clusters ( n =1–48) at the (100) MgO surface

PHYSICAL REVIEW B VOLUME 51, NUMBER 19 15 MAY 1995-I

Properties of small clusters at ionic surfaces: (NaC1)„clusters (n = 1 —48) at the (100) Mgo surface

Alexander L. ShlugerThe Royal Institution of Great Britain, 21 Albemarle Street, London &IX4BS, United Kingdomand Institute of Chemical Physics, University ofLatvia, 19Rainis Boulevard, Riga 1098, Latvia

Andrew L. Rohl and David H. GayThe Royal Institution of Great Britain, 21 Albemarle Street, London WIX4BS, United Kingdom

(Received 17 October 1994; revised manuscript received 13 January 1995)

We have studied the geometry, binding energy, interaction with the surface, barriers for diffusion, op-tical absorption, and the possibility for their observation using atomic force microscopy of (NaCl)„clus-ters (n =1—48) on the (100) MgO surface. We address the questions at which cluster size do the ad-sorbed molecules lose their identity and how do strained clusters accommodate the strain. The relationbetween the structure of initial molecular fragments adsorbed at the surface and the structure of the cor-responding thick 61m is discussed. The results are compared with the calculated structures of the freeclusters and the experimental data on the molecular-beam epitaxy of alkali halides.

I. INTRODUCTION

Elemental and molecular clusters are often consideredto bridge the gap between the properties and behavior ofsingle molecules in the gas phase and bulk condensedmatter. ' In particular, small molecular clusters of ion-ic compounds are the focus of many experimental andtheoretical studies. Interest in these systems is promptedfirst of all by their relatively simple structure. Free clus-ters of various sizes can be readily produced in laser-vaporization sources, by gas aggregation, and particlesputtering. The formation, size, structure, and chemicalproperties of charged clusters of alkali halides '

(AH's) and cubic oxides such as MgO and CaO (Refs.6, 13—16) have been investigated experimentally in greatdetail. Atomic and electronic structures of free small al-kali halide and oxide clusters have been calculated usingdifFerent techniques. '

Despite the increasing number of studies of the proper-ties of free molecular clusters, the experimental tech-niques applicable for the study of their geometric andelectronic structures are limited. In particular, theanalysis of the mass spectra of charged nonstoichiometricclusters provides only qualitative information regardingthe geometric models of these clusters. Recent advancesin the molecular-beam epitaxy (MBE) growth of insula-tors and the possibility of cluster deposition at insulatingsurfaces offer new opportunities in this respect: clusterscan be deposited directly on surfaces from the clustersource' or formed during homo- and heteroepitaxialgrowth of ionic compounds. Their structure can bestudied using a wide range of techniques including suchmethods as atomic force microscopy ' (AFM) and ionand metastable-atom impact electron spectroscopy(IIES and MIES). Apart from the clear fundamental in-terest, information obtained in these studies could pro-vide an insight into the mechanisms of crystal filmgrowth, homo- and heteroepitaxy, structural and elec-tronic characteristics of nanoscopic species and their in-teraction with surfaces, and properties of surface inhomo-

geneities, including steps and kinks. This informationcould be used in order to estimate the accuracy and reso-lution of the surface-sensitive techniques such as AFM,IIES, and MIES.

Theoretically, Bjorklund and co-workers considered re-actions between alkali halide molecular beams (mainlyAH molecules and dimers) and single crystals of alkalihalides. Shluger, Gale, and Catlow studied absorp-tion of MgO and LiC1 molecules at the (100) MgO sur-face. Celli and Urzua calculated the configurations ofadsorbed NaC1 and KBr molecules on the (100) NaCl and(100) KBr surfaces. No theoretical studies have exam-ined larger clusters on surfaces to our knowledge.

In this study we consider some general features con-cerning the structure and diffusion of molecular clustersat ionic surfaces with large geometric misfit using atomis-tic simulation and quantum-chemical techniques. A goodgeometric and chemical contrast between the host surfaceand point defects, such as impurities and adsorbed mole-cules and clusters, is necessary for their observation usingAFM and spectroscopic techniques. As a particular sys-tem we have chosen NaCl clusters on the (100) MgO sur-face. MgO is known as a good support for deposition ofdifferent molecules and heteroepitaxial growth of othermaterials. It provides an example of a cubic ionic materi-al with a strong crystalline field near the surface. Thelatter should make the adsorption energies large and em-phasize the strain effects in which we are interested inthis study. Formal misfit between the lattice constants ofMgO and NaC1 is about 6:8. We analyze the calculatedproperties of (NaC1)„clusters (n =1—48) at the MgOsurface from the point of view of their geometry, bindingenergy, barriers for diffusion, optical absorption, and thepossibility of their observation using AFM. Dynamicprocesses of the interaction of molecules and clusterswith surfaces are beyond the scope of this paper. Al-though at real surfaces the molecules often fill the defec-tive sites such as kinks, vacancies, and step edges first, inthis study we will focus on their interaction with perfectsurfaces.

0163-1829/95/51(19)/13631(14)/$06. 00 51 13 631 1995 The American Physical Society

13 632 SHLUCxER, ROHL, AND GAY 51

II. CALCULATIDN TECHNIQUES

%'e report the results of static calculations based ontotal-energy minimization to obtain the equilibrium clus-ter geometries and adiabatic barrier energies for clusterdiffusion, which do not take into account energy dissipa-tion in the adsorption and cluster formation process. Thelarge number of relaxing ions requires a semiempiricaltechnique for energy calculation. Therefore we have em-

ployed both an atomistic simulation technique (MARVINcode" ' '), and a semiempirical quantum-chemicalmethod (CLUSTER code ).

A. MARVIN atomistic simulation technique

This technique adopts a multiblock approach, with pla-nar two-dimensional periodic boundary conditions paral-lel to the interface. Each block consists of two regions asshown in Fig. 1. The first contains the ions which are re-laxed explicitly until there is zero force on each of them,while the ions in the second are fixed. Thus the model ofa surface consists of a block ( A) with its region 1 con-taining the ions closest to the surface, and the ions in re-gion 2 are fixed to reproduce the potential of the remain-ing lattice (see Fig. 1). The methodology is closely relat-ed to that used by Tasker in the M&DAs code. In the ad-sorption calculations, the adsorbing molecules are placedin the same block as the surface ions and all ions of theadsorbed cluster and the surface layer are allowed to re-lax. In all the MARvIN calculations of adsorption in thisstudy, the surface region 1 contained one plane of 200surface ions and the cluster ions. Region 2 containedfour planes of frozen ions. The surface deformation dueto the cluster-surface interaction in the present work didnot exceed 0.08a (a is the lattice interionic distance) per-pendicular to the surface plane. Based on our previousAFM calculations ' using the same technique, in whichthe surface deformation was much larger, we believe thatadditional unfrozen surface planes would not change ourqualitative conclusions.

