We present the results of over 90 tight-binding molecular-dynamics
simulations of collisions between three- and five-atom silicon clusters, at a
system temperature of 2000K. Much the most likely products are found to be two
'magic' four-atom clusters. We show that previous studies, which focused on the
equilibrium binding energies of clusters of different sizes, are of limited
relevance, and introduce a new effective binding energy which incorporates the
highly anharmonic dynamics of the clusters. The inclusion of dynamics enhances
the magic nature of both Si4 and Si6 and destroys that of Si7.Comment: 12 pages, 3 Figures, 500 KB gzipped PostScript fil

Godby, R. W.

Porter, A. R.

English

arXiv

We present the results of over 90 tight-binding molecular-dynamics simulations of collisions between three- and five-atom silicon clusters, at a system temperature of 2000K. Much the most likely products are found to be two 'magic' four-atom clusters. We show that previous studies, which focused on the equilibrium binding energies of clusters of different sizes, are of limited relevance, and introduce a new effective binding energy which incorporates the highly anharmonic dynamics of the clusters. The inclusion of dynamics enhances the magic nature of both Si4 and Si6 and destroys that of Si7

Porter, A.R.

Godby, R.W.

White Rose Research Online

This is a repository copy of Influence of dynamics on magic numbers for silicon clusters.White Rose Research Online URL for this paper:https://eprints.whiterose.ac.uk/200161/Preprint:Porter, A.R. and Godby, R.W. orcid.org/0000-0002-1012-4176 (1997) Influence of dynamics on magic numbers for silicon clusters. [Preprint] eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. promoting access to White Rose research papers      Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/4004  Published paper Porter, AR, Godby, RW (1998) Influence of dynamics on magic numbers for silicon clusters arxiv.org/abs/physics/9711005v1    White Rose Research Online eprints@whiterose.ac.uk  physics/9711005   6 Nov 19971Influence of Dynamics on Magic Numbers for Silicon ClustersA. R. Porter* and R. W. Godby Department of Physics, University of York, York YO1 5DD, United KingdomABSTRACT:  We present the results of over 90 tight-binding molecular-dynamics simulationsof collisions between three- and five-atom silicon clusters, at a system temperature of 2000K.Much the most likely products are found to be two ÔmagicÕ four-atom clusters.  We show thatprevious studies, which focused on the equilibrium binding energies of clusters of different sizes,are of limited relevance, and introduce a new effective binding energy which incorporates thehighly anharmonic dynamics of the clusters.  The inclusion of dynamics enhances the magicnature of both Si4 and Si6 and destroys that of Si7.In recent years much work has been done on developing the understanding of the structure ofatomic clusters and their transition to the structure of a bulk solid with increasing size.  Forsilicon in particular, the existence of magic number clusters, consisting of 4, 6 and 10 atoms, hasbeen the subject of much study using both ab initio and tight-binding (TB) methods [1,2,3].These magic numbers correspond to especially abundant cluster sizes arising during theproduction process, where laser evaporation of a solid silicon surface produces a plasma whichis then swept away in a stream of inert gas.  As this plasma cools, the clusters undergo a seriesof collisions as the system attains dynamic equilibrium.  It is found [4] that the magic numbersindicated by the mass spectrum of the clusters so produced are independent of the details of themethod of production, depending only on the fact that the clusters have reached dynamicequilibrium.2Previous work [1,2,3,5] has attempted to explain the existence of these magic numbers throughthe study of the equilibrium structures of the clusters. However, during their formation theclusters are at a high temperature and are consequently exploring geometrical configurations farfrom the potential-energy minimum.  The relevance of the minimum energy is therefore unclear.The aim of this work is to simulate a large number of collisions, analyzing the results in terms ofthe dynamics of the reactants, intermediate clusters and products, rather than simply the staticsof cluster equilibrium.Use of a tight-binding (TB) scheme makes such simulations computationally feasible whilstincorporating the quantum-mechanical description of the sp directional bonding in silicon that isessential for reproducing the correct cluster structures.  The TB scheme used here was devisedby Goodwin et al. [6] to give a good representation of a large database of the volume-dependenttotal energy of solid silicon in various structures, ensuring accuracy over a wide range of bondingconfigurations.  