Nuclear Instruments and Methods in Physics Research B62 (1991) 181-190North-Holland

Energy distributions of constituent atoms of cluster impactson solid surfaceYasunori YamamuraOkayama University of Science, Ridai-cho, Okayama 700, Japan

Received 7 June 1991 and in revised form 16 August 1991

Using the time-evolution Monte Carlo simulation code DYACAT, the energy distributions of constituent atoms due to bigcluster impacts on amorphous targets have been investigated, where the (Ag)� and (All, cluster (n being 10 to 500) with energies afew 100 eV/atom to keV/atom are bombarded on amorphous carbon and gold targets, respectively . It is found that the energydistribution of constituent atoms is strongly affected by the mass ratio M2/M1 (M I and M Z being the atomic masses of theconstituent atom and the target atom, respectively), the size of the cluster, and the cluster energy .

In the case of the I keV/atom (Ag)st () cluster impacts on C (M2/M1 < 1) the shape of the energy distribution of constituentatoms is trapezoidal, while in the case of the At cluster impacts on Au (M, /M, > 1) the high-energy tail of the energy distributionof AI atoms due to the big cluster impact (n > 100) can be well described in terms of the Maxwell-Boltzmann function, and itstemperature is linearly proportional to the energy. In the case of 1 keV/atom (AI)S,s, cluster impact on An, the quasi-equilibriumstate continues for more than 0.6x 10 -t3 s, but the temperature of the cluster impact region decreases as time passes. The presentsimulation supports the recent Echenique, Manson and Ritchie's Ansatz in their theory of cluster impact fusion .

1. Introduction

Cluster bombardment onto solid surfaces is one ofthe most interesting up-to-date subjects in the interac-tion of charged particles with solids [1-8] . Motivationsfor studies come from the needs of surface analysis,controlled thin-film growth by ionized cluster beamdeposition, and biomolecular mass spectrometry to sur-face modification of materials that are of interest inmicroengineering and space research . There are a lotof interesting and unsolved problems in the clusterimpact phenomenon, i .e ., the range, the energy loss,the energy spectrum of constituent atoms and recoils insolids, radiation damage, sputtering, deposition and soon.

About ten years ago, Yamamura [9] developed thetime-evolution Monte Carlo simulation code based onthe binary collision approximation in order to investi-gate the nonlinear sputtering due to high-energy heavyions . The concept of the time-evolution Monte Carlosimulation has been successfully applied to relativisticnuclear collisions in order to estimate the forwardsuppression of low-energy protons in Ne + U collisions[10] and the pion multiplicity in Àr + KCI central colli-sion [11]. Recently, the time-evolution has been incor-porated in the new version of the MARLOWE code[11].

0168-583X/91/$03.50 © 1991 - Elsevier Science Publishers B .V. All rights reserved

Nuclear Instruments8c Methods

in Physics ResearchSection 6

Three years ago, using the time-evolution MonteCarlo code DYACAT, Yamamura [8] investigated thesputtering by 100 eV/atom (Ar) � cluster ions (n beingmore than 100) bombarding a carbon target, and foundvery interesting effects in the cluster-impact phenom-ena, i.e ., the frontrunner effect and the accelerationeffect, which are not observed in the case ofmonoatomic ion bombardment . The frontrunner effectis that the frontrunners of the lower parts of thecluster clear away the target atoms for constituentatoms coming afterward, which results in a larger pro-jected range than that of the monoatomic ion bom-bardment. When the cluster ion are bombarded andtrapped in the solid, there are a lot of collisions be-tween constituent atoms in the cluster. Due to this newtype of collisions some of constituent atoms have largerenergies than the incident energy. This effect is calledhere as the acceleration effect.

Recently these two effects have been verified byShulga and Sigmund [13] using the molecular dynamicstechnique, with which they have investigated the pene-tration of a small (Au) t ; cluster with energies 0.1 and 1keV/atom through a thin silicon target, in which theycall the frontrunner effect the clear-the-way effect.Using the same computer technique, Pan and Sigmund[14] have investigated the interaction of small carbonclusters with thin gold targets, and found that recoiling

182

target atoms achieved kinetic energies up to more thantwice the maximum recoil energy in a single elasticcollision .