To speed up the calculations, we have incorporated theidea of "freezing. " This involves taking an ion out of theminimization if the force on it is less than a given value.

There is no guarantee that the force on this ion will notincrease as the rest of the ions relax around it, and this ischecked for at the end of each calculation. This tech-nique has been found to be extremely effective for the cal-culations presented here.

The functional forms of the potentials used to describethe clusters and surface and their interaction are based onan ionic model. The major interaction is between chargescentered on the positions of the ions, with electronic po-larizability incorporated via the Dick-Overhauser shellmodel. " The Coulomb interaction is long range and thesum over the crystal lattice is conditionally convergent.Hence a two-dimensional Ewald summation techniquehas been employed. ' Two-body potentials were usedto represent the non-Coulombic interactions between theions. The range over which these non-Coulombic in-teractions operate is determined by a potential cutoff.

The parameters of the pair potentials used in this workar summarized in Table I. The short-range potentials formost of the ions were specially optimized. The aim ofthis optimization was to maintain the consistency be-tween the different interactions and to reproduce thecharacteristics of the perfect lattices correctly but stillproduce a potential whose functional form is robust withrespect to significant distortions in the bond lengths awayfrom equilibrium values. In particular, the oxygen-oxygen potential was determined by simultaneouslyfitting the short-range parameters to reproduce the latticeparameters of a range of different binary and ternary ox-ides. " A11 other potentials were initially derived using anelectron-gas method and further refined by fitting to theperfect-lattice properties of the appropriate materials,e.g. , MgO, NaC1, Na20, MgC12. Since these are fitted toexperimental data, they can be described as "empirical"potentials. In cases where no lattice data were available(i.e., for O-C1, Mg-Na), an empiricizing procedure wasemployed described as follows using O-Cl as an exam-ple. The short-range energies between 0-Cl and between

TABLE I. Short-range potentials used. The potentials havethe functional forms 8'(r)= A exp( —r/p) (Born-Mayer) and8'(r) = A exp( —r /p)+ C/r {Buckingham).

Between ions Potential type A (eV) p A C(eVA )

Na+

CI

Mg2+I eglCXl 1

O-ONa-0Mg-0Cl-0Cl-Cl'Na-ClMg-ClNa-NaNa-Mg

BuckinghamBorn-MayerBorn-MayerBuckinghamBuckinghamBuckinghamBuckinghamBuckinghamBuckingham

9 574.961 677.831 284.384 393.12 021.33 046.42 511.516 927.8

28 261.4

0.21920.29340.29970.27210.35880.28360.28570.18360.1510

32.0

62.288.9812.826.224.432.10

FIG. 1. Part of the simulation cell for the calculation of theadsorption of a molecule of NaC1 onto the (100) plane of MgO.The cell is repeated infinitely in the x and y directions.

'The geometries of the small clusters are better reproducedwithout the short-range Cl-Cl interaction as it is fitted for theinfinite lattice. Therefore this potential was only used to derivethe Cl-0 short-range potential. It was not used in the calcula-tions of the cluster geometries, where only Coulomb repulsionwas included.

PROPERTIES OF SMALL CLUSTERS AT IONIC SURFACES: 13 633

Cl-Cl were calculated using an electron-gas method. '

Although the absolute energies of these potentials are notconsistent with the empirically derived potentials, thedifference between electron-gas-derived potentials is as-sumed to be useful. As such, if the difference between theO-Cl and Cl-Cl electron-gas potentials is added to theempirical Cl-Cl potential, the resulting "empiricized" 0-Cl potential is representative and consistent with theempirical potentials. For all anion-anion potentials, theC6 dispersion terms were determined using theKirkwood-Slater formulas. Only Coulomb repulsionwas taken into account for the Mg-Mg interaction. Allthe ions had their full formal charges and only the anions0 and Cl were treated as polarizable. Their shell-modelparameters ' were the following: Yo = —2.04e;Ko 6.3 Yc& 1 984 'Kc& 13 209 VA

Using the MARVIN code and this set of parameters wecalculated the reconstructions of the perfect (001) sur-faces of NaC1 and MgO. The results are qualitativelysimilar for both crystals: the cations displace slightly in-ward and the anions outward from the crystal perpendic-ular to the ideal surface plane. The magnitudes of thesedisplacements are about 0.01a (a is the bulk interionicseparation) for MgO, and about 0.02a for NaC1. A verysimilar surface relaxation was calculated for MgO usingthe MIDAS code and the CLUSTER code.

To calculate the cluster-surface interaction energy andthe cluster distortion at the surface we have also calculat-ed the energetic and geometric parameters of the freeclusters using interatomic potentials and the GULPcode. For comparison with the studies of free clusterswe used the results of extensive Hartree-Fock ab initiocalculations reported in Ref. 22 as they contain all thenecessary numerical information. The latter is alsonecessary to check the parameters of our interatomic po-tentials. Agreement of the geometric parameters ob-tained is better than S%%uo. However, the binding energiescalculated using interatomic potentials are generallyabout 10%%uo bigger than those obtained using theHartree-Fock method in Ref. 22. More detailed compar-ison of these results is presented in the next section.

B. cLUsTER quantum-chemical technique

The program employs both the embedded molecularcluster (EMC) model and the periodic large unit cell(LUC) method ' and is based on the intermediateneglect of difFerential overlap (INDO) approximation ofthe unrestricted Hartree-Fock-Roothaan method. ' Itallows us to determine the electronic structure of aquantum-mechanically described cluster that containsseveral tens of ions. It employs a minimal valence Slaterbasis set and uses a single-determinantal approximationfor the wave function and therefore does not take into ac-count van der Waals interactions. The lattice relaxationin the present study includes only ionic displacementsfrom the lattice sites. The optical-absorption energieswere calculated using the configuration-interaction tech-nique and included all single-electron excitations.

The calculation scheme of the CLUSTER code and theparametrization of the INDO method are described in

Ref. 44. The procedure for a defect study on a crystalsurface using the CLUSTER code includes the followingsteps. (1) Calculation of the electronic structure and re-laxation of the perfect surface are made using periodicboundary conditions where the crystal surface is treatedas a slab comprising several atomic planes. For this pur-pose the LUC method is used, as both sides of the two-dimensional infinite slab are equivalent. As has beenshown both in our previous semiempirical calculations,and in recent ab initio studies, a slab comprising 5 —7atomic planes of the rocksalt ionic crystal lattice satisfiesthese requirements. (2) A study of adsorbtion on the sur-face is performed in the EMC model for a cluster embed-ded in a slab and adsorbed molecules. The atomic struc-ture of the slab outside the cluster is treated as in the firststep of the study. The ions outside the cluster carry thesame basis of atomic orbitals (AO's) as inside the cluster,but with the Lowdin populations of those AO's frozento the AO populations in the slab simulating the perfectsurface. The Coulomb interaction with these ions is cal-culated exactly up to a distance R where the Coulomb in-tegral between two interacting points becomes practicallyequal to 1/R. The potential of the crystalline field pro-duced by the rest of the slab is then calculated using theEwald method. The crystal surface in the EMC calcula-tions was simulated by a Na48C148 molecular clustercomprising one plane of 96 ions embedded in the slab offive lattice planes.