The energy and forces obtained from the TB model were used in a classicalmolecular-dynamics (MD) simulation [7] of the atomic motion.Since all previous work has concentrated on the equilibrium structures of the Si clusters, weaddress these first.  Hence, we used our TB MD, to perform simulated annealing for clusters of 2to 8 atoms in order to identify their minimum-energy structures.  All of the structures identifiedin this way were found to agree with previous results using this scheme [6] except for the case ofSi5.  For this cluster we find a rather asymmetric, face-capped planar rhombus form for theminimum energy structure, in agreement with the results of Laasonen and Nieminen [5].  It issignificant that the search for the equilibrium configuration of Si8 was quite difficult, it being3found necessary to reduce the time step to one twentieth of the 2.42 × 10-4 ps used during thecooling of all of the other clusters.  This indicates that the Si8 minimum energy structure is anarrow and rather inaccessible valley in the potential energy surface of the cluster.Although the TB scheme used here achieves better agreement with ab initio results for theequilibrium structures of the clusters than other TB schemes [1,2] (typically bond lengths arewithin 10% of ab initio values), the associated binding energies per atom were consistentlyoverestimated by (1.03 ± 0.07) eV.  This may be due in part to the fact that Goodwin et al. [6]fitted the scheme to the results of an ab initio density-functional-theory calculation in the local-density approximation (LDA), since calculations of this type overestimate the cohesive energiesof bulk silicon and molecules.  However, as the overbinding is virtually independent of clustersize (and thus of the different types of bonding present in the different structures), it is unlikelyto have had a significant effect on the simulated dynamics and the outcomes of the collisions.  Itshould be noted that these errors are of the same order of magnitude as would be obtained froman ab initio LDA calculation.The results of the simulated annealing indicated that the equilibrium structure of the Si4 clusterwas considerably more stable than either the 3 or 5 atom forms.  By simulating the collision ofSi3 with Si5, we aim to investigate the effects, if any, of this property of Si4 on the outcome ofthe cluster-cluster collision process.  The collisions were carried out for impact parameters, b, of0 to 5.82  with an initial system temperature of approximately 2000K and were followed for atime corresponding to 7.26 ps. Although locating the minimum energy structure of Si8 required amuch reduced time step, its highly localized nature means that it will not significantly affect thedynamics during the collisions simulated here [8].  It is therefore acceptable to use the standard4time step of 4.84 × 10-4 ps.  Prior to a collision, the clusters were separately heated to 2000Kand assigned appropriate centre-of-mass velocities.  In order that the results may be consideredas an average over the many possible relative orientations of the colliding clusters, we rotatedeach cluster through three random angles about its internal axes before every collision.A summary of the results of all of the collisions simulated is given in Table I.  For all caseswhere the clusters came into contact a reaction resulted, such as the transfer of a single atom toproduce Si4 + Si4.  Although twelve simulations were performed for each impact parameter, theclusters failed to make contact on three occasions in simulations with the largest value of b andthus did not react.  The eight reactions that produced Si3 + Si5 all involved significant rebonding.The results in Table I clearly show the dominance of the Si4 + Si4 reaction product in thesecollisions, with this configuration being the result in almost 70% of the reactive collisions.  20%of the reactions produced a single Si8 cluster that did not fragment, making this the next mostabundant product.  It is interesting to note that the loss of a single atom by the highly mobile Si8cluster formed during the collisions was never observed.We emphasize the irrelevance of the equilibrium potential energy values of the clusters indetermining the outcomes of these collision reactions.  As an alternative, we have calculated theeffective binding energies (EBEs) of the clusters under the conditions simulated here.  Theeffective binding energy was defined to be −1 times the minimum value of potential energy thatwould exist if the potential-energy surface for the actual mean potential energy and kineticenergy were harmonic about its minimum (see Figure 1).  