When the energy per atom of the ionized clusterbeam is sufficiently low, various thin-film materialshave been prepared by ionized cluster beam deposi-tion. Yamada and coworkers [15,46] have found thatthick AI films can be cpitaxially grown by this tech-nique on SO I I) at room temperaurc. Recently, Yama-mura and his coworker [17] have investigated this clus-ter-beam deposition mechanism of (AI) � onto Si(I11)by applying the time-evolution binary collision latticesimulation code DYACOCT which is the dynamicalsimulation code of the binary collision lattice simula-tion code ACOCT [18].

Recently, Beuhler, Friedlander, and Friedman(BFF) showed that deuteron-deuteron fusion occurwhen singly charged clusters of 25 to 1300 D,Omolecules, accelerated to 200 to 325 keV, impinge onTO targets [IS] . These reports arc very interesting,because the tpanslational energy of deuterons in theteam is too low for any fusion . Similar yields arcobserved when (C,D,) �, TO and ZrD, .,S targets arcbombarded with D_O cluster ions [20]. Thus, cluster-impact fusion provides a new field for theories ofinteraction of atomic and molecular clusters with solids.

Echenique, Manson and Ritchie (EMR) have pro-posed an explanation for cluster-impact fusion basedon energy fluctuations of the deuterium atoms in acluster [21]. Under the assumption that the tail of theenergy distribution of the constituent D atoms is expo-nentially decreasing in energy, their choice of parame-ters gives reasonable agreement with the experimentalyields for D,O clusters on TiD. Recently Crawford [22]have developed the theory for dd fusion resulting fromthe impact of accelerated clusters of D,O on deuter-ated solid targets, where sequences of collisions be-tween D atoms and heavy atoms of the beam andtarget are assumed as a mechanism for quickly increas-ing the relative velocities of pairs of deuterons . Thecalculated fusion yields are many orders of magnitudesmaller than the results reported by BFF.

Some molecular dynamics calculations have beenperformed to investigate the possibilities of clusterimpact fusion . Shapiro and Tombrcllo [23] have investi-gated energy, number density, and confinement-timeproperties of 1 keV/atom AI,, and A1 6 , cluster im-pacts on Al and Au targets. They found that transientcompression of material in and near the point of im-pact is significant . But the compressed material re-mains confined for an extremely short period of time.It was found that the confinement parameter (nr) isabout 3 x 10 14 m - ' s for Al t,.3 -Au case . This is abouteight orders of magnitude less than that needed for theinitiation of d-d fusion by self-heating of compressedmaterial .

Y. Yarrrarnrrra / Clustrr 6nparts orr solirl srrrjaees

Molecular dynamics is a powerful method to investi-gate clusicr impact phenomena, and it gives preciseinformations about the interactions of the cluster beamwith soliJ< if the proper interatomic potential is used .But the Molecular dynamics requires a lot of comput-ing time . Then, at present, the biggest size of thecluster is less than 100 [13,14,23]. From the viewpointof the ac cleration effect of big cluster impacts, theimpact region should have much larger dimensionsthan the mean free path between the collision. Thismeans that the direct extension of the knowledges ofthe small cluster impact phenomena to the large clus-ter impact phenomenon is dangerous.

In order to treat the collision between low energyatoms more reasonably, in this paper, we refine theprevious time-evolution Monte Carlo simulation codeDYACAT [8,9], and take into account the many bodyeffect of a very low-energy particle penetrating a solid.The main concern of this paper is the energy spectrumof the constituent atoms due to big cluster impacts ona solid surface ; this is closely related to the under-standing of cluster-impact fusion .

