Stress generation mechanisms in carbon thin ﬁlms grown by

Acta Materialia 51 (2003) 5211–5222www.actamat-journals.com

Stress generation mechanisms in carbon thin films grown byion-beam deposition

Sulin Zhanga,∗, Harley T. Johnsonb, Gregory J. Wagnerc, Wing Kam Liua,K. Jimmy Hsiad

a Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USAb Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 60801, USA

c Sandia National Lab, P.O. Box 969, MS 9950, Livermore, CA 94551, USAd Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 60801, USA

Received 6 May 2003; received in revised form 7 July 2003; accepted 8 July 2003

Abstract

A three-dimensional molecular dynamics (MD) simulation is performed to study the stress generation mechanismsin carbon thin films grown by ion-beam deposition. The relationship between the kinetic energy of incident ions andthe steady-state film stress is established. Examination of the atomic stress and film microstructures reveals that thegrown films contain a significant fraction of vacancies, contradicting the presumption of the subplantation model. Bytaking into account both interstitials and vacancies, an analytical model is developed, in which the formation of thecompressive stress is attributed to competing mechanisms between generation and recovery of the defects. This modelcan satisfactorily explain the numerical observation in which compressive stress prevails in films in the presence ofvacancies. The present study provides useful insights into tailoring residual stress to control thin film curvature inmicroelectromechanical systems (MEMS) by ion-beam machining. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Carbon thin films; Residual stress; Interstitial; Vacancy

1. Introduction

Diamond-like carbon (DLC) films have foundwidespread use in the coating industry because oftheir structural and functional advantages. Interfa-cial delamination and substrate bending due to the

∗ Corresponding author. Tel.:+1-847-491-7065; fax:+1-847-491-3915.

E-mail address: [email protected] (S. Zhang).

1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.doi:10.1016/S1359-6454(03)00385-9

high compressive stress in thin films have becomemajor concerns limiting their performance. Despiteextensive studies of the delamination mechanismsand bending in compressively stressed thin filmson the micron scale[1,2], the origin of the com-pressive stress in thin films grown by ion-beamdeposition1 is not well understood and is the focusof the present study.

1 For the purpose of discussion, all energetic particleswhether charged or neutral are referred to here as ions.

5212 S. Zhang et al. / Acta Materialia 51 (2003) 5211–5222

When high-energy ions bombard a crystal dur-ing the ion-beam deposition, a cascade collisionprocess takes place. On the path of the impact ions,host atoms are displaced from their lattice sites,forming a displacement spike in which defects,such as interstitials and vacancies, are produced.Simultaneously, a significant fraction of the ionkinetic energy is transmitted to the solid in theform of concentrated lattice vibration, forming hotspots—thermal spikes. The thermal energy mayactivate the migration and recovery of defects. Thisprocess leads to a permanent rearrangement oflocal atoms in the impact area, thus substantiallymodifying the stress state of the solid.

To date, theoretical models for stress generationdue to ion bombardment are highly phenomeno-logical. Within the framework of elasticity, Guinan[3] obtained a rough estimation of thermal stressinduced by a single impinging ion in terms of thedissipated energy in the damaged zone. However,it is not straightforward to apply this model to thecases where a number of energetic ions sequen-tially impact the substrates, since the existing dam-aged regions may have a strong influence on thesubsequent production of defects. Based on theknock-on linear cascade theory, Windischmann [4]predicted a square-root dependence of compressivestress of films prepared by ion-beam sputtering onthe incident ion energy. This model, however, isnot applicable for sufficiently high ion energy.Davis [5] and Robertson [6] developed a subplant-ation model in which the compressive stress forma-tion is attributed to the implantation of energeticions into a subsurface layer of the film. Animportant implication of this model is that the filmis over-dense containing no voids, whereas filmsare usually under-dense based on experimentalmeasurements and numerical simulations.

A variety of extremely sensitive techniques havebeen used to monitor the stress evolution in situ ingrowing films during ion radiations [7–10]. Inthese experiments, curvature of the films is meas-ured and the stress in a thin film is calculated usingStoney’s equation. Volkert [7] performed in situwafer curvature measurements during amorphiz-ation of silicon by MeV ion bombardment. Sheobserved that the compressive stress increases andreaches a peak as amorphous regions are formed.

The stress then decreases and reaches a steady-state value as amorphization continues and eventu-ally saturates. Lee et al. [8] used a laser reflectionmethod to monitor the curvature change in DLCfilms during ion irradiations. They observed thatthe irradiations result in a decrease in the com-pressive stress in the DLC films, with final stressstate being mildly tensile. Van Dillen et al. [9]measured the stress change in alkali-borosilicateglass samples irradiated by 2 MeV Xe ions. A tran-sition from tensile to compressive stress wasobserved during the radiation. More recently, ion-beam machining techniques were successfullyemployed by Bifano et al. [10] to eliminate the cur-vature of thin films with thickness on the order ofseveral microns in microelectromechanical systems(MEMS). However, the atomic scale mechanismsof stress-state modification in the process of ion-beam machining are not addressed in their study.