Since, for a harmonic oscillator,5PE KE BEvib vib eff= − , (where  indicates a time average) and KE vib  can be calculatedfrom the total kinetic energy by using the equipartition theorem, this assumption enables anestimate of the minimum binding energy to be obtained from the simulation [9]:( )BE PE KEeff vib sim sim= − −−3 53NN(1)where N is the number of atoms in the cluster.  All of the time averaged values except that for Si7were obtained from the results of actual collision runs.  Since Si7 was never produced in thesecollisions, its effective binding energy was obtained by heating an example to approximately4000K (typical of the temperature attained by Si8 during a reaction) and then letting it evolve inisolation.  Since a clusterÕs temperature determines the extent to which it explores what is, inreality, an anharmonic potential energy surface, the appropriate EBE to characterize its motionwill also be a function of temperature.  Therefore, because the temperature of the reacting Si3 andSi5 clusters tended to differ from the temperatures of such clusters produced during a reaction,two values for the EBEs of these clusters have been calculated.The resulting EBEs differed significantly from the corresponding equilibrium values in a numberof ways.  For instance, the EBEs of both Si7 and Si8 are considerably less than the equilibriumvalues, demonstrating the inaccessibility of their equilibrium structures in the dynamicssimulated here.  In contrast, for Si4, the effective binding energy is greater than the equilibriumbinding energy, indicating a flattening, rather than a steepening, of the potential-energy surfacenear its minimum.The energy changes associated with different reaction outcomes may be estimated by subtractingthe binding energies of the reacting Si3 and Si5 clusters from those of the products.  The estimates6obtained with both the equilibrium and effective values for the cluster binding energies predict Si8to be the most favourable product, followed by Si4 + Si4.  However, the difference between thesetwo products is much less in the EBE estimate (1.3 eV compared with the equilibrium value of3.9 eV).  In general, the estimates produced using the EBEÕs give predictions in much closeragreement with both the results of our simulations and those of experiment [4] than do thecorresponding equilibrium estimates.  First, use of the effective values reduces the predictedincrease in binding energy associated with the production of Si + Si7 from 1.7 eV (the equilibriumestimate) to −0.9 eV.  This indicates that it is not an energetically-favourable outcome; hence itsabsence from Table I.  Second, they also give the correct ordering of the remaining products inthat Si3 + Si5 is predicted to be marginally (0.1 eV) more favourable than Si2 + Si6, in directcontrast to the equilibrium values which predict that Si2 + Si6 is the more favourable of the twoby 0.7 eV.  The fact that a product of Si8 is not as common as suggested by this simple estimateis due to the fact that the single cluster cannot accommodate all of the kinetic energy of thereaction.  Hence, it becomes simply  a matter of time as the cluster explores its configurationspace before it will fragment.Another way of comparing the equilibrium and effective binding energies is to plot them peratom as a function of cluster size (Fig. 2).  The additional stability of the points at N=4 and N=6,in comparison to what would be expected from a linear combination of neighbouring values,indicates the magic status of Si4 and Si6.  In contrast, the corresponding curve for the equilibriumvalues has no local maximum at N=6, predicting instead that Si7 is magic.  These features areemphasized in the plot of cluster fragmentation energy (defined as the binding energy of an N-atom cluster less that of the N−1-atom cluster) also shown in Figure 2.  The peaks at N=4 andN=6 in the curve obtained with the effective values show the enhanced stability of these clusters7under the dynamic conditions simulated here.  The absence of a peak for the Si7 cluster reflectsthe removal of its magic nature.As a typical example of the type of the behaviour observed during the collision processes weshall briefly discuss a particular collision with an impact parameter of 5.82 .  The variationwith time of the system potential energy and mean coordination number during this collision isshown in Figure 3.  (The definition of coordination for the latter quantity included a linear cut-off, chosen to maintain a nearest-neighbour character and thus a coordination number of 4 indiamond-structure bulk silicon.)  The time of first contact of the clusters is indicated by an arrowin the figure and corresponds to the sharp drop in PE as the clusters begin to mix.  It should benoted however, that the new Si8 cluster never attains its minimum energy of −37.2 eV during themixing phase, again emphasising the fact that its equilibrium structure is not significant underthese conditions. At t=3.8 ps, the mean coordination number drops briefly to its initial value asthe Si8 almost fragments into Si3 + Si5.  