2. Model off the DYACAT program

The D"ACAT program was developed for the dy-namical simulation of atomic collisions in amorphoustargets within the framework of the binary collisionapproximation . It is the dynamical mode of the ACATcode [241, and the main feature of the DYACAT codeis the same as that of the ACAT code . Unlike theusual Monte Carlo simulation code such as theTRIM.SP code [25], the ACAT program does not usethe concept of the mean-free-path of a projectile. Inthe ACAT and DYACAT programs an amorphoustarget is simulated employing the so-called cell model,where a target atom is assumed to be randomly dis-tributed in a simple cubic cell with an average latticeconstant R � = (1IN ) 1 i' . The treatments of elastic scat-tering and inelastic energy loss are very similar to thoseof ACAT -ode .

The G "ACAT program can treat three types ofcollisions: .) the collision between two moving parti-cles, 2) the collision between a moving particle and atarget ator; at rest, and 3) the collision between amoving par'.iclc and an interstitial . Here, the intersti-tial means the moderated atom whose energy is lessthan the cutoff energy . These three collisions will besimply denoted by MM, MT, and MI collisions, respec-tively, in the ensuing discussions . When a projectilemeets with a vacancy which was produced in the earlierstage, it goes straight forward without any collision .Consider an encounter between two moving particlesof masses M, and M_ The kinematics of MT and MIcollisions are special cases of a two moving particle

Fig. 1 . Space diagram of a two moving particles encounter.The relative motion of each particles takes place in the orbitalplane . The dependence on the law of interaction is completely

determined by the scattering angle in the orbital plane.

encounter [261. As a result of the collision, the veloci-ties of both particles change in direction and in magni-tude . We denote their velocity before the encounter byv, and v2 , respectively, and after the encounter by v,and v2 .

The initial velocity vectors of the two particles be-fore the collision define the plane which is called thefundamental plane, and the relative motion of bothparticles takes place in the plane which is called theorbital plane [26] . As is shown in fig. 1, the dependenceof the law of interaction is completely determined bythe scattering angle in the orbital plane, where therelative velocity V before the collision is given asV=V, -v 2 . The equation of motion in the orbitalplane can be manipulated in the usual manner to yieldthe scatting angle in the center-of-mass system

0='tr-2pf adr[r 2 g(rij _ 'r�

and the time integral

Orbital plane

a

_, /2rn-p2-~ ~(g(r)}_

- (I -P 2Ir

2)

dr,ro

,

(2)

where

g(r) = [I -p 2/r2- V(r)/ErJ 1/2,

Y. Yamamura / Cluster impacts on solid surfaces

p is the impact parameter, Er is the relative kineticenergy, r is the interatomic separation, V(r) is thepotential of interatomic force and r� is the apsis of thecollision. The time integral -r is the very importantparameter which determines the trajectory and thetime spent during the binary collision.

183

If the scattering angle in the center-of-mass systemis known, it is easy to obtain v, and v2, according tothe elementary theory of the binary collision of twomoving atoms which are given as

M,v, +M 2V2

M2v,

M, +M2

+ M, + M,V '(4)

M,V, + M,v 2

M,v;=

--M, +M,

-Mt +M2 V '

where V' is the relative velocity after the collisionwhich is represented in term of the scattering angle inthe center-of-mass system 0 and the absolute value ofthe relative velocity V as

V'=1 V I (e. sin 0+ep cos 0) .

(6)

The unit vectors e p = V/ I V I and eA are the directionof a projectile M, in the orbital plane and the direc-tion perpendicular to e p in the orbital pla,, , ~, respec-tively (see fig. 1). If the electronic energy loss at onecollision event is AE, the relative velocity of eq . (6)must be replaced by I V I

1 - DEIEr .The collision partner of a slowing down particle is

selected among many candidates of moving atoms,interstitial atoms and lattice atoms . At each collisionevent, we calculate the distances in time between amoving atom and many candidates, and pick up a realcollision partner which has the minimum distance intime . At the same time we check the applicability ofthe binary collision approximation . At each collisionwe calculate the collision diameter which is defined asthe apsidal distance at zero impact parameter . If thereare several atoms in the collision diameter, many bodyencounters are numerically solved by the moleculardynamics technique, in order to overcome the difficul-ties of the binary collision approximation for a low-en-ergy projectile, i .e .,

d'rpMp dt2

d2 r,M,z =d,2

where 0(r,) is the lattice potential and is assumed tobe given as

O(rJ = EB(exp [ - a1r, - ri � 1) -1 } 2,

and r,� is the original position of the ith atom, and E uis the bulk binding energy . In eqs . (7) and (8), m is thenumber of particles in the collision diameter of theprojectile, rp is the position vector of the projectile andr, is that of the ith particle, and r,p= I rp - r; I is thedistance between the projectile and the ith particle.Eqs . (7) and (8) are numerically solved by the Euler-Cauchy method .