Finally we should make some comments regarding theaccuracy of the results of the present calculations. First,the interaction energy between the periodically locatedclusters can be considerable. However, it was not biggerthan 0.03 eV in our calculations even for the largest 48-molecule cluster. Secondly, the energy of the van derWaals interaction between the cluster and their surfacedepends on the number of surface layers as it decreasesvery slowly. However, as is shown below, the geometryof the clusters at the surface does not depend significantlyon the van der Waals interaction. Therefore a five-layerslab was used in all MARVIN calculations. Thirdly, thecharges on the ions are fixed in all MARVIN calculations.However, they may be different for the cluster ions withdifferent coordination, and for the surface ions interact-ing with the cluster. As was obtained in the quantum-chemical calculations of the clusters at the surface usingthe CLUSTER code, the variation in the charge values doesnot exceed 0.05e (e is the electron charge). Therefore webelieve that this cannot change our qualitative con-clusions.

III. RESULTS OF CALCULATIONS

The cluster structure to a certain extent is determinedby the surface, and the variety of possible cluster struc-tures formed at surfaces, although large for large numberof molecules, can be much smaller than that for free clus-ters. In this study we did not consider all possible clusterstructures for a given number of molecules, but rather ad-dressed some specific issues regarding clusters at surfaces.

13 634 SHLUGER, ROHL, AND GAY

A. %'hich structural fragments can one expect at the veryinitial stages of molecule aggregations

Adsorption of individual AH and MgO molecules atthe (100) surfaces of AH and Mgo has been considered ina number of studies using diferent calculation tech-niques. ' ' At long surface-molecule distances their in-teraction is determined by the exponentially decayingelectrostatic field of the whole surface. At short (aboutthe crystal lattice constant) distances the ions of the mol-ecule interact with the individual surface ions by thestrong Coulomb and short-range forces. Finally, the mol-ecule adopts a distorted configuration with the positivelycharged ion roughly above the surface anion and the neg-atively charged anion above the surface cation (see Fig.1). This configuration refiects the predominance of theelectrostatic forces.

The properties of the whole system "the surface withadsorbed molecular cluster" are determined by its free en-ergy. ' In this study we approximately determined therelative stability of the difFerent configurations of theclusters comprising the same number of molecules at thesurface by comparing the total energies of the system.Another useful energetic parameter which characterizesthe cluster-surface interaction is the molecule (cluster)adsorption energy E,d at the surface. It is defined withrespect to the free state of the cluster with the same shape(if it exists) as stabilized at the surface. This energy in-cludes the surface, E,"„'„andmolecule (cluster), E',i', dis-tortion energies with respect to their free states, and themolecule-surface interaction energy E;„,at the equilibri-um configuration of the adsorption:

E„=E„E—,'„,E—,', =E,„,+(E „',+E; ),where E„is the total energy of the surface with n ad-sorbed molecules, and E,„,and E,&

are the energies of therelaxed free surface and cluster, respectively. The ad-sorption energy of the single NaC1 molecule at the (100)MgO surface obtained in our calculations is equal to 1.01eV. As can be seen in Table II, the surface distortion en-ergy due to adsorption of the NaC1 molecule is about tentimes bigger than the molecule distortion energy. Thenearest surface ions are considerably displaced from theirpositions at the perfect surface (see Fig. 1) due to thestrong dipole-surface and short-range interactions withthe molecule, whereas the intramolecular distance in-creases from 2.47 to 2.53 A. The biggest displacementsof the surface ions are those perpendicular to the surfaceplane and comprise 0.07a and 0.08a for the O and Mgion, respectively, where a is the Mgo bulk interionic sepa-ration (a =2. 1058 A in our calculations).

Displaced surface ions and adjacent vacancies consid-erably change the surface electrostatic potential whichdecreases much more slowly than that of the perfect sur-face. The interaction of the molecule dipole momentwith this potential determines the large molecule-surfaceinteraction energy. For instance, the latter is equal toonly —0.3S eV for the molecule interacting with the rigidperfect surface. It is interesting to note that the gain inthe molecule-surface interaction energy due to the sur-face distortion considerably outweighs the surface distor-tion energy. This explains why the surface of hard MgOdistorts so readily in the field of a molecule.

The mobility of the molecules at the surface is one of

TABLE II. Adsorption, cluster, surface relaxation, and cluster-surface interaction energies in eV forselected clusters. E,l'/X is the cluster relaxation per molecule. Missing energies do not have corre-sponding free clusters with the same shape as at the surface. Entries marked with an asterisk corre-spond to the most stable cluster configurations at the surface calculated in this study for a given numberof molecules.

4

8

88+

111212*161624 g

242432

48+

Shape

linearcubiclinearlinearplanarcubicplanarplanar3D linear3D linear3D squareparallelepipedrotation 1

rotation 2parallelepipedparallelepipedparallelepiped

Figure

3(b)3(c)

4(b)

4(c)

6(a)6(b)6(c)

—1.00—1.22—1.58—1.81—1.00—1.96

—1.48

—1.55—2.39—2.05—2.14—1.70—1.60—2.19—2.81—3.63

Erelcl

0.080.070.090.100.070.15

0.12

0.220.240.190.220.310.210.220.380.58

0.080.040.030.030.020.03

0.02

0.020.020.010.010.010.010.010.010.01

relEsur

0.760.640.810.900.480.991.551.400.661 ~ 871.830.761.010.910.900.870.750.861 ~ 121.51

—1.85—1.92—2.48—2.81—1.56—3.11—4.89—4.21—2.26—5.67—5.76—2.53—3.63—3.14—3.27—2.88—2.56—3.27—4.31—5.72


the main factors which determine the mechanism of theiraggregation. Three mechanisms for diffusion of the LiC1and MgO molecules along the (100) MgO surface wereconsidered in Ref. 38. It was demonstrated that two ofthem, successive "90' reorientations" in the surface planeand "180 out-of-plane rotations" (see Fig. 2), have thelowest and comparable energy barriers. Our present cal-

(a)

FICx. 2. Schematic diagrams showing diftusion mechanisms(a) 90' in-plane reorientation for one NaC1 molecule, (b) 180'out-of-plane rotation for a single NaC1 molecule, and (c) rota-tion of cube around an edge for four NaCl molecules. Barrier-point configurations are shown by bold broken lines. The curvedarrows indicate the reorientation pathways. The small arrowsin (a) shown the lattice relaxation at the barrier point.

culations for the NaC1 molecule diffusion gave qualita-tively similar results. In particular, in-plane 90 reorien-tations of the NaCl molecule have the smallest adiabaticbarriers: reorientation "around" the Cl ion requires us toovercome the barrier of about 0.29 eV, whereas "around"the Na ion requires 0.12 eV. As was discussed in Ref. 38for the LiC1 and MgO molecules, there is a considerablemolecular displacement from an ideal rotationconfiguration and further surface distortion at the transi-tion state (see Fig. 2). The energy barriers for the 180out-of-plane rotations are equal to about 0.5 and 0.57 eVaround the Na and Cl ions, respectively. These energiesare much smaller than the molecule adsorption energy atthe surface. Therefore one can expect that, in the processof relaxation into the stable adsorption state and energydissipation, the molecule can make several jumps alongthe surface.

