Microscopic theory for nanoparticle-surface collisions in crystalline silicon

P. Valentini and T. Dumitrică*Department of Mechanical Engineering, Institute of Technology, University of Minnesota, Minneapolis, Minnesota 55455, USA

�Received 14 March 2007; published 8 June 2007�

We present an atomic-scale description for the impact of silicon nanospheres onto a silicon substrate undervarious temperature conditions. In spite of the relatively low impacting kinetic energies considered, of below1 eV/atom, the process exhibits a rich behavior. The rigid Hertzian model is valid for speeds below �500 m/s,while a quasiellipsoidal deformation regime is encountered at larger speeds. No nanoparticle bouncing wasobserved. For speeds up to �1000 m/s the result is the deposition of a crystalline Si nanostructure and creationof a long-lived coherent surface phonon. Higher speeds result in a rapid attenuation of the coherent phonon dueto an unexpected partial cubic diamond→�-tin phase transformation occurring in the particle.

DOI: 10.1103/PhysRevB.75.224106 PACS number�s�: 64.70.Nd, 82.20.Wt, 62.25.�g, 62.50.�p

INTRODUCTION

Motivated by various technological applications, molecu-lar dynamics �MD� studies of nanoparticle-surface collisionsare of considerable importance for understanding the funda-mental physics of the impact process.1–6 A main ingredientfor a successful MD simulation is the atomic potential em-ployed. In many situations, a simplistic Lennard-Jones- �LJ-�based atomistic description suffices. For example, the inter-est in nanoparticle filtration prompted investigations of small�0.5–2 nm in diameter� Ag, Ni, and silica clusters impactingon substrates with low thermal incident speeds7 �of up to afew hundred meters per second, �0.6 eV/atom�. It wasfound that particles preserve their initial structure and do notbounce off the surface. Other LJ-based MD simulations of asoft nanoparticle impacting on a rigid surface or a hard par-ticle on a surface have been performed with the perspectiveof using aerosol bombardment for removing “killer” con-taminant particles from microelectronic surfaces.2

Many MD impact studies involving nanoparticles are re-lated to materials growth applications, where typically moreadvanced atomistic potentials were used, including potentialsbased on the concept of coordination. The advances in thesynthesis of nanophase materials, i.e., solids composed ofnanoscale structural units, motivated simulations of �NaCl�32

�Ref. 3� and ichosahedral Cu147 �Ref. 4� clusters with speedsof 2 km/s or greater. In this regime, the direct impact on thesolid target resulted in significant disordering and spreadingof the cluster, while the presence of a liquid film can stronglyalleviate the damage of the impact with the solid substrate.Thin film formation by energetic cluster impact8 has stimu-lated several studies,5,6 mainly focusing on the high-kinetic-energy �KE� impact ��10 eV/atom� of small clusters con-taining up to a few thousand atoms. Systems studied haveincluded Mo1043 clusters impacting on Mo �001� surfaces5

and Cu2000 clusters on Cu �001� surfaces.6 Impact at theseenergies leads to massive structural disruptions. For instance,a Mo1043 cluster with KE 10 eV/atom ends up being burieddeep below the first surface layer, forming a shallow crater.5

New findings often come by exploring new regimes. Herewe are concerned with relatively large �up to 5 nm in radius�crystalline Si nanospheres9 impacting on a Si substrate withrelatively low KEs �up to 1 eV/atom�. MD simulations of the

tetrahedral network of Si requires more sophisticated poten-tials based on the concept of bond order.

The impacting conditions of interest here are experimen-tally achieved in hypersonic plasma particle deposition10–13

�HPPD� and are of interest for technologies such as themanufacturing of nanostructured layers with sprays of nano-particles. It was experimentally confirmed that the depositedmaterial in HPPD maintains a particulate structure. There is astrong correlation between physical and chemical propertiesof a nanomaterial and its ultrafine structure, and many rel-evant examples can be found among natural or synthetic ma-terials. For example, recent studies on outstanding naturalmaterials with a nanocomposite structure �bone, tooth, andnacre� revealed the existence of an optimum length scale forthe nanoconstituents that ensures a maximum strength androbustness.14 Thus, understanding the basic deposition pro-cess in HPPD, i.e., the impact of a single particle onto asurface, is an important step toward the further understand-ing and controlling of the deposited nanophase.

