensional config-uration for the even clusters betweenNel = 8 and 20 versusrelative strength. Straight lines result from an energy changewithout structural change for both configurations. Kinks, onthe other hand, are produced by a change in cluster structureof one of the competitors when increasing the strength of theinterface potential.

The systems with the more stable three-dimensional con-figurations are collected in the upper part of Fig. 8. All ener-gies proceed as straight lines with the exception ofNel =20which produces a kink at a relative strength of 80%. Thiskink is related to a structural change amongst two planarconfigurations. The one belonging to strengths above 80%

235

Fig. 8. Difference in binding energy per particle between the planar groundstate and the best three dimensional configuration for various even clustersas a function of the strength of the interface potential. The strength is givenrelative to the standard strength of the optimized effective interface poten-tial. The short verticaldotted linesdenote values of the surface potentialwhere fundamental shape changes occur

is shown on the left column of Fig. 3 whereas the shapefor the weaker interface potentials is the same which wasobtained without corrugation shown on the right column ofFig. 3. All other configurations remain stable and the linesrepresent a steady trend in the energy difference betweentwo nearly unchanged clusters. In these cases, the interfaceenergy has no effect on the structure. It just adds a con-stant amount of energy per unit interface area which scaleslinearly with the interaction strength.

The systems with soft three-dimensional configurationsare collected in the lower part of Fig. 8. Linear trends stillprevail. But each system shows at least one kink hintingat structural changes there. The more frequent structuralchanges are not surprising for theses cases as we are deal-ing with soft three-dimensional electron configurations. Andit is indeed the structures of the three-dimensional clusterswhich undergo these changes, as can be seen from the down-shift after the kink. The structural optimization of the three-dimensional shape regains some energy for this variant andthus lowers the difference. The two configurations which areexchanged can be seen in Figs. 4 and 5. There we show inthe right panels for each case the two next isomers and thesetwo are interchanging their energetic order with varying in-terface energy due to their different interface area. One canalso deduce from these figures that the shape changes arelarger for the larger systems and this relates to the featurein Fig. 8 that the larger systems show the larger kinks. Onecase amongst this group owes its kink to a shape transition inthe planar groundstate, similar as forNel = 20. It is the 16-

Fig. 9. Critical strength of the interface potential, i.e. the strength fromFig. 8 for which the planar configuration becomes energetically favorable,as a function of the cluster size. Thehorizontal dashed lineshows an asymp-totic critical surface potential for the transition to planar clusters

electron cluster which undergoes a transition from the shapeobtained without corrugation to the one with four-fold sym-metry shown in Fig. 3, resulting in a small up-shift after akink around 45%.

In any case, the general trend of the energy differenceis always increasing with increasing interface energy. Forzero interface energy, the three-dimensional clusters are, ofcourse, energetically preferred, indicated by a negative valueof ∆E. That complies with the experience from free Na clus-ters. This preference fades away with increasing interfaceenergy and sooner or later the planar configurations take thelead. The transition point depends on the system size and itmoves to a lower interaction strength with increasingNel.The interaction strength where the energy difference crossesthe zero line can be called the critical strength for the tran-sitions from three-dimensional to planar shapes. Figure 9shows these critical strengths deduced from Fig. 8 versussystem size. The caseNel = 8 with its pronounced spher-ical magic electron configuration resists the longest time.But then the critical strength drops rapidly and stabilizes forNel≥12 at around 40% of the given strength which meansthat the combination Na on Na-Cl has a rather large inter-face energy forcing clearly a monolayer growth. For relativestrengths below 40%, the clusters form droplets correspond-ing to the 2nd best isomer shapes from Figs. 4 and 5. Thismeans that for a weak interface potential the magic num-bers of free spherical clusters are expected to survive, evenif the free geometry is slightly distorted. It is interesting tonote that the critical strength levels off at exactly the samepoint where the binding energies per particle can be fittedvery well by a constant interface energy per unit area, i.e. atNel = 12 as seen in the lower plot of Fig. 7. This seems tobe the system size from which on a more global descriptioncould be possible.