The second molecule can be attached to the first one,already adsorbed at the surface, in three different ways:antiparallel, perpendicular, and linear (along the samesurface axis). The most stable configuration is the anti-parallel rhombic configuration in which the ions makethe largest number of bonds. The attachment energy E,«can be defined as the difference E tt =E E

&E

where E is the total energy of the cluster of m (m =n orn —1) molecules and the surface, and E

„

is the energyof the free NaC1 molecule. It characterizes the energygain due to attachment of the next molecule from the freestate to the cluster already adsorbed at the surface. Inthe rhombic configuration it is about 2.53 eV. For com-parison, this is 0.75 eV larger than for the linearconfiguration in which the molecules are adsorbed inneighboring positions along the (100) surface axis. Thebinding energy per molecule between the rnolecules in theadsorbed rhombic dimer is 1.13 eV which is only 0.03 eVsmaller than in the free dimer. The rhombicconfiguration is very close to that of the free dimer and isslightly distorted perpendicular to the surface. In partic-ular, the Na-Cl-Na angle increases from 77.9 to 78.7,whereas the Na-Cl distance increases from 2.59 to 2.62

0A. For comparison, the experimentally determinedgeometric parameters of the free dimer are 78.6 and2.58 A, respectively. The surface distortion energy issignificantly smaller than that for the single molecule be-cause of the much smaller electrostatic interaction. Thelatter increases considerably after attachment of a thirdmolecule as the cluster reacquires a dipole moment.

The attachment energy for the third molecule in themost stable configuration antiparallel to the adjacentmolecule [this geometry is similar to that shown in Fig.3(a) if one ignores the fourth molecule attached on theleft side] is 2.18 eV. However, the energy of thisconfiguration is lower than the energy of the dimer andthe individual molecule separately adsorbed at the sur-face by about 1.18 eV. We note that reaction energyNaC1+(NaC1)z —+(NaC1)3 calculated for the free mole-cules is 1.82 eV (1.6 eV according to Ref. 22). Thedifference is caused predominantly by the large surfacedistortion. The linear-chain configuration of the three al-most antiparallel molecules is one of the two stableconfigurations which are established for the free NaC1 tri-

13 636 SHLUGER, ROHL, AND GAY 51

mer. The other configuration for a free cluster is a ring,which has a very close or even lower energy. For threemolecules it has a structure close to the linear chain. Thestrong interaction with the squared lattice template di-minishes the difference between the two configurationsand in fact they converge into one.

For the four different configurations of four moleculesshown in Fig. 3, the attachment energy is only useful forthe first two planar configurations. The first of them hasE,« = 1.84 eV, whereas the second configuration has

E,«=2. 13 eV. The rearrangement of the cluster fromthe first into the second configuration requires severalsuccessive reorientations of a molecule. However, due tothe high attachment energy, the barrier for rotation ofthe molecule, which limits the rearrangement process, isabout 1.0 eV. This makes the rearrangement highly un-likely once the adsorbed molecule(s) are in thermal equi-librium with the substrate. Part of the attachment ener-gy, though, could be used for this rearrangement duringthe equilibration process. The cluster in the second

(b)

(c)

FIG. 3. Optimized geometries for fourdifferent configurations of four NaCl moleculeson the (100) surface of MgO; (a) planar cluster,(b) linear chain of antiparallel molecules, (c)cube, and (d) ring. Large circles correspond tothe Cl ions and small circles to the Na+ ions.The approximate geometry for the (NaC1)3cluster can be visualized from (a) and (b) by theremoval of one molecule.

51 PROPERTIES OF SMALL CLUSTERS AT IONIC SURFACES: 13 637

configuration makes a bridgelike strained structure [seeFig. 3(b)]. The energy of this structure is only O. ll eVhigher than the energy of the most stable cubic structureshown in Fig. 3(c). The latter is strongly distorted withinterionic distances in the upper rhomb of 2.66 A and inthe lower rhomb of 2.72 A. The energy of the ring struc-ture in Fig. 3(d) is 0.59 eV higher than that of the cube.The relative stability of the free clusters and the ad-sorbed clusters in the same order [Fig. 3(c), 3(b), 3(d)], butthe energy differences are much larger for the free clus-ters. In particular, the energy of the cubic structure islower than that for the linear chain of antiparallel mole-cules by 0.83 eV (0.9 eV in Ref. 22). This clearly resultsfrom the interaction with the surface. Note that E,'„",an

E;„,for the cubic structure are much smaller than for thelinear (NaCl)& structure [Fig. 3(b)] and close to the values

(a)

for the (NaC1)z cluster (see Table II). The latter suggeststhat the molecular pair closest to the surface makes themain contribution to both energies.

These results already reAect several common featureswhich ersist for all the clusters studied in this paper.First, the barriers for diffusion of individual moleculesalong the surface are much smaller than the attachmentenergies o exis int 'sting clusters and intracluster interactionenergies. Further rearrangement of the cluster structureafter a molecule sticks to a particular cluster position inmany cases seems to be unlikely. This implies thatdiffusion-limited aggregation can be the main mechanismof cluster growth and that clusters may have branchedfractal features Therefore, in principle, for a largenumber of molecules, the number of possible initia yformed (meta)stable configurations can be very large.The relative abundance of these structures at real grow thconditions depends on the speed of molecular deposition,substrate temperature, and structure. Secondly, theconngurations s own inhown in Fig. 3 reveal three characteristicqualitative shapes of the most stable structural fragmentsobt

'ed in this study a linear chain of antiparallel-

aligned molecules which we will call further a "linearchain" [see Figs. 3(b) and 4(a)], a planar monolayer struc-ture [see Figs. 3(a) and 4(b)], and a three-dimensionaltwo-layered parallelepipedlike structure [see Figs. 3 c,4(c), 5, and 6]. We will call the latter type "paral-lelepiped. " Finally, the surface deformation produced byclusters extends wider than the projection of the clusteronto the surface. For example, for the linear (NaCI)z

J,

(

FICx. 4. Equilibrium cluster and surface geometries for (a)linear antiparallel chain of eight molecules, (b) planar array of11 molecules with kink, and (c) linear three-dimensional two-layered parallelepipedlike structure of 16 molecules.

FIG. 5. Two views of the equilibrium geometry of the(NaCl)36 cluster on the MgO surface. (a) shows the large clusterdistortion perpendicular to the surface. In (b) the areas wherethere is a correspondence between surface and cluster sites aremarked by broken lines.


