Dependence of cluster ranges on target cohesive energy: Molecular-dynamics study of energetic Au402 cluster impacts

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NIM B Beam Interactions with Materials & Atoms

Nuclear Instruments and Methods in Physics Research B 266 (2008) 44–48 www.elsevier.com/locate/nimb

Dependence of cluster ranges on target cohesive energy: Molecular-dynamics study of energetic Au402 cluster impacts Christian Anders, Herbert M. Urbassek * Fachbereich Physik, Universita¨t Kaiserslautern, Erwin-Schro¨dinger-Straße, D-67663 Kaiserslautern, Germany Received 18 July 2007; received in revised form 8 October 2007 Available online 30 October 2007

Abstract It has long been known that the stopping and ranges of atoms and clusters depends on the projectile-target atom mass ratio. Recently, Carroll et al. [S.J. Carroll, P.D. Nellist, R.E. Palmer, S. Hobday, R. Smith, Phys. Rev. Lett. 84 (2000) 2654] proposed that the stopping of clusters also depends on the cohesive energy of the target. We investigate this dependence using a series of molecular-dynamics simulations, in which we systematically change the target cohesive energy, while keeping all other parameters fixed. We focus on the specific case of Au402 cluster impact on van-der-Waals bonded targets. As target, we employ Lennard–Jones materials based on the parameters of Ar, but for which we vary the cohesive energy artificially up to a factor of 20. We show that for small impact energies, E0 [ 100 eV/ atom, the range D depends on the target cohesive energy U, D / Ub. The exponent b increases with decreasing projectile energy and assumes values up to b = 0.25 for E0 = 10 eV/atom. For higher impact energies, the cluster range becomes independent of the target cohesive energy. These results have their origin in the so-called ‘clearing-the way’ effect of the heavy Au402 cluster; this effect is strongly reduced for E0 J 100 eV/atom when projectile fragmentation sets in, and the fragments are stopped independently of each other. These results are relevant for studies of cluster stopping and ranges in soft matter. Ó 2007 Elsevier B.V. All rights reserved. PACS: 79.20.Rf; 79.20.Ap; 61.80.Lj Keywords: Cluster impact; Ranges; Molecular-dynamics simulation

1. Introduction The interaction of clusters with surfaces has received increased attention in recent years. Cluster stopping and the range of clusters in matter are both of a fundamental interest and relevant for applications such as thin-film deposition [1,2], secondary ion and neutral mass spectrometry [3] or as desorption method for biomolecules [4]. Here, also the fate of the cluster and in particular its range are of interest. Atom ranges have been studied both experimentally and theoretically intensely [5]. They are known to depend on the projectile-target interaction potential, the target atom density and the mass ratio of projectile and target atoms, *

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but not on the target cohesive energy. Less is known about the stopping and range of clusters. From a theoretical point of view, the stopping of heavy clusters in a light material has been analyzed by Shulga and Sigmund [6]. They found that the range of a heavy cluster is increased beyond that of an equi-velocity atom. The underlying reason is the so-called clearing-the-way effect [7]. The front atoms in the cluster collide with target atoms and convey them sufficient momentum to clear the way for the following cluster atoms. As a consequence the stopping power of the entire cluster is reduced, and hence its range is increased. Recently, Carroll et al. [8,9] inquired into the stopping of Agn clusters (n = 20–200) impacting with total energies in the range of 0.5–6 keV into graphite. Their moleculardynamics results gave evidence that the range D of a cluster

C. Anders, H.M. Urbassek / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 44–48

D / E0 n1=3 =U :