4.2 A simple model of the critical strength

The above results indicate that many features of clusters onsurfaces can be described in terms of simple global quan-

236

tities as surface energy and interface energy. This can beparameterized in a Liquid-Drop-Model (LDM) formula forthe energy as

E = Evol − aiσi + asσt ,

ai = 0.02992eV/a02 , as = 0.00443eV/a0

2 ,(13)

whereai is the numerical coefficient of the interface energyandas stands for the surface energy of the free clusters perunit area. The surface area in contact with the substrate isσiand the total surface area isσt (embracing surface to vacuumplus surface to interface). The signs have been chosen suchthat the coefficients are all positive, i.e. the the interface en-ergy is attractive whereas the free surface energy is repulsivetrying to minimize the surface area. Consequently the inter-face energy drives deformations towards a planar geometrywhereas the surface energy prefers a spherical shape.

We now want to use this simple LDM model to estimatethe transition from three-dimensional to planar clusters. Tothis end, we need to parameterize the shapes of a clustertouching on a surface. As a very crude and most simpleapproach we assume a sphere of radiusr which is cut by aplane such that a segment of heightx is stripped off. Thetotal volume of the cluster is to be kept constant at

V =4π3R3 (14)

whereR = rsN1/3 is the radius of the corresponding free

cluster. This fixes the actual radiusr of the cut sphere independence ofx, i.e.r = r(x,R). The heightx of the missingsegment remains as the crucial deformation parameter inthe model. Useful values for it are inx ∈ {0, 2R − rs}wherex = 0 is the limit of an unperturbed sphere andx =2R−rs represents the limit of a planar cluster. With standardgeometrical relations for the volume and surface of this cutsphere, we find

σi = π3x(

2Rx2R−x − x + 6R

)σt = 4πR2 + π

3x2(

4R2R−x + 1

) (15)

leading to the energy

E = Efree +∆E

Efree = Evol + as4πR2

∆E = aiπ

3x

(2Rx

2R− x− x + 6R

)+as

π

3x2

(4R

2R− x+ 1

)(16)

where∆E summarizes all modifications through deforma-tion and contact with the surface. Deformation occurs if∆Eis negative.

The onset of deformations is checked at smallx. Ex-panding up to second order inx gives

∆E ≈ −2πRaix + πasx2 . (17)

Thus∆E drives always towards some contact with the inter-face for nonzero (and positive) interface energy because theinterface term is the only linear term in the expansion andthus dominates for very smallx. The process is stopped withgrowing deformation by the counteracting surface energy

such that finally some equilibrium value forx is arranged.From the smallx expansion (17) we find the equilibriumpoint atxeq = Rai/as which nicely shows the interplay be-tween interface and surface energy. This extremely simplerelation, however, holds only if the restoring surface en-ergy is strong enough to maintain a small deformationxeq.In general one has to deal with the full∆E(x), as givenin Eq. (16). A closer analysis of (16) shows that basicallytwo cases have to be distinguished: The first one deals witha weak interface energy where the minimum is far awayfrom the limit of a planar cluster. There we expect slightlydeformed three-dimensional clusters with anxeq which isconsiderably smaller than 2R − rs. The second case corre-sponds to a strong interface energy. Here we expect planarclusters, meaningxeq ≥ 2R − rs. The critical strength isreached if the minimum appears exactly at the planar limitof xeq = 2R−rs. The energy (16) is still simple enough thatthe position of the minimum can be evaluated analytically.We find

xeq = R

(2 +

(4(ai + 2as)ai − as

)1/3)

. (18)

Note that this result is still independent of the choice of thecluster-surface combination. As argued above, the criticalstrengthacrit

i is reached ifxeq = 2R−rs. For sodium clusters(rs = 4.1a0), this happens at

acriti = 2as . (19)

The actual parameters from (13) for the system Na on Na-Cl(100) deliverai = 6.75as which is high above the criti-cal value in agreement with the experience that this systemalways prefers the planar clusters. Reading the relation 2to 6.75 the other way round, we conclude that the criti-cal strength from our simple estimate is 30% of the actualstrength for the system Na on Na-Cl(100). From Fig. 9 wehave deduced that the critical strength in the detailed cal-culations comes at 40%, which corresponds to a value ofacrit

i = 2.7as. This is in excellent agreement in view of theextreme simplicity of our present LDM estimate (remind, inparticular, the crude deformation model). Thus we see thatsimple LDM models may account very well for the globalenergy balance of clusters on surfaces.