FIG. 6. An overlay of the borders of diferent orientations ofthe (NaCl)24 cluster on the (100) surface of MgO obtained by ro-tation about a vertex of cluster (a) ~

a template which favors quadrangular structures overhexagonal. Apart from n =3,5, the most stable structuresof all other clusters adsorbed at the surface obtained inour calculations correspond to the most stable free-cluster structures. The strong interaction with the sur-face makes the energy of the linear (NaC1)z structureabout 0.1 eV lower than that of the (NaC1)4 cube with anattached NaC1 molecule. For the free clusters, the latterwas calculated to have the energy 0.66 eV lower than thatof the linear chain (0.76 eV in Ref. 22). The linear-chainconfiguration remains the most stable only for three andfive molecules. However, even for eight molecules thetotal-energy di6'erence between the three types of struc-tures is only about 0.3 eV with the parallelepipedtypestructure being the most stable. The planar structuresconsidered in this study included up to 12 molecules.Starting from six molecules this structure is already lessenergetically profitable than the two-layered paral-lelepiped structure. These structures may have kink(s)[see, for example, Figs. 7(a) and 4(b)]. The attachmentenergy of the next molecule to the kink is much largerthan that to the linear edge of the cluster. Therefore pla-nar clusters of an even number of molecules and regularshape are more stable than those of odd number. Howev-er, starting from six molecules, the parallelepiped struc-

cluster, the surface ions at the edges of Fig. 3(b) are dis-placed away froID the cluster by about 0.04—0.09 A.Further from the cluster, the displacements are close tozero. For other clusters, the surface deformation typical-ly extends about 1 —2 lattice constants wider than theprojection of the cluster onto the surface [see, for exam-ple, Figs. 4(a) and 4(b)].

B. Comparison with free clusters

The cluster-surface interaction is rejected in the shapeof the most stable adsorbed clusters compared to the freeclusters of the same number of molecules. The structuresof the free (NaC1)„clusters obtained in Refs. 19 and 22can be approximately described as parallelepipeds or po-lygons constructed from NaCl, (NaC1)2, and (NaC1)3 frag-ments. The most stable configuration of (NaC1)2 is arhomb. That of (NaC1)3 is a hexagonal ring. The energyof the latter is extremely close to that of the linear-chainconfiguration with C2„symmetry in which the three mol-ecules are arranged almost antiparallel. For an evennumber of molecules, the parallelepipedlike structuresare the most energetically stable starting at n =4. Thehexagon-based cylindrical structures in some cases havevery close energies to the parallelepipeds, especially whenn is divisible by 3. Many of the most stable structures foran odd number of molecules resemble the parallelepipedswith n —1 molecules with an attached molecule. Alter-natively they may be smaller parallelepipeds which in-corporate rhombs and hexagonal rings. ' '

As was pointed out previously, the MgO surface makes

FIG. 7. Three-dimensional electrostatic equipotential sur-faces produced by the clusters. (a) Planar cluster of 11 mole-cules with kink and (b) parallelepiped cluster of 36 molecules.The displayed surfaces correspond to 1.5 eV. In (b), for clarity,only the potential around the top layer of the cluster is drawn.


tures dominating the free-cluster structure also begin todominate the most stable structures of the adsorbed clus-ters. This results both from the accumulating misfit andthe intracluster interaction. In particular, the binding en-ergy per molecule in the free and adsorbed clusters in-creases with the cluster size whereas the cluster-surfaceinteraction per interface molecule decreases (see TableII). We discuss these issues in more detail in the next sec-tion.

Based on this tendency we considered further only themost compact three-dimensional two-layered paral-lelepiped clusters of 16, 24, 32, 36, and 48 NaC1 mole-cules. We should stress, however, that many of the linearand planar structures of adsorbed clusters, which do notexist in the free state and are stabilized due to the interac-tion with the surface, have energies which are only slight-ly different from those of the adsorbed parallelepipedstructures.

C. How does the interaction of the clusters with the surfacechange with the cluster size?

More detailed information regarding the cluster-surface interaction can be gained from Table II. It can beseen that the total cluster-surface interaction energy gen-erally increases with the cluster size. To compare theE;„,for the linear, planar, and parallelepiped structureswe divided this energy by the number of cluster mole-cules in direct contact with the surface. Although weshould note that due to the cluster bending [see Figs. 4(a)and 4(b) and the next section for discussion] this charac-teristic is approximate, we believe that it reAects the gen-eral tendency. As is clear from Table II, the E;„," de-creases as the interface area increases. The total surfacedistortion energy on average changes much less than thecluster-surface interaction energy despite the strikingdifference in cluster shape and size. Since the surface de-formation distributes over a larger area than the projec-tion of the cluster, it is difficult to define the interfacearea. However, it is clear that the surface distortion en-ergy per interface area also decreases rapidly as the clus-ter size increases. These features reAect the transitionfrom the strong interaction for a single molecule to an in-terfacial interaction between two crystal surfaces withlarge misfit.

The electrostatic interaction between a cluster and thesurface makes one of the major contributions to the in-teraction energy. The interference of the Coulomb con-tributions from the individual ions decreases the electro-static potential at a particular distance from a cluster asits size increases. However, the potential near low-coordinated ions at edges and corners decays more slowlythan that below the middle of the cluster where the ionsare coordinated more strongly. This point is illustratedin Fig. 7 where the surfaces of equivalent Coulomb poten-tial around the clusters of 11 and 36 molecules arepresented. It can be seen that the potential in the internal

part of the regular 36-molecule cluster clearly resemblesthe periodic exponentially decaying potential characteris-tic for the perfect infinite surface. Its numerical value ata distance of 2.8 A from the cluster is only about 10%higher than that for the perfect lattice of the samecharges. The strong interactions of the corner and edgeions with the surface can bend the cluster shape, and thestrong electrostatic potential produced by the kink mustimpose anisotropy for molecules migrating towards thecluster. Finally, we should note that the van der Waalscontribution to the adsorption energy is significant evenfor small clusters. For clusters larger than those con-sidered in this paper it can become bigger than the elec-trostatic contribution. However, the van der Waals in-teraction begins to depend on the ion position withrespect to the surface only when the ion-surface distanceis less than approximately the surface lattice constant.Therefore it does not affect strongly the cluster geometryat the surface. In particular, for the three different orien-tations of the (NaC1)24 cluster shown in Fig. 6 the van derWaals contribution to the cluster-surface interaction en-ergy changes by only 0.3 eV, whereas the electrostaticcontribution changes by 1.45 eV (see the next section).

The energetic characteristics provide, however, onlyaveraged information concerning the cluster-surface in-teraction. To get a deeper insight into the individualfeatures of particular clusters we will now turn to theirgeometric structure.