ð1Þ

The availability of a scaling law such as Eq. (1) would be of immediate interest to the applications mentioned above, since available computer codes like SRIM [10], which work well for monatomic ion ranges, cannot predict cluster ranges. The applicability of the cluster range formula (1) to a more general cluster stopping scenario has been tested in [11] with a particular emphasis on the cluster size dependence. There, the impact of 100 eV/atom Aun clusters with n varying in the range of 1 6 n 6 402 was simulated on a variety of targets, including graphite and a condensed Ar target. It was found that also in this case, ranges follow a power law, D / na, where a = 0.3–0.4. Quite recently, [12] the nuclear stopping of Aun clusters was evaluated as a function of cluster size n and impact energy per atom E0. Since there a larger range of cluster energies was investigated, it could be found that the exponent a – which describes the n-dependence of stopping in analogy to that of the ranges – depends slightly on the impact energy such that with increasing E0, a goes to zero. In the present paper, we want to test the dependence of cluster ranges on the target cohesive energy. We focus on a particular system, the impact of Au402 clusters on a vander-Waals bonded target. This situation is representative of irradiation experiments on soft (biological or polymeric) matter. 2. Method The simulations presented in this paper closely follow our cluster size simulations published previously [11]. The target is modelled as a van-der-Waals-bonded material with a pair interaction potential of the Lennard–Jones form    r 12  r 6 UðrÞ ¼ 4  ; ð2Þ r r which is cut off at rc = 2.5r. The length parameter is ˚ . The dimer bond energy  appropriate for Ar is r = 3.4 A Ar = 10.3 meV. We note that with our cutoff radius, the cohesive energy of an fcc crystal is U = 7.9 = 81 meV. ˚ , we spline the Lennard–Jones potential to For r < 3.4 A the KrC potential [13] valid for close Ar–Ar encounters. The cohesive energy of our target material is easily changed by choosing different values of . For our systematic investigations, we chose /Ar in the range of 0.5–10, but also performed a few test simulations with increased bonding strength . Fig. 1 shows the interatomic potentials employed in our study. We note that with increasing , the spline to the KrC potential is no longer performed so

1

V (eV)

of energy E0 per atom and containing n atoms scales as D / E0n1/3. If one assumes [8] that E0 is mainly spent in breaking bonds of the target solid while penetrating into it, the penetration depth becomes inversely proportional to the cohesive energy U of the target material,

45

ε/εAr 0.5 1 2 3 10

0.1

0.01 0 –0.02 –0.04 –0.06 –0.08 –0.1 0

1

2

3

5

4

6

7

8

9

r (Å) Fig. 1. Interatomic potential V(r) between target atoms as function of interatomic distance r. Note that the ordinate changes from linear to logarithmic scale at positive energies. The parameter /Ar characterizes the well depth of the Lennard–Jones potential, Eq. (2), with respect to that of Ar.

easily. Our crystalline fcc target contains 160,000 atoms, ˚; ˚ 2 and a depth of 522 A has a lateral size of 102  102 A 3 ˚ its atomic number density is 0.0295 A . The 402-atom Au cluster is spherical; the Au–Au interaction potential is of a many-body form [14]. The Au–Ar interaction is purely repulsive according to the ZBL potential [5]. We performed for each impact energy E0 and for each target a number of five simulations to obtain sufficient statistics. Each simulation uses a different cluster impact point on the target. Since the projectile is quite large, range fluctuations are considerably smaller than for monatomic projectiles, and the accuracy achieved is sufficient for our purpose. We performed simulations both for an amorphous and a crystalline (fcc) target. While the ranges are systematically larger in the amorphous targets, the trends are analogous, and hence we concentrate on the presentation of the crystalline target [15]. Furthermore, we performed a number of further test simulations – increased target thickness and lateral size, varied projectile shape, fifth-order (instead of third-order) spline between Lennard–Jones and KrC potential – which corroborate our results. 3. Results Fig. 2 gives a cross-sectional view of the penetration of the cluster into the target at a time when the cluster has reached about its maximum penetration depth. The straight ‘cannon-ball-like’ trajectory of the projectile is evident. Clearly, the effect on the target material is considerably more spatially constrained for larger target binding energies. For U/UAr = 30, Fig. 2(a), only a small crater has developed at the surface, while the material along the projectile trajectory has become amorphous or liquid. In contrast, for the small binding energy, U/UAr = 2, Fig. 2(c), a huge tunnel has developed along the cluster