The relatively large size of the impacting nanoparticlesmakes it meaningful to inquire if aspects pertinent to theimpact of macroscopic particles, such as the contact mechan-ics and plastic deformations, are significant at the nanoscale.Hertz theory15 is a reference theory for the macroscopic col-lision of nonconforming bodies. In this model, the collidingbody is described in the lumped approximation of a perfectrigid body equipped frontally with a spring. The elastic de-formation upon impact is local and is captured by this spring,while the rigid part will move with the velocity of the centerof mass. It is interesting to see the extent to which the Hertzmodel is relevant at the nanoscale.

A manifestation of the roughness in the contact of realsurfaces is that the impact of macroscopic particles oftenresults in bouncing. This is because, on the loading stage,contact is developing just at selected points, through junc-tions between asperities. Such junctions can easily get bro-ken during unloading. The studying of the particle bouncingmode16 is of great interest for filtration and it was experimen-tally identified in impact studies involving micron-sizedparticles.17 However, it was not found in previous MD stud-ies of nanoparticle impact.18–20 The MD simulations willreveal whether the bouncing mode occurs in our collisionregime.

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Concerning the energy loss mechanism, many studies21,22

on the impact of micron-sized particles have attributed it tothe occurrence of plastic flow in the particle that has theeffect of reducing the contact pressure. Recent nanoindenta-tion data9 revealed that Si nanoparticles can accommodateplastic deformations. Thus, following conventional wisdom,one conjunctures that plastic yield may occur upon impactleading to a loss of energy, in agreement with macroscopicpredictions.15 However, one concern is that Hertz theorypredicts23 a decrease of the impact duration ��� with the par-ticle radius �r0� as ��r0

6/5. Thus, the resulting strain rateexperienced by the particle could be too high for inducingplastic deformations.

This paper provides a detailed microscopic description ofthe impact and shows that, in spite of the relatively low KEs,the process exhibits a surprisingly rich behavior. The reportis organized as follows. Section I gives a description of theMD scheme employed and details the quantities we monitorduring simulation. Section II presents the numerical resultsfor nanoparticle-surface impact. The main results are sum-marized in the last section.

I. METHODOLOGY

We conducted detailed atomistic simulations to describethe impact of Si nanoparticles on Si �001� surfaces. Themethod of investigation is classical MD,24,25 where the mo-tion of the Si cores is described with the potential proposedby Tersoff,26 incorporated in the computational packageTROCADERO.27 For comparison, selected runs were repeatedusing the more recent environment-dependent interatomicpotential28 �EDIP�. Note that both potentials are based on theconcept of bond order. Their strength is that both can be usedto describe the most stable Si phase with diamond structure,as well as various Si phases emerging under pressure ��-tin,simple cubic, fcc�. We also note that use of a classical treat-ment is justified as electronic excitation effects can be ne-glected due to both the low impact KEs and the semicon-ducting nature of Si. One advantage is that classicaltreatment of covalent bonding allows for simulations ofrather substantial systems. Here we could simulate twospherical Si particles of radii r01=2.5 nm and r02=5 nm con-taining Np1=3870 and Np2=31 075 atoms, respectively, anda Si substrate with dimensions 16.3�16.3�5.4 nm3 con-taining 72 000 atoms. A linked-cell list24 O�N� strategy tosearch for the interacting atoms was implemented, and thesystem was evolved in time with a 2 fs time step with asymplectic velocity Verlet algorithm.

The dynamical treatment of the finite-size substrate re-quires special care as artificial absorbing boundaries5,29 areneeded outside the region of interest. During the simulation,the two bottom layers �3600 atoms� of the substrate werekept fixed in time, the next ten layers �18 000 atoms� wereevolved with Langevin dynamics,5,25 while the rest of thesubstrate is treated with standard MD dynamics.25 The damp-ing coefficient in the Langevin dynamics was set to ��D /6,where �D is the Debye frequency of silicon. Periodic bound-ary conditions are applied to the horizontal x and y directions�the z impact direction is vertical�.