D. How do strained clusters accommodate the strain?

This question is closely related to another question: atwhich cluster size do the adsorbed molecules lose theiridentity? If we look at the perfect surface as a probe ofcluster structure we can restate this question as follows:at which cluster size does the molecule-surface interac-tion characteristic for adsorption transfer into the solid-solid interaction characteristic for an interface? We ad-dress these questions by analyzing the character of thecluster deformation.

There is a striking difference between the structures ofthe linear, planar, and three-dimensional clusters of thesame number of molecules in direct contact with the sur-face. In the first two types of cluster, the corner ions tendto maintain their correspondence with the surface ions ofopposite charge. Because of the strong misfit with thesurface along the length of linear clusters, they releasethe strain by forming rooAike structures in which somemolecules are repelled up out from the plane as shown forthe eight-molecule cluster in Fig. 4(a). For a large num-ber of molecules, the planar structures stabilized by bend-ing perpendicular to the surface similarly to the linearstructures, as shown in Fig. 4(b) for 11 molecules. Al-though we only considered linear and planar clusters upto 12 molecules, we believe that much larger clusters canbe stabilized in this way as well.

The internal structure of the three-dimensional clustersis more rigid. The upper layer in these clusters weakly

13 640 SHLUGER, ROHL, AND GAY

interacts with the surface and bends in order to accom-modate the strain [see, for example, Fig. 5(a)]. This isenough to minimize the strain for the clusters of six andeight molecules. However, the larger linear clusters bendalong the surface as shown in Fig. 4(c) for 16 molecules.Although they already interact with the surface like crys-tal blocks they still bear strong molecular-clusterfeatures. To clarify this and other points we will considerthe cluster (NaC1)&6 in more detail.

The ratio between the bulk lattice constant of NaC1and MgO is equal to 1.337 which approximately corre-sponds to 6 lattice units of NaC1 per 8 lattice units ofMgO along the (100) surface axis. Therefore formallyone could expect a good geometric correspondence be-tween the corners of the squared (NaC1)~6 cluster and thesurface lattice sites as the side of the cluster is 6 interion-ic distances long. However, as one can see in Fig. 5(b),the correspondence between the cluster corners and thesurface sites is in fact approximately 6:7. This illustratesthe point as to how fast the geometric parameters of thecluster approach the bulk crystal values as the clustersize increases. The distances between the nearest ions inthe cluster interior are approximately equivalent andequal about 2.71 A. Those between the inner ions andthose at the cluster edges are equal to 2.69 A. The dis-tances between the nearest ions at the corners on theupper plane are 2.67 A and these on the lower plane are2.62 A. These parameters are still much smaller than thecalculated bulk lattice constant, which is equal to 2.815A. They also reflect the considerable cluster distortion atthe corners which is notable in Fig. 5. Due to the strongmisfit there is relatively good correspondence between theopposite-sign cluster and surface ions only at the twocluster corners shown by the broken lines. The two othercorners are located approximately above the sites withthe same ion charge. Because of the electrostatic repul-sion at these corners the cluster is considerably twisted[see Fig. 5(a)]. In particular, the difference in height fromthe surface for the two types of corners is equal to 0.65 A.

The strong distortion of the (NaC1)i6 cluster due to theelectrostatic repulsion at the corners, the peculiar distor-tion of the planar clusters [see Figs. 4(a) and 4(b)], andthe difference in the electrostatic potential between thecluster edges, corners, and interior (see Fig. 7) seem tosuggest that the interaction of the low-coordinated clus-ter sites with the surface can strongly influence the clus-ter geometry and orientation at the surface. In order tounderstand this point better and to study other possiblecluster configurations at the surface we considereddifferent orientations of the less symmetric (NaC1)z4 clus-ter shown in Fig. 6. The first configuration of this clusterhas its edges roughly parallel to the (100) surface axisand a quite poor correspondence between the cluster andthe surface ions of the opposite sign. The cluster bendsstrongly both parallel and perpendicular to the surfaceplane in order to improve the correspondence and toreduce the electrostatic repulsion. In particular, the dis-tance from the surface for the ions with a "good"correspondence with the surface ions is about 2.89 A,whereas the ions in the opposite cluster corner are 3.69 Afrom the surface. To find a better correspondence be-

tween the corner ions of the cluster with the surface ionswe rotated the cluster with respect to the surface andminimized the total energy. Two stable configurationswhich satisfy both criteria are shown in Figs. 6(b) and6(c). The first of these configurations has the total energy0.44 eV higher, and the second configuration 0.55 eVhigher than the energy of the initial configuration. Asone can see from Table II, both of the rotatedconfigurations have lower surface distortion energies.However, the cluster-surface interaction energies are inboth cases much smaller. This results from the fact thatafter rotation there is no area of a good cluster-surfaceion correspondence except for the few scattered ions andthe cluster corners (see Fig. 6). As a result the electro-static contribution to the interaction energy decreasesfrom —3.44 eV for the initial configuration to —1.99 eVfor the configuration shown in Fig. 6(c).

Turning back to Table II and Figs. 3—7 it becomesmore apparent that qualitatively speaking the bigger thearea of a good correspondence between the cluster andsurface ions of the opposite sign the larger the cluster-surface interaction energy. Similar electrostatic require-ments for anion-cation near-neighbor pairs at the inter-face were discussed recently in Ref. 63. We should notethat because of the misfit only an approximate correspon-dence is possible for large areas. For instance, the bestsuch correspondence exists for the (NaC1), 6 cluster whichcan be viewed as four cubes arranged in a squareconfiguration. However, as one can see in Table II, thecluster-surface interaction energy for this cluster is lessthan two times larger than that for the cube. The same istrue for the dimer and the linear configuration of the te-tr amer.

These results demonstrate also that the bulk-crystalgeometric parameters can be achieved only at muchlarger cluster sizes. It seems tempting to interpolate ourdata in order to find approximately this cluster size.However, closer analysis shows that the cluster sizes con-sidered in this work are still too small for reliable con-clusions. First, the interionic distance changes only from2.66 to 2.71 A as the cluster size increases from 4 to 48molecules. Secondly, the number of interior ions in theclusters smaller than 36 molecules is small and theirshape is strongly afFected by the size and their interactionwith the surface.

E. Cluster diffusion

Another way to probe the cluster interaction with thesurface is to calculate the adiabatic potential-energy sur-face for cluster displacements along the surface. In par-ticular, one can expect that the adiabatic potential fortheir displacement in certain directions can be very flatsince the electrostatic potential of the large clusters is in-coherent with that of the surface. For comparison wecalculated the adiabatic potentials for the displacementsof the cubic (NaCl)4 cluster shown in Fig. 3(c) and thosefor the parallelepiped (NaC1)~6 cluster (see Fig. 5) alongthe (100) and (110) surface axes. In these calculationsone of the cluster ions was fixed at a certain trajectorypoint whereas all other cluster and surface ions were al-

51 PROPERTIES OF SMALL CLUSTERS AT IONIC SURFACES: 13 641

lowed to find new positions in which the forces at theseions were zero. The barrier point was found at zeroforces exerted at all cluster and surface ions including thefixed ion.