a m

arp { F V(",)t= J 1

(7)

a(V(r,p)+~(r) }, (8)

ar,

184

The present DYACAT program was applied to the(Ag)� cluster bombardment onto an amorphous carbontarget, and the (AI)� onto an amorphous gold target,where u is the number of constituent atoms in acluster, and the Molii:re potential [27] is used as theinteratomic potential . The cluster shape is assumed tobe spherical, and the average lattice constant of thecluster is set to be equal to that of the bulk medium .The binding energy of the spherical cluster is neglectedin the present calculation., because the cluster energyemployed here is large enough. The electronic energyloss is estimated by using the Oen-Robinson localmodel [28). similarly to the ACAT code . In the presentcalculation the bulk binding energy E t; in eq . (9) is afree parameter which is the order of the vacancy for-mation energy, and the incident particle and the cner-gótic recoiling atom arc followed throughout theirslowing-down process until their energies fall belowpredetermined cutoff energies E, and E, where E,,and E._ arc those for the constituent atom and thetarget atom, respectively. The displacement thresholdenergy is set to be zero during the slowing-down pe-riod, but if T- En < E,=, the target atom is not addedto the cascade, where T is the kinetic energy trans-ferred to a target atom after the collision. In thepresent calculations, E t; and a in ß(r) are set to 1 .0eV and 3/R,u, respectively.

3. Results and discussions

In order to investigate the main mechanism of thecluster impact phenomena, we have performed thesimulation under two different conditions which arcdenoted by Model A and Model B . The Model Acalculation is the linear simulation and it gives thesame result as the monoatomic ion which is slowingdown in the unirradiated target, and the term "linear"implies that every constituent atom behaves indepen-dently and it is slowed down only by the collision witha target atom at rest . On the other hand, the Model Bcalculation is the usual DYACAT simulation. In theDYACAT program the time-dependent materialchanges are stored and used for tbc next step calcula-tion. In the present calculation the cluster beam isincident normally .

3.1. Ag cluster impacts on an amorphous carbon targets

First of all let us consider the case of M, > M, i.e .,a Ag cluster impact on an amorphous carbon target .Figs . 2 show the time-dependent phenomena of theenergy distribution (fig. 2a), the angular distribution(fig. 2b) and the depth profile (fig. 20 of Ag atoms forI keV/atom (Ag)%) bombarded on C, where the peakvalues of these distributions at t = 0 arc normalized to

Y. Yarunanrra / Cluster impacts orr solid surfaces

Fig. 2 . The normalized angular distributions and the energydistrihutions of constituent Ag atoms for I keV/atom (A0 5Mcluster impacts on an amorphous carbon targets are pl-rstcd as.r function of time. where the peak values of the energydistribution and the angular distribution at t = 0 are set to beequal to unity. (a) The energy distribution (b) the angular

distribution (c) the depth profile

Table 1The projected range of 1 keV (Ag) 5 �� cluster impact on anamorphous carbon target at t = 0.15 ps

be equal to unity, and E,, and E, 2 are set to be 10 cV.As is shown in fig. 2a, a small fraction of constituentatoms are accelerated due to many MM collisionsbetween constituent Ag atoms or between a Ag atomand an energetic recoiling C atom, and fig. 2b showsthat almost all constituent atoms have inward direc-tions even after the cluster penetrate deeply enough .From fig. 2c we can see the cluster beam is compressedat t = 0.11 ps, because we use the relatively largecut-off energies E,, and E,2 . After t = 0 .1 ps smallfraction of Ag atoms have relatively large angles due tomany body effect, which cannot be expected within thelinear simulation . After 0 .1 ps there are a lot of Agatoms with energies less than 100 eV, as is shown infig. la . These slow Ag atoms have large collision diam-eters, and collide with many atoms at the same time .As a result, the slow Ag atoms were deflected withlarge angles .