In preparation for the simulations, two nanoclusters havebeen spherically cut from bulk Si, annealed at Tp=1000 or300 K, and optimized with a conjugate gradient algorithm.The particular choice for a spherical geometry was motivatedby both the experimental observations of the particles pro-duced in HPPD,9 and by previous theoretical calculations30

that demonstrated the advantage of this shape. A silicon sub-strate was cut from bulk along the �001� direction and asimilar annealing-optimization procedure was applied. Theprocedure led to the expected Si �001� surface reconstructionby dimerization. Table I, which reports the cohesive energiesEp

0 for these structures, shows a relatively small decrease incohesion with increase in the surface-to-bulk ratio of thestructure. Recalling that the impact KEs are below1 eV/atom, we conjuncture that the particles will not frag-ment during the collision.

We attempted to choose simulation parameters �particlesizes, impact velocities, particle and substrate temperatures�and measure quantities that are relevant for the impact ex-periments of Rao et al.10,11 In a series of simulations, thespeed of the incident nanoparticle was taken in the range of550–1640 m/s. To explore the role of temperature, we car-ried simulations on combinations of cold �Tp=300 K� andhot �Tp=1000 K� particles impacting on a cold �Ts=300 K�or hot �Ts=800 K� substrate. For a better understanding ofthe process we found it necessary to conduct several simula-tions outside this main velocity-temperature range.

All simulations were conducted for a duration of 40 psand were followed by structural optimizations with a conju-gate gradient scheme. During the simulation, we have moni-tored a set of indicators. When pieced together, they shouldconverge to provide a microscopic picture for the dynamicsof the impact. The specific indicators are described below.

�a� Instantaneous global and local particle temperatureTp. To help monitor the energy flow, an instantaneous par-ticle temperature was defined as

Tp�t� =m

3NpkB�i=1

Np

vi2�t� , �1�

where vi2�t� is the square of the norm of the ith particle

velocity as measured in a coordinate system comoving withthe particle’s center of mass, kB is the Boltzmann constant,and m is the Si mass. Further, to account for the partitioningof the kinetic energy during the collision we superimposedonto the system a three-dimensional �3D� xyz grid with binsof 10�10�5 Å3 in size. Atoms were assigned to bins ateach time step based on their location. The local temperaturewas then calculated by averaging the thermal KE containedin each bin over 20 time steps �which proved to be a short

TABLE I. Initial cohesive energies �Ep0� for the two employed

particles, the substrate, and for bulk Si.

System

Si3,870 Si31,075 Si substrate Si bulk

Ep0 �eV/atom� −4.41 −4.51 −4.54 −4.62

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time compared to the characteristic time scale of the impactbased on the impact velocity� and using Eq. �1�.

�b� The contact radius a. The particle-surface contact ra-dius is a quantity of interest in macroscopic theories ofcollision.15 We have measured the same quantity at thenanoscale. For the definition of the contact radius, we fol-lowed the approach of Vergeles et al.:31 we imagined thecontact surface defined by the atoms in the particle �Na� di-rectly interacting with the substrate. Once the Na atoms wereidentified, the position of their center of mass Xc.m. and Yc.m.was computed. The radius a of the contact area is written

a2 =2

Na�i=1

Na

��xi − Xc.m.�2 + �yi − Yc.m.�2� . �2�

Here xi and yi are the coordinates of the Na atoms. We couldestablish an uncertainty of approximately 5% on a by vary-ing the cutoff parameter that identifies interacting atoms.

�c� The particle’s horizontal diameter �d. To get a quan-titative idea of the horizontal deformations in the nano-sphere, we recorded the deviations from the reference diam-eter 2r0. This was done by identifying the atoms withextreme xy coordinates, i.e., xmin, xmax, and ymin, ymax, andaveraging as

�d =1

2��xmax − xmin� + �ymax − ymin�� − 2r0. �3�

�d� The mean distributed contact stress pm. The stress atthe nanoparticle/substrate interface can indicate if stress-relieving mechanisms �such as plastic flow� occur during thecollision. Note that the stress distribution is not uniform overthe contact. An average contact stress was evaluated by com-puting the net vertical force acting on the nanoparticle anddividing that by the corresponding value for the circular areaof contact.