The bottom ions of the (NaC1)4 cubic cluster in its moststable configuration at the surface strongly interact withthe corresponding surface ions with the opposite sign.This configuration has a symmetry axis perpendicular tothe surface and passing through the middle of the Mg202surface unit and the cluster. Therefore all Mg202 surfaceunits in the (100) direction are equivalent with respectto the (NaC1)4 cluster translations. The barrier pointoccurs when the center of the cluster lies above the cationor anion site at 0.71a from the initial cluster position,where a is the interionic surface distance. The value ofthe calculated barrier energy for the displacement of the(NaC1)4 cluster between the equivalent surface positionsis 0.39 eV.

The length of the vector corresponding to the transla-tion of the cluster between equivalent positions in the(100) direction is equal to 2a. However, in fact thedirect translation does not take place but the clustertends to rotate over a corner or edge. The latter processis schematically depicted in Fig. 2(c). It provides theshortest path for the cluster displacement along the(100) axis and has a barrier energy equal to 0.36 eV.The rotation over the corner in fact leads to the clusterdisplacement along the (110) axis. The calculated bar-rier for this rotation is about 0.73 eV. It is interesting tonote that the barrier energies for the translations of the(NaCI)~ cluster along the surface are not much higherthan those for the individual molecule.

Due to the symmetry of the adsorption state of the(NaC1)36 cluster at the surface [see Fig. 5(b)] all clustertranslations on the intersite distance along the ( 110) axisare equivalent. The translational displacement of the(NaC1)36 cluster along the (110) direction requires over-coming a barrier of 0.37 eV. The cluster displacement onthe surface lattice constant along the (100) axis alsobrings it into the equivalent position. However, thecorrespondence of the cluster ions with those at the sur-face changes between the different cluster corners.Therefore the displacement of the cluster is accompaniedby its twisting. The calculated barrier for the clustertranslation along the ( 100) direction is only 0.14 eV.

IV. DISCUSSION

In this section we will consider possible implications ofthe results of this study with respect to the different prop-erties of insulator-insulator adsorption and interfaces.First, we will compare the structural and energetic prop-erties of the clusters studied in this paper with the resultsof other cluster studies and MBE of AH's. Then we will

briefly consider their electronic structure and opticalcharacteristics, and the possibility of their observationusing AFM.

A. Cluster structural properties and 61m growth

Epitaxial growth of wide-gap insulating materials suchas alkali halides and simple ionic oxides provides a very

interesting example of the MBE of nonmonatomicspecies. In particular, it has been demonstrated experi-mentally that thin films and excellent single crystals ofseveral alkali halides and MgO can be grown by MBEeven at temperatures as low as about 100 K. ' It hasbeen suggested that this unusually low-temperaturegrowth may be due to very high mobility of surfacespecies. An activation energy of the order of 0.1 eV hasbeen estimated for the mobility of KC1 molecules on asurface terrace of KC1. The calculated adiabatic bar-riers for diffusion of LiC1 and MgO single moleculesand NaC1 molecules on the surface of MgO, which areabout 0.3 eV, confirm the generally high mobility ofmolecular species on surface terraces of cubic ionic ma-terials. Large adsorption energies of these molecules andbig surface deformation energies obtained in this study(see Table II) suggest that the high mobility of the pri-mary adsorbed species may be partly due to their "hot"jumps during energy dissipation. This point requiresmore careful study using molecular-dynamics techniqueswhich are currently in progress in this group.

Heteroepitaxy of alkali halides on the surface of alkalihalides has been studied recently using high-resolutionHe-atom scattering. In particular, oscillations in theintensity of the specular beam due to layer-by-layergrowth were observed for the first several layers of KBrgrowth on the (100) NaC1 surface. However, thediffraction pattern after the deposition of two layersshows peaks with both the NaC1 and superstructure cor-rugation and very small specular intensity. By six layersthe specular is much larger, indicating a considerablybetter surface. The surface lattice spacings for NaC1 andKBr have a large mismatch of about 17'. Therefore toexplain the data for the first two layers it was suggestedthat they should contain many defects or very small is-lands. In particular, it has been proposed that the KBrmolecules initially deposit oriented normal to the surface,forming small bilayer patches that buckle and give rise tothe superstructure, but in subsequent layers they alignparallel to the surface. The results of the present studyqualitatively support this model. They suggest that atlarge mismatch and strong intermolecule interaction ini-tial formation of the bilayer structures is more plausiblethan that of the monolayer. The cluster bending bothalong and perpendicular to the surface shown in this pa-per can give rise to formation of defects and dislocations.

We should note that the clusters considered in thisstudy are most probably much smaller than the "small is-lands'* proposed in Ref. 28. However, another interestingquestion is how are these islands and the next layers ofadsorbed material oriented with respect to the substratelattice axis, and how does the structure of initial frag-ments determine the structure of the thick film? Thisquestion is related to a near-coincident-site-lattice modeldeveloped for noncommensurate heteroepitaxial inter-faces. According to this model various two-dimensionalcoincidence-site lattices can be produced by rotating onelattice with respect to the other about the axis normal tothe interface until three nonlinear lattice sites of the twomaterials are in coincidence. This procedure gives a setof initial formal geometries of the interface which should


be further optimized taking into account the interionicinteractions and using energy-minimization techniques.This procedure seems to be appropriate for the calcula-tion of the interaction between two macroscopic surfacesand grain boundaries and was recently applied to thestudy of the BaO/MgO interface. It concludes that forthe best site coincidence at large misfits two lattices canbe rotated with respect to each other.

Unfortunately, both the experimental data on theKBr/NaC1 interface and the theoretical results of thepresent work do not give a conclusive answer to the ques-tion posed in the previous paragraph. From the He-atom-scattering (HAS) data one can construct the surfacephonon dispersion curves. The experimental data andthe results of analysis of phonon spectra performed inRefs. 28 and 29 seem to suggest that for several layers ofKBr the main surface axes of both lattices are parallel.However, the experimental details given in Refs. 28 and29 are not enough to reach a final conclusion. The resultsof the present calculations demonstrate that at moleculardeposition very small clusters are already very stable atthe surface and can act as nucleation sites. The latter willgrow further by the attachment of new molecules. At ini-tial stages of this growth (at least until 16 molecules inthis study) the clusters tend to align with there edgesparallel to the surface axis. However, there is a clear ten-dency for the larger clusters to bend along the surface.Therefore the shapes of the real clusters and islands canbe complicated. This point requires statistical simulationof the cluster growth which is out of the scope of this pa-per.