5 Conclusions

We have investigated the structure and energetics of Na clus-ters on Na-Cl(100). Starting from the assumption that theelectronic structure dominates Na clusters, we have used theultimate jellium model (UJM) which omits the ions as ex-plicit degrees-of-freedom and fixes them tightly to the elec-tron density. The interaction with the substrate is incorpo-rated by an effective electron-surface potential which is ad-justed by comparison with ab initio calculations. The modelprovides good agreement with the results of the ab initiocalculations concerning shapes and energies.

The interface energy of the system Na on Na-Cl is verystrong such that planar configurations are always energeti-cally preferred because they produce the largest overlap withthe attractive interface potential. Three-dimensional clusters

237

are regularly found as next isomers, often with large en-ergy distances, but stabilized also by large barriers. Onlythe 8-electron system has a particularly small energy differ-ence between the planar ground state and a distorted sphereas first isomer, which is in perfect agreement with MD-calculations. The interface energy has led to a disappearanceof magic numbers as the energy gain through a perfect sur-face matching washes out the small shell energies of theseplanar systems. A variation of the interface energy leaves thebasic shapes more or less invariant and merely changes therelative energies according to the different area of contactwith the substrate (and thus different energy gain from theinterface). This leads to “shape transitions”, i.e. exchange ofthe role of ground state and first isomer, as a function of thepotential strength. The critical strength depends on the clus-tersize for small clusters and stabilizes at about 40% of theactual strength atNel≥12. For these larger clusters, the en-ergetic effect of the contact with the substrate can be reliablyparameterized in terms of a constant interface energy timesthe area of contact. This allowed us to construct a simple liq-uid drop model which has reproduced nicely these findingsand which is applicable to other cluster-surface combina-tions.

The interface potential consists of a generally attractivewell and it is superposed by a periodic corrugation which isrelated to regions of more (Cl atoms) and less (Na atoms)attraction. The energetic effect of the corrugation can stillbe accounted for in a global fashion by a constant corruga-tion energy per unit area (which is, of course, part of thetotal interface energy). The shapes, however, change sub-stantially when switching from a potential with corrugationto one without. This means that energetics and the dimen-sionality of the sodium clusters on a substrate can be esti-mated in a model of simple structure averaged energies. Butthe detailed density distribution at the interface area requirescorrespondingly the more detailed models.

We thank Hannu Hakkinen and Matti Manninen from the University ofJyvaskyla for their cooperation and their help.

References

1. Schaefer, D.M., Patil, A., Andres, R.P., Reifenberger, R.: Phys.Rev.B51, 5322 (1995)

2. Castro, T., Li, Y.Z., Reifenberger, R., Choi, E., Park, S.B., An-dres, R.P.: J.Vac.Sci.Technol.A7, 2845 (1989)

3. Li, Y.Z., Reifenberger, R., Andres, R.P.: Surf.Sci.250, 1 (1991)

4. Kuhrt, Ch., Harsdorff, M.: Surf.Sci.245, 252 (1995)5. Barnett, R.N., Landman, U.: Phys. Rev. Lett.67, 727 (1991)6. Luedtke, W.D., Landman, U.: Phys. Rev. Lett.73, 596 (1994)7. Head, J.D., Silva, S.J.: J.Chem.Phys.104, 3244 (1996)8. Mejias, J.A., Sanz, J.F.: J.Chem.Phys.102, 327 (1995)9. Erkoc, S., Halicioglu, T., Tiller, W.A.: Surf.Sci.274, 359 (1992)