�e� The particle energy Ep and the particle-surface adhe-sion energy Ea. In discussing the energetics of the boundsystem Es-p, it is instructive to distinguish between theparticle-substrate adhesion energy and the individual cohe-sive energies. For this purpose, following the concluding re-laxation, we have artificially detached the particle and sepa-rately computed the particle �Ep� and substrate �Es� energies.The magnitude of the energetic deviations from the startingvalues entered in Table I will reflect the degree of structuralchange caused by the impact. As a result of the partial par-ticle surface coalescence, there is a gain in energy due to anoverall smaller total surface. The energy related with thissurface elimination process is a good measure of the particle-surface adhesion. Thus, we define the particle-surface adhe-sion energy as

Ea = Es + Ep − Es-p. �4�

Further, we define a specific adhesion energy as

�a =Ea

2�a2 , �5�

which can be seen as the energy needed to create two newsurfaces of unit area.

In the following, we define the maximum penetrationpoint as the moment in time corresponding to the zero trans-lational velocity of the incident particle.

II. SIMULATION RESULTS

In all our simulations, the impacting particles remainedattached to the surface. Figure 1 depicts three representativestages from the impact process of the r02=5 nm particle: theinitially undistorted structure, the instant of maximum pen-etration into the substrate, and the structure after 40 ps. Theselected outmost impacting conditions vividly illustrate thateven in this relatively low-energy impact regime, there is astrong qualitative dependence on the impact speed. At anincident speed of 550 m/s �Fig. 1�a�� the particle penetratesvery little into the room temperature substrate and sufferslittle structural damage following the impact. After the maxi-mum penetration point, on the rebound stage, a small junc-tion with a neck is formed at the particle-surface interface, ascan be observed in the last sequence. The overall result is thedeposition of a crystalline Si nanostructure. On the otherhand, at an incident speed of 1640 m/s �Fig. 1�b��, it is clearthat a bigger amount of the incident KE is transformed intodeformation and disruption of order. Indeed, the initiallycrystalline particle shows evident signs of structural changesat the maximum penetration point, as the red �dark gray�colored atoms now have a coordination of six. Additionally,the spherical shape is partially lost in favor of a quasiellip-soidal shape, and a junction with a large interfacial contactsurface is formed. In both cases, the substrate appears com-pliant and does not show any sign of significant structuraldeformation or any degree of implantation with foreignatoms.

FIG. 1. �Color online� Collision of a 5-nm-radius Si nanopar-ticle onto a Si �001� surface. �a� A slow cold impact �vp=546 m/s,Tp=Ts=300 K� and a fast hot one �vp=1640 m/s, Tp=1000 K, Ts

=800 K� �Ref. 32�. The yellow atoms �light gray� have a coordina-tion of 4, while red �dark gray� have a coordination of 6. Thesurface atoms �in blue� have a coordination of 3.

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A. The nanoparticle-surface adhesion

An interesting aspect of the dynamics of the collision pro-cess transpires from Fig. 2, which displays the time evolutionof the contact radius a in the same two impact conditions. Asthe particle impinges onto the substrate, a increases up to themaximum penetration point, marked with arrows. Next, forthe lowest impinging vp, a practically saturates at the valueattained at the maximum penetration distance into substrate.The higher impact vp leads to a less quasistatic behavior, asafter the maximum penetration point the contact area slightlydecreases on the rebound. Most importantly, this dynamicalresult suggests that the contact area is largely determined byvp and the mechanical properties of both particle and sub-strate, and to a lesser extent by the subsequent events thatfollow the maximum penetration.

A broader view about the role of the impacting KEemerges from Fig. 3, which plots the final contact radius a asa function of vp for the 5 nm impacting particle under roomtemperature conditions. To fully elucidate the vp dependenceit was rewarding to perform additional low-KE-impact simu-lations, outside our main range of interest. After a carefulanalysis we concluded that for speeds below 500 m/s theimpact largely follows the Hertzian predictions. �i� The mea-sured contact radii �empty circles� fit into the typical �vp

2/5

dependence, which assumes a localized particle deformationat the maximum penetration point. �The free term correctionreflects the atomistic aspect of the contact of nonconformingshapes.� �ii� The dynamically measured a values are veryclose with the predictions �empty triangles� made with theformula15 a=�pmr02/2E�, where pm represents the meancontact stress at the maximum penetration and E�=65 GPa isthe computed reduced particle-substrate modulus.33 Addi-tionally, �iii� Fig. 4, which plots the horizontal deviation �dfrom the original particle diameter, suggests that the particleexperiences little extended deformation during impact.