On the numeric side, the cascade process by ionirradiation was simulated by Gibson et al. [11].Using a pair potential, they studied the orbits ofthe knock-on atoms and the subsequent damage. Itwas observed from their simulation that theresulting damaged configurations consist of a var-iety of interstitial–vacancy pairs, that is, Frenkeldefects. However, the relationship between thenumber of Frenkel defects and the knock-onenergy was not established due to insufficient sam-ple energies for the impact ions. Molecular dynam-ics (MD) simulations of stress evolution due to ionflux [12,13] have been limited to the two-dimen-sional cases. Muller [12] monitored microstructureevolution of Ni films bombarded by Ar neutrals. Itwas shown that the grown films are under tensilestress. The magnitude of the tensile stress isstrongly dependent on the defect concentration andvoid size distribution in the film, which is in turndetermined by the kinetic energy of the impingingparticles. Marks et al. [13] studied the relationshipbetween the ion-beam energy and the stress in agrowing film on a graphite substrate using MDsimulations. A transition from tensile to compress-ive stress was observed as the kinetic energy ofimpacting ions changes from 1 to 75 eV. Accord-ing to their simulation, incident ions insert them-selves onto the surface of the growing film ratherthan penetrate into the substrate, thus contradicting

5213S. Zhang et al. / Acta Materialia 51 (2003) 5211–5222

the subplantation model. Based on this obser-vation, they proposed a new model in which thecompressive stress in the grown films is due to abalance between impact-induced compression andthermal spike annealing. However, due to the con-straint of the third dimension in their simulation,their MD simulations may not be capable of cap-turing the realistic dynamics of atomic motion infilm growth.

The present study is motivated by the limitationsof the theoretical models as well as the discrep-ancies between theoretical modeling, MD simula-tions, and experimental observations. In the presentwork, a three-dimensional MD simulation is per-formed to simulate deposition of carbon atomsonto an initially perfect (1 1 1) diamond lattice.The residual stress in the growing films is evalu-ated during the deposition process. A relationshipbetween the steady-state film stress and kineticenergy of incident ions is established. Based on theatomic stress and the structure of grown films fromthe simulations, an analytical model, which takesinto account the competing mechanisms of defectproduction and recombination, is proposed to inter-pret stress generation in the deposition process.Results of the MD simulation and the model pro-vide guidelines for reducing the compressive stressin thin films.

The rest of the paper is organized as follows:Section 2 presents MD simulations of growing car-bon thin films by an ion-beam deposition process.The stress in the films grown by bombardment ofions with varying kinetic energies is calculated. InSection 3, basic material properties, such as themass density, the sp3 fraction, the microstructuresand stress distribution of the grown films are exam-ined. Stress generation mechanisms are discussed.Based on the numerical observation, an analyticalmodel is presented in Section 4 to interpret stressgeneration mechanisms. Discussions and con-clusions are presented in Section 5.

2. MD simulations: methods and results

2.1. Film growth

The method of using MD simulations for thinfilm growth by ion-beam deposition is described

elsewhere [14,15], and will therefore be outlinedhere briefly. The interatomic potential, �, betweencarbon atoms is of Tersoff–Brenner form [16,17].The total potential of a system containing N atoms,�, is the discrete summation of all the potentialsamong atoms, as

� �12�

i

�j � i

�ij(rij) (1)

where index i runs over all the atoms in the system,while index j runs over all the neighbors of atomi within a specified cut-off radius, rij is the spatialdistance between atoms i and j.

The initial substrate is of a perfect diamond lat-tice consisting of 672 carbon atoms with 56 atomsper layer. Prior to the deposition, the lattice isquasi-statically relaxed to its minimum potentialconfiguration using a conjugate–gradient method.Energetic carbon neutrals are then sequentiallydeposited onto the substrate with normal incidenceand randomly chosen locations. Periodic boundaryconditions are applied in the plane perpendicularto the film growth direction. The bottom two layersof the substrate are held fixed. Upon deposition,substrate atoms within a prescribed radius centeredat the depositing atom are completely free, whileall the other atoms are coupled to an external heatbath employing the method developed byBerendsen et al. [18], as illustrated in Fig. 1. Thetime constant in the Berendsen method is 125 fs.Based on the actual maximum velocity of theatoms in the system, a varying time-step strategyis introduced to accelerate the simulation. Betweentwo consecutive depositions, the system is thermo-stated for a relaxation time of approximately 10 ps.Over 900 carbon atoms are deposited to ensure theattainment of a steady-state film growth condition.