The cluster energy 1 keV/atom is large enough forthe frontrunners of a penetrating (Ag) s(s, cluster tokick off target atoms and so constituent atoms comingafterwards can easily penetrate into the solid . In table1 we compare the Model A range and the Model Brange at t = 0 .15 ps, and the effect of E.2 on the rangeis also shown . In the case of the monoatomic bombard-ment (Model A), the cutoff energy Ec2 does not influ-ence the range of the projectile, but in the case of thebig cluster impact the smaller E c2 leads to a largerrange of the constituent atom, because the frontrun-ners kick off the target atoms and clear the way for theconstituent atoms coming afterwards. In other words,the frontrunner effect of high energy cluster impactsincrease the range . If we adopt the very large cutoffenergy E,2 artificially, the cluster could not penetratethe solid so deeply and its range is less than that ofModel A, because the frontrunners make the verydense region on the front of the cluster. In this case,the fronrunner effect decreases the range [8,13] . Thesame situation will be realized if the cluster energy isof the order of a few eV/atom, and this negativecontribution of the frontrunner effect is the mainmechanism of the ionized cluster beam deposition [17].

Y. Yamamura / Cluster impacts oa solid surfaces

NC0

wâ

dNE0â

Fig. 3. Plots of the mean value (E) and the straggling (S) ofthe energy distributions of Ag atoms as a function of time,where 1 keV/atom (Ag)s�� clusters are incident on C targets.As a comparison, the mean value (E) and the straggling (S)

of Model ?: ire ::!r^ pl-ad i- broken lines .

In fig . 3 we plot the average energy (E) and thestraggling (S) of the energy distribution of fig . 2a as afunction of time, where for comparison those of theenergy distribution of the monoatomic bombardment(Model A) are also plotted. As a matter of course, theaverage energy (E) is a decreasing function of time,while the straggling (S) has a maximum. In the ensu-ing discussions the time at which the energy distribu-tion has the maximum broadness will be denoted bytopt . The optimal time t.P, of the Model A energydistribution is 0.03 ps, and that of the Model B energydistribution is 0.09 ps. There is a big difference be-tween (E) of model A and that of Model B at theearly stage when due to the frontrunner effect theenergies of constituent atoms coming afterward are notslowed down so much.

185

Fig . 4 . The energy dependences of the energy distributions ofAg atoms and C atoms at t = 0 .09 ps in the case of 1keV/atom (AI)5w cluster impact on C, where the energydistribution of Ag atoms at t = 0 .09 ps have the maximum

broadness .

Model E,,(eV)

E,(eV)

E n(eV)

R p(À)

dAVlV

A 10 .0 10 .0 1 .0 16 .8 vcwB 10.0 1000.0 1 .0 7 .38 tû

10.0 10.0 1 .0 22 .5 v10.0 2.0 1 .0 22 .9

18 6

Fig. 4 shows the energy distributions of constituentAg atoms and recoiling C atoms at t = /opt and thecenter of the cluster is located near x = 1 I A. It isinteresting that there is an appreciable amount ofenergetic recoiling carbon atoms whose energies arcmuch larger than the maximum recoil energy yE in asingle elastic collision, where yE = 361 cV, and y =4M,M_l(M, + M,)2. The acceleration effect of recoil-ing carbon atoms is much larger than that of Ag atoms[141 . The energy distribution of carbon atoms has along tail as compared with that of Ag atoms, which isdue to the fact that since the carbon atom is lighterthan the Ag atom the carbon atom will be randomizeddue to MM collisions between a Ag atom and a recoil-ing C atom even at the early stage. As a result, violentMM encounters between the energetic recoiling car-bon atoms take place often . On the other hand, almostall Ag atoms at l =0.09 ps have the inward directions,and few violent MM encounters between the energeticrecoiling Ag atoms can take place, and the speed ofthe slowing down is very low due to the frontrunnereffect. This is why the energy distribution of Ag atomshas the trapezoidal shape . Therefore, it is importantfor the acceleration effect that energetic atoms shouldbe randomized, and the frontrunner effect affects theenergy property of slowing down particles as well asthe projected range.