A qualitative scaling change can be seen in the a dataabove 500 m/s, where the Hertzian predictions clearly be-come less valid. The agreement with the stronger power�vp

1/2 scaling suggests the impact enters into an extendeddeformation regime. As noticed in Figs. 1�b� and 4, the par-ticle exhibits a quasiellipsoidal deformation at the maximumpenetration point, similar with the observed experimental be-havior of impacting gel balls.34,35

Turning now to the speed range of interest, Fig. 5�a� plots,for both particles, the vp dependence of the square of thecontact radius �rescaled by the square of the particle radius�as obtained under different temperature conditions. For bothparticles a rather linear dependence, with a similar slope, canbe noted. Small differences appear when the impact is themost energetic: For the smaller particle, a reaches the refer-ence particle radius r01. Under the same conditions, thelarger particle suffers slightly lower or larger deformations,depending on the temperature conditions. In summary, the�a /r0�2 ratio appears to be independent both of the particlesize and the temperatures that characterize the process, as thelarge overlap among the error bars suggests. A best fit to ourdata gives �a /r0�2=0.086+5.5�10−4 �s /m��vp.

The energetically favorable aspect of the deposition pro-cess is portrayed in the linear trend of the adhesion energy Eawith the incident kinetic energy of the particle, as presentedin Fig. 5�b�. A higher impact speed causes a larger contactsurface, thus “eliminating” more unfavorable surface. Onceagain, we notice that the thermal conditions are secondaryand Ea is mainly determined by the incident KE.

Encouraged by these results, we have next computed thespecific adhesion energy, as defined by Eq. �5�. The results

t [ps]

a[Å

]

0 10 20 30 400

20

40

60 vp = 1640 m/s

vp = 550 m/s

FIG. 2. �Color online� Time dependence of the contact radius forthe two simulations of Fig. 1. Down arrows mark the maximumpenetration instant.

vp [m/s]

a[Å

]

0 500 1000 1500 2000

10

20

30

40

50

601.65 vp

2/5 + 8.13

0.98 vp1/2 + 9.00

FIG. 3. Final a values �open circles� for the 5 nm particle androom temperature conditions as a function of vp. Empty trianglesmark Hertzian predictions for a.

t [ps]

δd[Å

]

0 5 10 15 20

0

2

4

6

8

10 vp = 330 m/svp = 440 m/svp = 550 m/svp = 820 m/svp = 1090 m/svp = 1640 m/s

FIG. 4. �Color online� Time evolution of the horizontal devia-tion �d from the r02=5 nm radius under different vp.

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presented in Fig. 5�c� show that, regardless of the particlespeed, �a is nearly constant: While the incident velocityshows a variation of more than 200%, �a remains onlywithin 20% of the average value, which we computed to be0.10 eV/Å2.

B. Collision modes

We next turn to a detailed analysis of the dynamics of theimpact, aiming to identify characteristic collision modes.Particularly useful for this purpose is the Ep data presented inFig. 6, where a thresholdlike dependance on vp can be noted.For collision speeds of up to �1000 m/s, variation with thenanoparticle speed exhibits a plateau and the differencesfrom the initial cohesive values of Table I can be attributed tothe local structural changes related to the neck formation atthe interface. Beyond the �1000 m/s value, there is a rapid

increase in Ep which clearly indicates that the nanoparticlesuffered a significant structural change after landing. Al-though a clear-cut critical speed value was not identified, ourdata suggest that, as a function of the incident speed, thereare two rather distinct impact modes, leading to differentfinal structures of the nanoparticle. Additionally, the thermalconditions of the impact appear to influence the impact out-come very little.

To better understand these two distinct collision modes,we have concentrated next on the dynamical features exhib-ited by two extreme cases of Fig. 1, namely, a slow impact atvp=546 m/s with Tp=Ts=300 K, and an energetic hot im-pact vp=1640 m/s with Tp=1000 K and Ts=800 K.