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C. Anders, H.M. Urbassek / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 44–48

a

500 400

Dmax (Å)

300

Fig. 2. Cross-sectional view of a E0 = 100 eV/atom Au402 cluster stopped in van-der-Waals materials with various cohesive energies U. Dark: projectile atoms; Light: target atoms. (a) Strongly-bonded target, U/ UAr = 30, at time t = 7 ps after impact, (b) U/UAr = 10, at time t = 13 ps after impact and (c) weakly-bonded target, U/UAr = 2, at time t = 20 ps after impact.

U/U Ar 0.5 1 2 3 10

100 10

b

100 E0 (eV/atom)

1000

300 250 200

Dcm (Å)

trajectory, in which the material has been gasified and now flows out into the vacuum above the surface. For the intermediate case, U/UAr = 10, displayed in Fig. 2(b), the central track region has become gaseous, while the track rims are covered by a thick amorphous or liquid layer. Evidently, in particular the weakly-bonded target has not yet cooled enough to evaluate its final state of damage. However, the cluster has already reached its final position in the target, and hence the penetration depth can be deduced. The projectile has been fragmented to some degree, but a largest fragment carrying the major part of the projectile mass is clearly discernible in Fig. 2. A closer inspection of the data shows that the amount of fragmentation – as measured both by the number of fragments formed and by the size of the largest fragment – increases with decreasing target cohesion. In the strongly-bonded target, U/UAr = 30, the projectile shatters into only six fragments; the largest fragment still contains 393 atoms. In the weakly-bonded target, U/UAr = 1, on the other hand, 28 fragments are generated, and the largest fragment contains 308 atoms. Thus the more strongly bound targets contain the penetrating cluster more strongly and prevent it from excessive fragmentation. The range of a cluster is not as uniquely defined as the range of an ion. We introduce two measures to characterize it. Dmax is the largest range of a cluster atom, i.e. the maximum depth which any cluster atom reaches during the impact. Dcm is the range of the centre of mass of the cluster; for further discussion see [11]. Fig. 3(a) assembles the maximum ranges of Au402 clusters for impact energies E0 = 10–1000 eV/atom in our various targets. The ranges increase with impact energy. Note that at the highest impact energies, the ranges in the various materials investigated coincide within statistical accuracy, with the exception of the weakest-bonded target material. However, with decreasing E0, the ranges become strongly dependent on the target cohesive energy. This behaviour is due to the clearing-the way effect described in Section 1 above: The efficacy, with which the front atoms of the Au402 cluster can move target material away depends on the target strength and hence cohesive energy; it will be more pronounced in soft than in hard materials.

200

150 U/U Ar 0.5 1 2 3 10

100 10

100 E0 (eV/atom)

1000

Fig. 3. Range of Au402 clusters as function of the projectile energy per atom, E0. The parameter U/UAr denotes the cohesive energy of the target material. (a) Maximum range, Dmax and (b) centre-of-mass range, Dcm.

An analogous, but not so pronounced result is seen for the centre-of-mass range in Fig. 3(b). However, this quantity is non-monotonic with E0, such that Dcm decreases with increasing projectile energy for E0 > 100 eV/atom. This is due to projectile fragmentation, which sets in roughly at 100 eV/atom. A closer analysis shows that for Au402 impacts in Ar, at E0 = 100 eV/atom the cluster fragments into roughly 20 pieces; however, the largest remaining cluster still contains 300 atoms. At E0 = 400 eV/atom, however, the cluster is shattered into more than 150 fragments, the largest of which contains only 20 atoms. Fragmentation has a strong effect on the centre-of mass range, since the individual fragments may come to rest at widely varying positions. In some instances – in particular for higher energies, we even observed that a few (up to 10) Au atoms were entrained in the liquid or gaseous flow displayed in Fig. 2 and reached the surface of the target. The onset of fragmentation is thus easily recognized as it leads to a considerable difference between the maximum and the mean range. Note, however, that also the maximum range is affected to some degree by the onset of fragmentation: The steepness of the Dmax(E0) curve