2.2. Stress at atomic scale

Within the framework of continuum mechanics,the Cauchy stress at a material point is defined interms of a resolved force on an area in the limitwhere the area tends to zero. Such a mechanicaldefinition of stress breaks down at the atomic scale.One of the commonly used definitions of stress formolecular systems is the virial stress. This defi-


Fig. 1. Initial (1 1 1) diamond lattice as a deposition substrate(red atoms are coupling to an external bath; green atoms areheld fixed; blue atoms are completely free). (a) side view, theblue atom on the top of the substrate is the depositing atom;(b) top view.

nition is based on a generalization of the virial the-orem of Clausius [19] for gas pressure. In thisdefinition, the average virial stress over an effec-tive volume can be written as a discrete sum of thecontribution from all the atoms in the domain ofvolume �:

s �1��N

i

�miri�ri �12�

j � i

rij�fij� (2)

where mi is the mass of atom i, ri is the timederivative of position ri, rij = ri�rj is the spatialvector between atoms i and j, and � denotes thetensor product of two vectors. The parameter N isthe total number of atoms in the domain. The

interatomic force fij applied on particle i by particlej is:

fij � �∂�ij

∂rij

rij

rij

(3)

The sign convention adopted here for force is posi-tive for repulsion and negative for attraction.Accordingly, a positive stress indicates com-pression and a negative stress indicates tension.

The stress expression in Eq. (2) includes twoparts. The first part stems from the kinetic energyof the atoms, while the second part is from theinteratomic forces. For solids, the kinetic energyterm is usually small compared to the interatomicforce term. Unless otherwise noted, this term isignored in our calculation.

In continuum mechanics, the volume of thedeformed configuration is generally calculatedusing the deformation gradient and the volume ofthe undeformed, initial configuration. For the depo-sition-formed volume, such initial atomic volumeis ambiguously defined. An alternative approach isto express the effective volume by � = N /r,where r is the atomic density in the computationaldomain, and is related to the average bond length,rav, by

rrdiamond

� �rdiamond

rav�3

(4)

where rdiamond is the atom number density per unitvolume of a perfect diamond lattice, and rdiamond isbond length of a diamond. The relation expressedin Eq. (4) is based on a comparison between anamorphous carbon structure and a perfect dia-mond lattice.

It should be noted that using Eq. (4) to estimatethe volume of an amorphous structure might causesystematic drift since it might not satisfy the sumrule.

2.3. Stress during film growth

Prior to deposition, the substrate is thermostatedat 300 K. Due to thermal expansion, a thermal mis-match exists between the bottom two rigid layersand those adjacent to them. Corresponding thermalmismatch stress can be calculated from the thermal


expansion coefficient of a diamond. In calculatingthe film stress, the thermal stress is ignored sincethe thermal stress is at least one order of magnitudesmaller than the overall stress caused by othereffects, as shall be seen later.

Simulations of film growth are performed withseven levels of ion energy: 1, 20, 40, 60, 80, 100and 150 eV, respectively. It is observed in oursimulations that surface sputtering occasionallyoccurs during depositions. Generally, the higherthe incident energy, the more frequent the sputter-ing. For example, the sputtering rate is approxi-mately 11% of the deposition rate when ion energyis 100 eV, while this number decreases to 6.8%when ion energy is 40 eV. Therefore, the net num-ber of atoms incorporated into the film is less thanthe number of atoms that are actually deposited. Incalculating the film stress, only atoms that fallabove the initial free surface of the substrate areconsidered. The film stress is calculated when thesystem is fully thermostated after each deposition.Our simulations show that the difference betweenthe in-plane stresses, sxx and syy is very small. Inaddition, the in-plane normal stresses sxx and syy

are at least one order of magnitude larger than theout-of-plane stress and all the shear stresses. Fig.2(a) shows stress evolution of a film grown with40 eV ions. In the initial stage of deposition, thebiaxial stress exhibits large fluctuations. Whenover 600 atoms are deposited, the stress graduallyapproaches a steady-state value of about 5.0 GPa.The film stress is taken to be the linear fit of thesteady-state portion of the stress evolution curve.Fig. 2(b) plots the mean biaxial stress, (sxx +syy) /2, as a function of kinetic energy of incidentions. It shows that the mean biaxial stress isstrongly dependent on the incident energy. Thefilm undergoes tensile stress at low ion energy. Atransition from tensile to compressive stress occursat around 10 eV. The compressive stress peakswhen the kinetic energy of the incident ion isaround 60 eV. Further increasing the ion energyleads to a gradual decrease in the compressivestress. The overall trend is similar to that predictedby Marks et al. [13], but the magnitude of the com-pressive stress predicted here is much less than thatreported by them. Moreover, the ion energy atwhich the compressive stress peaks is larger than

Fig. 2. (a) Compressive stress evolution in a growing film(ion energy = 40 eV); (b) mean biaxial stress versus incidentenergy (negative value stands for tensile stress, positive valuefor compressive stress).

their prediction. Ion energy larger than 150 eV isnot considered in the present simulation due to theincrease in the computational cost.