To gain insight about the transfer of the incident KE, wehave plotted in Fig. 7 the instantaneous speed of the nano-sphere’s center of mass. Figure 7 �left�, referring to the im-pact of the 550 m/s particle, indicates the creation of a trans-versal coherent phonon mode36 with a small dephasing time.Modeling the resulting oscillatory motion as an under-damped second-order system, i.e., vp�t��exp�−�nt�cos��n

�1−2t+�, where �n is the system’s naturalfrequency and is the dephasing parameter, we obtained �0.04. Thus, most of the incident KE is transferred to thecoherent transversal vibrations of the particle. On the otherhand, Fig. 7 �right�, corresponding to a 1640 m/s incidence,indicates the creation of the same type of mechanical surfaceoscillations with a more efficient energy dissipation. This isbest reflected in the dephasing parameter, which becomes �0.37.

We have next monitored the instantaneous particle tem-perature in Fig. 8 and the spatial distribution of temperatureacross the system in Fig. 9. Overall, in the low KE case theparticle temperature is simply enhanced to about 400 K. Thespatial decomposition of Fig. 9�a� reveals that at the touch-down instant only a small portion of the interface materialreaches a temperature of about 560 K. Next the temperatureis seen to rise rather uniformly, meaning that part of theinitial incident kinetic energy is converted into thermal en-ergy that moderately heats up the system.

A very different dependence of Tp is observed in the fastimpact case. Following the touch-down instant, Fig. 8 showsa sharp Tp rise to 1350 K. This ultrafast heating is followedby a slight cooling, and finally by an ample increase in Tp onthe rebound stage, up to a maximum of 1550 K. For theremaining 30 ps of the simulation, the particle temperatureslowly drops, as the system is driven towards equilibrium bythe Langevin dynamics of the substrate. Figure 9�b� clearlyindicates that the early temperature rise can be attributed to

FIG. 5. �Color online� vp dependence for the �a� square of thecontact radius �rescaled by the square of the particle radius�, �b�adhesion energy Ea, and �c� specific adhesion energy �a.

FIG. 6. �Color online� Final cohesive energy for the �a� 2.5 and�b� 5 nm particles.

FIG. 7. �Color online� Time dependence of particle speed vp.Arrows mark the maximum penetration instant.

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the localized increase of Tp at the interface, that can reachvalues up to 2400 K. A distinguished behavior is seen duringthe next few picoseconds: Instead of a homogenization, the2500 K front expands into the nanoparticle only, specificallyin the conical region corresponding to where the amorphiza-tion is found at the end of the simulation �see Fig. 1�. Next,as in the slow impact case, a rather uniform Tp distributioncan be seen towards the end of the simulation.

Particularly important are the results for the time evolu-tion for the contact stress pm, shown in Fig. 10. There are

obvious differences in the two cases: While pm exhibits largeoscillations for the slow impact, in the fast collision the firstpressure peak is quickly dampened and, shortly after the im-pact, the stress at the interface becomes zero. Interestingly,for the fast impact only the maximum pressure peak does notcorrespond to the maximum penetration point �marked bythe vertical arrow�, suggesting the early occurrence of astructural transformation able to relive the contact stress. Inboth cases, very large pressures are reached, around 13 GPafor the slow elastic regime, and around 18 GPa for the fastand damped one. Such stresses are amply within the range ofpressures capable of determining a phase transition for bulkSi �10–15 GPa for the first pressure-induced cubicdiamond→�-tin phase transformation37–39�.

C. Microscopic mechanism for the inelastic collision

What is the microscopic mechanism causing the above-threshold behavior? To answer this question, we have ana-lyzed in more detail the structural dynamics of our systemand investigated the possibility of an induced �-tin phase.We found that the extremely high pressures reached at thenanoparticle-substrate interface provide the explanation forwhy such a localized increase in temperature takes placeonly in the nanosphere, leaving the substrate almost unaf-fected. The microscopic picture is as follows. The nano-sphere responds to the high pressure rates experienced duringthe initial compressive stage of the impact with the creationin the most strained regions of a new �-tin phase. This way,a large amount of elastic energy accumulated during com-pression is not returned to the particle for rebounce. Thisstructural transformation is accompanied by a rise in Tp, asnoted in Fig. 8. Further, on the rebound stage the “hot” par-ticle undergoes a structural amorphization, leading to a fur-ther Tp increase. In fact, hardly any trace of the �-tin phase isfound in the nanoparticle at the end of the simulation. Theobserved amorphization during the rapid upward bouncingmotion appears in agreement with the amorphizationreported40,41 in bulk Si after a rapid retraction of a nanoin-dentor.