10. Roy, H.-V., Boschung, J., Fayet, P., Patthey, F., Schneider, W.-D.:Z.Phys. D26, 252 (1993)

11. Roy, H.-V., Fayet, P., Patthey, F., Schneider, W.-D., Delley, B., Mas-sobrio, C.: Phys.Rev.B49, 5611 (1994)

12. Vandoni, G., Felix, C., Goyhenex, C., Monot, R., Buttet, J., Har-brich, W.: Surf.Sci.331–333, 838 (1995)

13. Haberland, H., Insepov, Z., Moseler, M.: Phys. Rev.B51, 51 (1995);Z.Phys. D26, 229 (1993)

14. Fallis, M.C., Daw, M.S., Fong, C.Y.: Phys.Rev.B51, 7817 (1995)15. Massobrio, C., Fernandez, P.: J.Chem.Phys.102, 605 (1995)16. Munger, B.P., Chirita, V., Greene, J.E., Sundgren, J.-E.: Surf.Sci.355,

L325 (1995)17. Fernandez, P., Massobrio, C., Blandin, P., Buttet, J.: Surf.Sci.307–309,

608 (1994)18. St.John, P.M., Beck, R.D., Whetten, R.L.: Z.Phys. D26, 226 (1993)19. Schafer, D.M., Ramachandra, A., Andres, R.P., Reifenberger, R.:

Z.Phys. D26, 249 (1993)20. Bifone, A., Casalis, L., Riva, R.: Phys.Rev.B51, 11043 (1995)21. Moullet, I.: Surf. Sci.331–333, 697 (1995)22. Patil, A.N., Paithankar, D.Y., Otsuka, N., Andres, R.P.: Z.Phys D26,

135 (1993)23. Shluger, A.L., Rohl. A.L., Gay, D.H.: Phys.Rev.B51, 13631 (1995)24. Henry, C.R., Chapon, C., Duriez, C., Giorgio, S.: Surf.Sci.253, 177

(1991)25. Bellamy, B., Mechken, S., Massno, A.: Z.Phys D26, 61 (1993)26. Gotz, T., Hoheisel, W., Vollmer, M., Trager, F.: Z.Phys. D33, 229

(1995)27. Barnett, R.N., Landman, U.: Phys.Rev.B48, 2081 (1993)28. Cheng, H.P., Landman, U.: Science260, 1304 (1993)29. Hakkinen, H., Manninen, M.: Phys.Rev.Lett76, 1599 (1996)30. Hakkinen, H., Manninen, M.: submitted to J.Chem.Phys. (1996)31. Hakkinen, H., Manninen, M.: Europhys. Lett.34 (3), 177 (1996)32. Kohl, C., Montag, B., Reinhard, P.-G.: Z.Phys. D38, 81 (1996)33. Troullier, N., Martins, J.L.: Phys.Rev.B43, 1993 (1991)34. Koskinen, M., Lipas, P.O., Manninen, M.: Z.Phys. D35, 285 (1995)35. Montag, B., Reinhard, P.-G.: Z.Phys. D33, 265 (1995)36. Kolehmainen, J., Hakkinen, H., Manninen, M.: submitted to Z.Phys.

D (1996)37. Reimann, S., Koskinen, M.: submitted to Z.Phys. D (1996)38. Jones, R.O., Gunnarsson, O.: Rev.Mod.Phys.61, 689 (1989)39. Gunnarson, O., Lundqvist, B.I.: Phys.Rev.B13, 4274 (1976)40. Ju, N., Bulgac, A.: Phys.Rev.B48, 2721 (1993)41. Hakkinen, H.: (private communication 1996)42. Hakkinen, H.: (private communication 1996)43. Lauritsch, G.: PhD thesis, Erlangen (1992)44. Montag, B.: Z.Phys. D34, 263 (1995)45. Lang, N.D., Kohn, W.: Phys.Rev.B1, 4555 (1970)46. Montag, B., Reinhard, P.-G., Meyer, J.: Z.Phys. D32, 125 (1994)47. Brack, M.: Rev.Mod.Phys.65, 677 (1993)