C. Anders, H.M. Urbassek / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 44–48

considerably sinks at the fragmentation threshold, in particular for the weakest-bonded target materials. The reason for the range reduction lies in part in the fact that the projectile energy is used up for cluster fragmentation. But, more importantly, after fragmentation the projectile can no longer benefit from the clearing-the way effect, and hence its range is relatively smaller than at smaller energies, where the projectile is still intact. In order to analyze the ranges more quantitatively, we display in Fig. 4 the same data as in Fig. 3 in their dependence on the cohesive energy. Included are fit lines to a power law D / U b :

ð3Þ

The values of the fit parameters are indicated in Table 1. Apparently, for E0 = 100 eV/atom, a value of b = 0.10 describes the dependence of the cluster ranges on the target cohesive energy quite well. Thus, the bonding energy inside the target has a definite, albeit small, influence on cluster ranges. b decreases to zero with increasing impact energy, and hence for sufficiently high cluster energies, the influence of the target bonding on the ranges vanishes. We also note that in available theories of atom or ion (not cluster) 1000

D max (Å)

E 0 (eV/atom): 10 25 100 400 1000

47

stopping, such as implemented in the SRIM code, the influence of the target atom bonding is disregarded, amounting to b = 0. Carroll et al. [8,9] interpret the dependence of the cluster range on the target cohesive energy in the following way. The total energy Etot = nE0 of the cluster is used to displace target atoms out of the path of the cluster until it is stopped. Thus a ‘tunnel’ is created with a volume proportional to D n2/3, since the geometrical cross-section of the cluster scales as n2/3 with the cluster size n. Hence, Etot / U D n2/3 or D / E0n1/3/U, Eq. (1). Our simulations, which show that b decreases with increasing impact energy, suggest that the energy given to atoms in the tunnel does not only contain the potential energy of bond breaking, U, but also kinetic energy which is conveyed to target atoms in order to dislocate them from the projectile path. For high impact energies, the kinetic energy contribution dominates and hence the dependence of the ranges on U is reduced. We note that Carroll et al. [8] performed their simulations of 0.75–6 keV Agn (n = 20–200) cluster impact into a graphite target. These simulations correspond to an energy per atom of E0 = 4–300 eV, which is in the range of (0.5–40)  U, where U = 7.4 eV is the cohesive energy of graphite. In contrast, even our slowest clusters have an energy of E0 = 125U. Thus the simulations presented here extend the results of Carroll et al. [8,9] towards higher scaled impact energies E0/U. In Fig. 5 we display the time needed until the largest penetration is reached, i.e. until the ranges have assumed their values Dmax and Dcm, respectively. The time zero has been set when the downmost cluster atom is at a dis˚ from the surface, and starts feeling the intertance of 8.5 A action with the surface. We note that the times show a quite strong dependence on the target atom bonding, which can be fitted to t / U c ;

100

ð4Þ

10

1 U/U Ar

40

Fig. 4. Maximum range Dmax of Au402 clusters as function of the target cohesive energy U. The parameter E0 denotes the projectile energy per atom.

maximum depth mean depth

35 30

Table 1 Power exponents b describing the dependence of the Au402 cluster range on the target cohesive energy, Eq. (3), for various impact energies E0, in a crystalline target E0 (eV/atom)

bmax

bcm

10 25 100 400 1000

0.23 ± 0.01 0.10 ± 0.03 0.10 ± 0.01 0.03 ± 0.02 0.03 ± 0.03

0.25 ± 0.01 0.11 ± 0.02 0.09 ± 0.01 0.00 ± 0.01 0.005 ± 0.005

The subscripts relate to the maximum range, Dmax, and to the mean range, Dcm, respectively.

t (ps)

25 20 15

10

1

10 U/UAr

Fig. 5. Time t needed for the Au402 cluster, E0 = 100 eV/atom, to reach the deepest penetration, Dmax, and the mean range, Dcm, respectively. The lines give a power-law fit, Eq. (4).