However, the clusters recombine before finally splitting(indicated by second arrow) as Si4 + Si4 at approximately 4.8 ps.  The final potential energy ofthe system is considerably lower (by roughly 1.4 eV) than that of the reactants.In conclusion, we have simulated over ninety collisions of Si3 with Si5 at a system temperatureof approximately 2000K.  Over 70% of the reactive collisions resulted in the production of Si4 +Si4, whereas Si + Si7 was never produced.  The equilibrium binding energies and associated magicnumbers have been shown to be unsatisfactory in describing the behaviour of the clusters underthese dynamic conditions.  We have introduced the concept of an effective binding energy thatincorporates the highly anharmonic dynamics of the clusters.  The magic numbers predicted bythese effective binding energies are in agreement with both the results of our simulations and8experiment.  In particular,  Si4 and Si6 are both shown to be magic whilst the prediction of theequilibrium values that Si7 is magic is shown to be incorrect.  The energy changes predicted usingthe effective binding energies also account (with the exception of Si8 which is unable toaccommodate all of the energy of the collision) for the relative abundances of reaction productsobserved in the simulations.ARP acknowledges the financial support of the University of York.References* E-mail address: arp28@phy.cam.ac.uk  E-mail address: rwg3@york.ac.uk[1] F.S. Khan and J.Q. Broughton, Phys. Rev., B39, 3688 (1989).[2] D. Tomnek and M.A. Schlter, Phys. Rev., B36, 1208 (1987).[3] K. J. Raghavachari and C. M. Rohlfing, J. Chem. Phys., 89, 2219 (1988).[4] W. L. Brown, R. R. Freeman, K. Raghavachari and M. Schlter, Science, 235, 860 (1987).[5] K. Laasonen and R.M. Nieminen, J. Phys. Cond. Matt., 2, 1509 (1990).[6] L. Goodwin, A.J. Skinner and D.G. Pettifor, Europhys. Lett., 9, 701 (1989).[7] J. M. Haile, Molecular Dynamics Simulation; Elementary Methods, (John Wiley, New York,1992).[8] To verify this, we continued one run for a single Si8 cluster with the smaller time step.  Themean PE of this run was Ð30.5 ± 0.8 eV which does not differ significantly from the value ofÐ30.7 ± 1.1 eV obtained with the normal time step.[9] Examination of the actual centre-of-mass kinetic energy for sample clusters indicates that theassumption of equipartition typically underestimates the effective binding energy per atom byless than 0.1 eV, which is generally within our statistical error bars.  Slightly largerunderestimates occasionally occur for the 2-atom cluster.9Table I.  Summary of the collisions performed and relative abundances of reaction products.Reaction products - number of occurrencesImpactparameter,b ()Number ofreactivecollisionsSi8 Si + Si7 Si2 + Si6 Si3 + Si5 Si4 + Si40.00 12 1 0 1 1 90.27 12 1 0 0 2 90.53 12 6 0 1 1 41.06 12 2 0 0 1 91.59 12 3 0 0 0 92.65 12 3 0 0 1 84.76 12 0 0 0 2 105.82 9 2 0 0 0 7Total No.of reactivecollisions:93Abundances of reactionproducts as % of the No.of reactive collisions:19.4% 0.0% 2.2% 8.6% 69.9%10Figure 1.  An illustration of the concept of an effective harmonic potential, used to characterizethe dynamic behaviour of a cluster exploring what is actually an anharmonic potential.  Thepotential energy is plotted schematically as a function of atomic configuration.11Figure 2.  The top graph shows the variation of binding energy per atom as a function of clustersize.  For the effective binding energy curve, values calculated from thre reaction products wereused with the reactant values for Si3 and Si5 shown as additional solid points.  An enhanced valuecompared to neighbouring values indicates that a cluster is more stable than expected and henceÔmagicÕ.  The lower plot of the fragmentation energy, BE(N)−BE(N−1), emphasizes the magicstatus of the 4- and 6-atom clusters when the effective binding energies incorporating dynamiceffects are used.  This is in contrast to the superimposed equilibrium curve which predicts thatSi7 is magic.12Figure 3.  The variation of potential energy (top) and mean coordination number (bottom)during an example collision with b=5.82 .  The collision proper begins at t=1.3 ps (first arrow)and is followed by a rapid fall in potential energy as the clusters mix.  At t=3.8 ps the meancoordination number falls as the Si8 almost splits into Si3 + Si5 before coming back together andfinally splitting as two Si4 clusters at t=4.8 ps (second arrow).  Snapshots of the atomicpositions are shown at 0.73, 1.45, 2.42, 3.75, 4.72 and 5.32 ps.

Influence of dynamics on magic numbers for silicon clusters

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Influence of Dynamics on Magic Numbers for Silicon Clusters

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