3.2. Al cluster impacts on an amorphous gold targets

When the atomic mass M, if the constituent atomof a cluster is less than that of a target atom M,, wecan observe different phenomena from Ag cluster im-pacts on a C target which was discussed in the previoussection . As an example, the At and Au combination isadopted . Figs . 5 show the time-dependent phenomenaof the energy distribution (fig . 5a), the angular distribu-tion (fig . 5b) and the depth profile (fig 5c) of AI atomsfor 1 keV/atom (A]),(, ) clusters incident on a Autarget, where every distribution is also normalized att = 0, and Ec, and E,:z are set to be 10 cV. There arebig differences in the energy distributions, the angulardistributions and the depth profiles of constituentatoms between figs. 2 and figs. 5 . These differences aremainly due to the difference of the mass ratio M,/M2.The energy distribution, the angular distribution andthe depth profile of Al atoms are much broader thanthose of Ag atoms .

In the case of M, « M, constituent atoms trappedin the solid can change their directions without severeenergy loss due to the collisions with the target atoms.After that, as is shown in fig. 6, the MM collisions willtake place between an At atom and an At at ,jm orbetween an At atom and an energetic recoiling Auatom . Then, as is shown in fig. 5b, the constituentatoms are already randomized at the early stage and

Y. Yamamura / Cluster impacts on solid surfaces

Fig. 5. The angular distributions and the energy distributionsof constituent A1 atoms for 1 keV/atom (Al), )() cluster im-pacts on An are plotted as a function of time, where the peakvalues of the energy distribution and the angular distributionat t = 0 are normalized to be equal to unity. (a) The energydistribution (b) the angular distribution (c) the depth profile

E0ÁVQINO.

OCdO'vCON9

TWO (psec)Fig . 6. The collision frequencies of MT, MM and MI collisionsof I keV/atom (AD5�� cluster impacts on Au are plotted as afunction of time, where as a comparison that ofMT collisions

in the Model A calculation is also plotted in broken line.

they have a nearly isotropic angular distribution . As aresult, violent MM collisions between an AI atom andan AI atom or between an Al atom and an energeticrecoiling Au atom take place often . Namely, the clusterimpact region will be in quasi-equilibrium state, andthe constituent atoms have a broad energy distributionwhich is shown in fig. 5a . In fig . 6 the MM collisionincludes both the collision among constituent AI atomsand between an energetic Al atom and an energeticrecoiling Au atom, and the broken line means the MTcollision of the Model A calculation.

Fig. 7 shows the energy distributions of AI atomsand recoiling Au atoms at t = tov,, where t., is0 .05 ps . As is shown in fig. 5c, the depth profile isalso compressed near t = t op, and the center of thecluster is located near x = 20 A . It is interesting thatthere is an appreciable number of energetic recoiling

Energy (ev)

Fig . 7. The energy distributions of constituent Al atoms andrecoiling Au atoms at t=0 .05 ps, where 1 keV/atom (AD5(H)cluster are bombarded on Au target, and the energy distrilbu-tion of Al atoms at t = 0.05 ps have the maximum broadness.The broken line is a Maxwell-Boltzman function with the

temperature 187 eV .

Y. Yatnamura / Claster impacts on solidsurfaces

10 0

NC

N

aN

O

EOQ

_NC

m

aNE

21

0á

187

Fig. 8. The incident energy dependences of the energy distri-butions of AI atoms at t = t,, in the case of (AD5. clusterimpact on Au, where the best-fit Maxwell -Boltzmann func-

tions are plotted in solid lines .

gold atoms whose energies are larger than yE, whereyE =424 eV . The high energy tail of the Al energydistribution can be well described in terms of theMaxwell -Boltzmann function (broken line) whose tem-perature is 187 eV, i.e., N(E) a exp(-E/187). Thehigh-energy tail of Au atoms chows a Maxwell-Boltz-mann-like distribution . The cluster impact region be-comes a hot spot like a fireball which consists ofconstituent atoms and en.°.rgetic recoiling atoms.