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where the exponent c is around 0.4. This considerable reduction might be used for simulations of cluster ranges: by choosing a target material with deliberately increased cohesive energy, a considerable saving in computer time may be achieved, while the cluster range itself is only little affected. 4. Conclusions Using artificial target materials, which differ from each other only by their cohesive energy, we study the dependence of Au402 cluster ranges at fixed energy per atom, E0, on the target cohesive energy. In contrast to ranges of atoms, which do not depend on the target cohesive energy, we find that below a characteristic energy of E0 ffi 100 eV/atom, the cluster ranges become dependent of the target cohesive energy. This dependence can be described by a power-law, Eq. (3). The exponent b increases with smaller impact energy and assumes values up to b = 0.25 for E0 = 10 eV/atom. These results extend previous data by Carroll et al. towards higher (scaled) cluster impact energies E0/U. Our results have their origin in the clearing-the-way effect of the heavy Au402 cluster; this effect is strongly reduced for E0 J 100 eV/atom when projectile fragmentation sets in. Furthermore, we find that the time which clusters need to be stopped depends strongly on the target cohesive energy U, like t / U0.4. These results are useful for the modelling of cluster ranges in soft matter: (i) When the cluster range in complex materials such as polymers or bioorganic materials, is studied, the results will be quite independent of the detailed modelling of the target cohesive properties, such that simplified assumptions can be made here. (ii) Simulation time might be saved by judiciously choosing to model stopping

in a material with higher cohesive energy while the stopping results will be only slightly influenced. Acknowledgements We acknowledge discussions with Yvon LeBeyec and his group, which inspired us to perform this work. This work has been made possible by a grant of the Graduiertenfo¨rderung des Landes Rheinland-Pfalz. References [1] W.L. Brown, M.F. Jarrold, R.L. McEachern, M. Sosnowski, G. Takaoka, H. Usui, I. Yamada, Nucl. Instr. and Meth. B 59/60 (1991) 182. [2] H. Haberland, Z. Insepov, M. Moseler, Z. Phys. D 26 (1993) 229. [3] J.C. Vickerman, D. Briggs (Eds.), ToF-SIMS: Surface Analysis by Mass Spectrometry, IM Publications, Chichester, UK, 2001. [4] A. Tempez, J.A. Schultz, S. Della-Negra, J. Depauw, D. Jacquet, A. Novikov, Y. LeBeyec, M. Pautrat, M. Caroff, M. Ugarov, et al., Rap. Commun. Mass Spectrom. 18 (2004) 371. [5] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985. [6] V.I. Shulga, P. Sigmund, Nucl. Instr. and Meth. B 47 (1990) 236. [7] V.I. Shulga, M. Vicanek, P. Sigmund, Phys. Rev. A 39 (1989) 3360. [8] S.J. Carroll, P.D. Nellist, R.E. Palmer, S. Hobday, R. Smith, Phys. Rev. Lett. 84 (2000) 2654. [9] C.F. Sanz-Navarro, R. Smith, D.J. Kenny, S. Pratontep, R.E. Palmer, Phys. Rev. B 65 (2002) 165420. [10] J.F. Ziegler, Srim: http://www.srim.org/ (2000). [11] C. Anders, H.M. Urbassek, Nucl. Instr. and Meth. B 228 (2005) 57. [12] C. Anders, H.M. Urbassek, Nucl. Instr. and Meth. B 258 (2007) 497. [13] W.D. Wilson, L.G. Haggmark, J.P. Biersack, Phys. Rev. B 15 (1977) 2458. [14] T.J. Colla, H.M. Urbassek, Nucl. Instr. and Meth. B 164–165 (2000) 687. [15] C. Anders, PhD. thesis, University Kaiserslautern, 2007.