The above microscopic picture should be more evidentunder higher-KE and very low-temperature conditions. Toillustrate this point, we have performed an additional impact

FIG. 8. �Color online� Time dependence of 5 nm particle tem-perature Tp. Arrows mark the maximum penetration instant.

FIG. 9. �Color online� Temperature distribution at selected in-stants from the impact of the 5 nm particle: �a� slow cold and �b�fast hot impact. In both cases the first shown frame corresponds tothe touch-down instant.

t [ps]

p m[G

Pa]

0 10 20 30 40-15

-10

-5

0

5

10

15

20 vp = 546 m/svp = 1640 m/s

FIG. 10. �Color online� Time evolution of the mean contactstress pm for the 5 nm particle.

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simulation using the same r02=5 nm particle impacting at2200 m/s under very low temperature conditions �10 K forboth the particle and the substrate�. The evidence for a struc-tural transition in time is presented in Fig. 11, where thefraction of nanoparticle’s atoms having four, five, six, andseven nearest neighbors is plotted for the first 20 ps of theimpact. Almost 90% of the atoms are initially fourfold coor-dinated, as expected since most of the particle has a cubicdiamond structure; however, during the impact, the fractionof atoms having a higher coordination number �up to seven�increases rapidly, and at around 20 ps more than 40% of theparticle is made of atoms with a coordination number be-tween 5 and 7, with less than 15% of the atoms having onlyfour nearest neighbors. Note that the �-tin phase is charac-terized by a coordination number of 6.

To witness the phase transformation, we cut a slice of tenatomic layers from the particle center. Figure 12�a� showsthe touch-down instant, where the development of a strainfield in the nanosphere is visible, and the instant of maxi-mum penetration, where we found the largest amount of thenew �-tin phase. In addition to a detailed visual inspection,the precise evidence of an ephemeral �-tin phase at the3.9 ps instant is given in Fig. 12�b�, showing the radial dis-tribution function g�r� of the particle obtained after a struc-tural optimization. On the same figure we have plotted alsog�r� for the perfect infinite �-tin crystal computed based onthe lattice parameters of Ref. 42. Aside from the first peak,located at roughly 2.35 Å �i.e., the nearest-neighbor distancein the Si cubic diamond lattice�, caused by the residual origi-nal structure, the following spikes correspond well to thoseobtained from the perfect infinite crystal. On the other hand,the pair correlation at the end of the simulation, shown inFig. 12�c�, indicates the lack of �-tin phase as only peakscorresponding to the perfect Si cubic structure �vertical lines�are present.

D. Comparison with the EDIP potential

It is important to remember that, in spite of its ability todescribe the characteristics of both cubic and �-tin Si, theTersoff potential is lower performing at high temperatures,describing43 a melting point of bulk Si at 2547 K, well abovethe experimental one. This is a cause of concern since duringour simulation we see instantaneous local temperatures rais-

ing up to 2400 K. To evaluate the way in which this weak-ness in our potential affects the above impact outcomes, wehave repeated selected simulations with the more recentEDIP Si potential.28 This potential predicts44,45 a 1572 Kmelting temperature for Si, below the experimental 1683 Kvalue.

The EDIP simulations did not show any new qualitativefeatures, thus confirming the picture of the impact presentedabove. Quantitatively, we obtained the result that the EDIPpotential describes a somewhat softer character of the im-pact, with larger structural deformations in the nanoparticle.These differences are most obvious at high speeds and underhigh-temperature conditions. Figure 13�a� compares the ob-tained �a in the two models under the two outmost tempera-ture conditions. For room temperature deposition, both mod-els predict approximately the same �a. When the impactstake place at higher temperatures, the EDIP potential predictsa somewhat higher �a for all incident KEs. A similar pictureemerges from Fig. 13�b�. While room temperature differ-ences are negligible, the high-temperature conditions pro-

t [ps]

Fra

ctio

nof

atom

s[%

]

0 5 10 15 200

20

40

60

80

100

4567

FIG. 11. �Color online� Time evolution of the average numberof neighbors. The impact conditions were vp=2200 m/s, Tp=Ts

=10 K.