In order to know the dependence of the incidentenergy on the profile of the energy distribution of theconstituent atoms, in fig. 8, we plot the energy distribu-tions of 100 eV/atom, 500 eV/atom, 1 ke -J/atom and2 keV/atom Al cluster on Au, where the number of AIatoms is 500. The solid lines in fig. 8 are the best-fitMaxwell-Boltzman function . The best-fit temperaturesEo of the energy distributions of fig . 8 are plottedagainst the incident energy Ei � in fig. 9 . It is interestingthat as was already predicted in EMR's theory [21] thetemperature E u is linearly proportional to the clusterenergy, i.e., E�= 15.0 + 0.17E�,. In table 2, we list theaverage energy (E), the straggling (S> and the best-fittemperature Eu of the energy distributions of fig. 8.

188

Energy/atom (eV)Fig . 9. Plots of the best-fit temperatures of the high-energytails of the energy distributions of AI atoms at t = t�p,, whichare shown in solid lines in fig . 8, against the incident energy

E�,. The solid line is E� = 15.0+0.17 E�, .

For a fixed cluster size the relative straggling (S)/E �,becomes large with the decrease of E�,, because themean free path between collisions is much less thanthat between high-energy atoms .

The size of cluster is a very important factor whenwe discuss the energy distribution, because it is neces-sary for the quasi-equilibrium state that the clusterimpact region should be much larger than the mean-free-path between collisions. In fig. 10, the mean value(E) (fig. 10a) and the straggling (S) (fig. 10b) of theenergy distribution of AI atoms for (AD .. . (AD5, ) and(Al)S,x) cluster bombardment with on energy of 1iccV/atom are plotted as a function of time, where forcomparison those of the monoatomic bombardment(Model A) are also shown . Since we adopt the rela-tively large cutoff energies Ec , and E c, to save comput-ing time, the difference of the mean value (E) be-tween the monoatomic bombardment (Model A) and asmall cluster impact is small. But the straggling ( :,> ofthe energy distribution due to the small cluster (AI) �) ,and (AD 5,) impacts are large as compared with that ofthe monoatomic bombardment at t = 0 .05 ps. In thecase of big cluster impacts (n = 500) on Au both the

Table 2The incident energy dependence of the average energy, thestraggling and the temperature of the energy distribution ofAI atoms for (AI)5�e cluster impact on Au target at the timewhen the distribution has their maximum broadness width

Y. Yantamtra / Cluster impacts on solid surfaces

1000

ANV

Time (psec)Fig. 10. Plots of the mean value (E) and the straggling (S) ofthe energy distributions of AI atoms as a function of time,where I keV/atom AI cluster with different number of con-stituent atoms are bombarded on Au targets . As a comparison, the mean value (E) and the straggling (S) of Model A

;tr- also platted in broken lines.

mean value and the straggling are much larger thanthose of the monoatomic bombardment (Model A).

When we compare fig . 10 with fig. 3, we can observethe interesting difference between these two figures.The mean value <E) of fig . 3 is much larger than thatof Model A, and the straggling (S) of fig . 3 reaches amaximum at a later time than that of Model A. On theother hand, the difference between < E) of n = 500and <E) of Model A in fig . 10 is not so large, but thedifference between (S) of n=500 and (S) of ModelA in fig . 10 is very big. In the case of a Ag cluster on Cthe frontrunner effect determines the energy proper-ties of constituent atoms, while for an AI cluster on Authe acceleration effect is important for the energydistribution of constituent atoms .