However, we should point out that these resultsdemonstrate clearly that the formal "misfit" between thelattice constants of the substrate and the deposited ma-terial strongly depends on the cluster size. According toour results, rotation of the large clusters to satisfy newcoincidence-site lattice conditions appropriate for theirsize seems unlikely. It is also interesting to note in thisrespect that the cluster distortion energies obtained inour calculations are much smaller than the correspondingsurface distortion energies (see Table II). This is despitethe fact that some of the clusters are strongly deformedcompared to their geometry in the free state. Apart fromthe reasons discussed in the previous section, this iscaused by the fact that because of its continuity the lat-tice deformation spreads over a bigger area than that ofthe cluster. Therefore gaps between clusters can decreasethe overall strain energy and their existence is energeti-cally profitable. Thus island formation, their buckling,and formation of dislocations seems to be a realistic mod-el for the first MBE layers, as discussed in Ref. 28. How-ever, HAS cannot give any definite answers regarding thestructure of small islands. Some additional informationin this respect could be gained from spectroscopic andAFM measurements.

B. Cluster electronic structure

In this work we are particularly interested in the possi-bility of the identification of the clusters at surfaces usingoptical absorption and other spectroscopic techniques.This is only possible if there is a considerable dependence

of the cluster optical properties on their size. This depen-dence has been demonstrated experimentally for wide-gap semiconductors, such as CdS. The transition fromthe electronic structure of single adsorbed NaC1 mole-cules to the well-known bulk structure has been studiedrecently for thin adlayers deposited on W(110) using elec-tron energy-loss spectroscopy (EELS), MIES, and otherspectroscopic techniques. However, similar studies ondielectric substrates such as MgO are much more dificultbecause of surface charging. The question which arises iswhere the holes created in the Auger process will be lo-calized.

To address these questions we calculated the electronicstructure and optical-absorption energies for the clustersof n =1,2, 4, 32, 48 NaC1 molecules, the infinite bulk lat-tice of NaC1, and the (100) surface of MgO. The clusterone-electron states form a discrete energy spectrum. Wecalculated the one-electron transitions been these statesusing a configuration-interaction technique similar tothat discussed in Ref. 58. They form a spectrum whichincludes transitions with different transition matrix ele-ments. For our present purpose we are primarily in-terested in the "strong" transitions, i.e., in those with thelargest transition matrix elements, which fall in the opti-cal band gap of the MgO (100) surface. The latter wascalculated to be 7.9 eV. The energies of the lowest"strong" transitions for the free NaC1 clusters of n mole-cules with n = 1, 2, 4, 32, and 48 are 5.5, 6.8, 7.6, 7.2, and7.7 eV. The calculated optical band gap for bulk NaC1 is8.7 eV (8.6 eV is the experimental value ). For the 32-and 48-molecule clusters the lowest transitions corre-spond to the electronic states delocalized by the low-coordinated corner ions. The transitions correspondingto the "internal" cluster ions have energies about 8.1 and8.2 eV, respectively, i.e., much closer to the bulk value.These data demonstrate that the optical "band gap" in-creases as the cluster size increases and is distinctlydifferent from the bulk NaC1 value for the small clusters.

For these clusters adsorbed at the surface, there is a"tail" of cross transitions corresponding to the electrontransfer from the surface oxygen states perturbed by thecluster to the cluster sodium states perturbed by the sur-face. They have energies in the range of 6.1 —6.9 eV, andsmaller transition matrix elements than the intraclustertransitions. The cluster perturbation is rejected in theredshift of the strongest transition. In particular, in thefour-molecule cube the energy of this transition decreasesfrom 7.6 to 7.1 eV. The low-energy cross transitions per-sist for all clusters considered. We should note, however,that they appear because the bottom of the conductionband in the bulk of NaC1 in our calculations lies lowerthan that of the bulk and surface of MgO. There arepresently not enough theoretical or experimental data toconclude whether this is correct. Based on these dataand the results of previous calculations ' we can con-clude that, although the optical-absorption energies ofthe small clusters are smaller than the band gap of bulkMgO, they are likely to mix with the surface states of thereal MgO surface containing steps and kinks and with theexcitonic bands which have much lower energies. On theother hand, they could be observed in the case of the oth-


er substrates with higher surface excitation energies [9.2eV for NaF and 10.2 eV for LiF (Ref. 68)]. The fact thatchlorine valence states lie much lower than the top of theoxygen valence band implies that the holes produced inMIES and similar experimental techniques based onAuger processes will localize in the substrate. However,this point requires more careful calculation of the self-trapping energy.

C. Possibility of the AFM observation

It is tempting to suggest application of the AFM tech-nique to the study of small clusters at insulating surfaces.This could be the only direct way to study the geometryof individual clusters and the very initial stages of filmgrowth. The crucial issue which determines the very pos-sibility of the observation of clusters at surfaces is thetip-cluster interaction. A small cluster can be adsorbedon the tip or displaced by it along the surface. One can,at least in principle, try to avoid the first problem bychoosing the tip material. However, the second problemseems to be more fundamental. This can be understoodfrom the experimental fact that the friction forces exertedon the tip when it climbs the surface step are consider-ably larger than these when it scans down the step.

We have calculated the maximum forces necessary todisplace the 4- and 36-molecule clusters between theirstable configurations along the (100) and (110) surfaceaxes. They are equal to. 0.9 and 0.3 nN along the (100)axis, and 0.6 and 0.7 nN along the (110) axes, respec-tively, for the (NaC1)~ cluster and the (NaC1)36 cluster.The experimentally measured friction force exerted onthe AFM cantilever scanning up a two-layer step on the(100) NaC1 surface is about 9 nN, i.e., more than an or-

der of magnitude bigger. Although this is not exactly thesame system we believe that this comparison is meaning-ful. One can see that the tip can easily move even rela-tively large clusters along the surface during scanning.This implies that one can expect to observe clusters main-ly at surface steps and kinks.

Finally, we should note that many of the qualitative re-sults obtained in this study with respect to the structure,properties, and possibilities of experimental observationof clusters at ionic surfaces should be valid for a widerange of alkali halides and cubic ionic oxides where theintramolecular separation exceeds the substrate latticeparameter. We believe that further development of exper-imental techniques will allow the study of trapping ofelectrons and holes in these and similar adsorbed clusters,chemical reactions of adsorbed clusters with molecules,and photoinduced processes which are already in pro-gress for free clusters and surfaces. ' ' '

ACKNOWLEDGMENTS

A.L.S. acknowledges SERC, Latvian Scientific Council(Grant No. 93.270), and Deutsche Forschungsgemein-schaft (Contract No. 436 LET 113) for financial support.A.L.R. was supported by SERC and ICI. D.H.G. wouldlike to thank Biosym Technologies for their financial sup-port. The authors are indebted to R. W. Grimes and J.Binks for deriving the pair potentials used in this study.We are grateful to C. R. A. Catlow, D. C. Sayle, V.Kempter, L. Howald, R. Luethi, and A. Baratoff for use-ful discussions. We want to thank Biosym Technologies,San Diego and Cherwell Scientific, Oxford for supplyingthe computer codes used for the graphical representa-tions in this study.

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clusters ( n =1–48) at the (100) MgO surface