FIG. 12. �Color online� �a� Selected impact instants. The impactconditions were vp=2200 m/s, Tp=Ts=10 K. Pair-correlation func-tion of the particle at �b� the 3.9 ps instant and �c� the end of thesimulation.

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duce more damage in the particle as well as a less cleardelineation of the collision regimes. To visually illustrate thedifferences encountered under high KE and high tempera-tures, Fig. 13�c� displays the final optimized configurationsobtained with the two potentials under the same impact con-ditions �Tp=1000 K, Ts=800 K, and vp=1640 m/s�. Differ-ences are visible not only in the radius of the contact area butalso in the nanoparticle structure: With EDIP we obtained alarger conical volume containing amorphous Si material.Amorphization is present also on the particle’s surface, indi-cating the occurrence of surface melting phenomena. Addi-tionally, the amorphous state slightly extends into the sub-strate. This aspect is captured by Fig. 13�a� in the slightlydifferent values for �a under high temperatures.

SUMMARY AND CONCLUSION

We carried MD simulations using the transferable Tersoffpotential �and for selected runs, the EDIP� to study the non-equilibrium deposition process of a Si nanoparticle onto aSi�001� surface. Two representative impacting Si nano-spheres have been used in the simulations, with radii 2.5 and5 nm. All simulations indicated the deposition speed as acrucial factor and the particle and substrate temperature con-ditions as secondary.

It was particularly rewarding to examine at the nanoscalethe aspects pertinent to the macroscopic impact. We quanti-

fied the degree of adhesion through the microscopic contactradius a and revealed some interesting regularities: The con-tact radius at the end of the simulation is equal with contactradius attained at the maximum penetration point. For veloci-ties below 500 m/s, the obtained scaling a�vp

2/5 is in agree-ment with Hertzian predictions. Above this speed value thescaling law changes to a�vp

1/2 as the incident particle under-goes extensive ellipsoidal shape changes during the impact.Next the particle-surface adhesion energy showed a linearscaling Ea�vp. It follows that the specific adhesion energyattains a constant value �a0.1 eV/Å.

Concerning the aspect of incident KE dissipation, for bothparticles we have identified two distinct regimes: For speedsbelow �1000 m/s the structure and shape of the impactingnanoparticle are well preserved. A transversal coherent pho-non mode with a small dephasing time is produced. For amore violent impact �vp�1000 m/s�, there is an efficienttransfer of incident KE into the heating of the particle. Adistinguished nanoscale feature was revealed: the KE trans-fer is not mediated by a plastic yield as in the macroscopiccase but by an ephemeral cubic diamond→�-tin phase trans-formation taking place in the particle during the initial stagesof the impact.

Simulations with the EDIP potential showed qualitativeagreement in all important aspects. Overall, the EDIP poten-tial describes a somewhat softer impact, with larger contactarea and structural deformations in the nanoparticle. Moresignificant quantitative differences appear under highest tem-perature conditions.

Beside the fundamental aspect of describing, based on abond-order atomistic potential, the microscopic impact ofspherical Si nanoparticles with crystalline structure in a newvelocity-size regime, our results are particularly relevant forHPPD manufacturing. One practical goal was to provide in-sight into the particle adhesion and the structure of the de-posited unit. These two aspects are decoupled: While theadhesion energy decreases linearly with vp, the particle un-dergoes severe damage only above the vp�1000 m/s. Thus,deposition speeds of just below �1000 m/s ensures optimaladhesion while still preserving the size and crystalline struc-ture of the impacting nanoparticle units.

ACKNOWLEDGMENTS

We thank P. H. McMurry, S. L. Girshick, J. V. R. Heber-lein, D. Y. H. Pui, and W. W. Gerberich for stimulating dis-cussions, and R. Mukherjee for help. The work was sup-ported by NSF-NIRT CTS-0506748 and MRSEC-NSF underGrant No. DMR-0212302. Calculations were performed atthe University of Minnesota Supercomputing Institute.

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FIG. 13. �Color online� Comparison between the vp dependencefor �a� �a and �b� Ep with the two models. Comparison between thefinal nanoparticle structures described with Tersoff �left� and EDIP�right� potentials. Impact parameters were vp=1640 m/s, Tp

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