In fig . 11 we show the corresponding energy distri-butions of AI atoms at t = tops to figs . 10 . The clustereffect or the acceleration effect is already importanteven for (Al),() , but it is very difficult to fit the highenergy tail of the energy distribution of (AD tt) clusterimpacts in terms of the Maxwell- Boltzmann function,

Energy/atomE

Average energy( E) (cv)

Straggling(S) (CV)

TemperatureE� (CV)

100 66.3 37.3 27500 340 167 1091 keV 709 307 1872 keV t 180 540 360

d5

daEF

Y. Yamamura / Clusterimpacts on solidsurfaces

Size of cluster

Fig . 11 . The energy distributions of Al atoms at t =t�p, for 1keV (AI) � ,, (AD 5� , (AI) �K) , and (AD5�� clusters on Au. As acomparison, the Model A energy distribution at t = t o� , of a 1keV monoatomic At ion bombardment on Au is also shown.

because the cluster impact region is too small forconstituent atoms to become randomized due to enoughviolent MM collisions . In Fig. 12 we plot the best-fittemperature E � of the energy distributions of Al atomsin fig . 11 . The best-fit temperature E� increases withthe increase of the cluster size, and the solid linecorresponds to E,) = 93 .2 + 15 .5 log(n).

Form fig . 5a and fig . 9 it is known that the energydistributions have the long tail of the exponentiallydecreasing behaviour from t = 0.03 ps to t = 0.11 ps .At t = 0.11 ps the impact region is still hot and is in thequasi-equilibrium state . As is shown in fig. 11, thehigh-energy tail of the energy distribution at t = 0 .11ps can be well described in terms of the Maxwell-Boltzmann function exp(-E/156). Fig . 13 shows thetime-dependence of the temperature E,) after t� P, .Thus, in the case of 1 keV/atom (AI)S(Kl cluster impact

Fig. 12. Plots of the best-fit temperatures of the high-energytails of the energy distributions at t - t ap , as a function of the

size of cluster. The solid line is E�= 93.2+ 15.5 log(n).

250

aEd1-

150

189

1000 .00 0.05 0 .10 0 .15

Time (psec)Fig . 13 . Plots of the best-fit temperatures of the high-energytails of the energy distributions shown in fig . 5a as a functionof time. The solid line is E �=250-830 t, where the time r is

in ps .

on Au, the quasi-equilibrium state will continue formore than 0 .6 x 10 - 1; s, but the temperature of thecluster impact region will decrease as time passes .

4. Conclusions

Using the time-evolution Monte Carlo simulationcode DYACAT, we have simulated the energy proper-ties of big cluster impact phenomena. Several interest-ing features are found. In the case of 1 keV/atom(Ag)S,K) cluster impacts on Au, where the atomic massof Ag is much larger than that of carbon, the mostimportant cluster effect is the frontrunner effect whichseems the frontrunners of the lower parts of the clusterclear away target atoms for constituent atoms comingafterward. The frontrunner effect affects the energyproperty of slowing down particles as well as the pro-jected range . The energy distributions of constituentatoms have a trapezoidal shape, while those of recoil-ing carbon atoms have a long tail . The accelerationeffect of the cluster impact appears in the energydistribution of the lighter element, because the lighterelement is more easily randomized.

In the case of (AI) � cluster impacts on Au, wherethe atomic mass of Al is much smaller than that ofgold, the high-energy tail of the energy distribution ofAl atoms of the big cluster impact (n > 100) can bewell described in terms of the Maxwell-Boltztrlnnfunction. Its temperature is proportional to the energyand it is an increasing function of the size of a cluster ilthe number of constituent atoms is larger than 50. Itthe case of 1 keV/atom (AI)5 . cluster impact on Authe quasi-equilibrium state will continue for more that0.6 x 10 -t' s, but the temperature of the cluster impact region will decrease as time passes . For the smalcluster impact the quasi -equilibrium state does no

190

appear. This means that the direct extension of theknowledge of small cluster impact phenomena to largecluster impact phenomenon is dangerous.

Acknowledgements

This work was supported by a Grand-in-Aid ofTheMinistry of Education, Science and Culture. The au-thor is grateful to Dr. H. Tawara of the NationalInstitute for Plasma Science for valuable discussions.

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