3. Stress generation mechanisms

The subplantation model predicts that the com-pressive stress in the film is caused by ion implan-tation-induced subsurface densification [5,6]. Thismodel implies that all the atoms in the sublayer of


the film undergo compressive stress, and thus isonly applicable to over-dense films. Based on atwo-dimensional MD simulation, Marks et al. [13]asserted that the compressive stress in thin films isnot due to the implantation of energetic ions. Theyproposed a new model in which the steady-statecompressive stress is due to a balance betweenimpact-induced compression and thermal spikeannealing.

The two models mentioned above are inconsist-ent with the observations of our simulation. In test-ing the validity of these models, basic properties ofthe grown films are examined. Fig. 3 shows atomicstructures of four films grown with 1, 20, 60 and100 eV ions, respectively. In each film, the redatoms represent those originally in the substrate;the green ones are deposited atoms. All the grownfilms are amorphous. For 1 eV impact, the sub-strate atoms and deposited atoms have no inter-penetration. For 20 eV impact, the impact ionspenetrate into the substrate to a depth of 2–3 layers.Some of the atoms originally in the substrate aremixed with the film atoms. As ion energyincreases, the penetration depth and degree of mix-ing between substrate and film atoms progressivelyincreases. The difference between our numericalobservation and that by Marks et al. is obvious:the degree of mixing between substrate atoms anddeposited atoms in our simulation is much largerthan that in their simulation. Besides using differ-ent interatomic potentials, the unrealistic materialsand unphysical constraint in the third dimension intheir two-dimensional simulation may account forthe discrepancy in the results.

Mass densities of the grown films are calculatedand shown in Fig. 4. Starting from maximum pen-etration of the incident ions at which the depth ofthe film is defined by d = 0, the grown films arepartitioned into slices with the same thickness inthe film growth direction. The original diamondsurface is at the depth of d = 5.3 . The numberof atoms in each slice is counted and the mass den-sity for each slice is calculated, and normalized bythat of a perfect diamond lattice. From the massdensity profile, three regions can be identified: asurface region with the lowest mass density (III);an intrinsic region in which the mass densityremains almost constant (II); and a transition

region to the substrate (I) [15], as seen in Fig. 4(a).The mass density of the grown films is taken tobe the average value over region II. Generally, therelationship of mass density versus incident ionenergy follows the same trend as that of stress ver-sus incident ion energy, as seen in Fig. 4(b).

Extensive research over the last decade has beencarried out to numerically or experimentally deter-mine the sp3 fraction in the DLC films, focusingon identifying an optimal deposition condition thatpromotes sp3 bonding [15,20–21]. On the onehand, sp3 bonding is desirable since it makes thinfilms both mechanically hard and chemically inert.However, a large fraction of sp3 bonding likelyresults in high, local compressive stress in the thinfilms. It is generally accepted that sp3 fractiondirectly reflects the damage of the film. The sp3

bonding is associated with interstitials, while thesp2 bonding is associated with vacancies. Fig. 5(a)shows the change of number of sp3-bonded atomsduring the growth of film with 60 eV ions. Priorto deposition, there are 560 atoms in the systemwith sp3 bonding. The rest of the atoms at the freesurface are with sp2 bonding. At the initial stageof deposition, the sp3 fraction decreases due to thedamage caused by ion implantation. After enoughatoms are deposited, the number of sp3-bondedatoms increases and gradually reaches a steadystate, where the number of sp3-bonded atomsincreases linearly with the number of atomsdeposited. The sp3 fraction of the film is taken tobe the slope of the linear fit. Fig. 5(b) shows thesp3 fraction in the film as a function of the ionenergy. The results here are consistent with thosepredicted by previous studies [15,20]. The trend ofsp3 fraction versus ion energy is similar to that ofthe film stress in Fig. 2(b), indicating an inherentrelation between the sp3 fraction and film stress.

Besides sp3 fraction, the local atomic stresspresents a useful insight into the distribution of thedefects in the film. Fig. 6 shows the atomic stressin the film grown with 1 and 60 eV ions, respect-ively, where the atoms in red undergo local tensilestress, while the atoms in blue undergo local com-pressive stress. A significant fraction of subsurfaceatoms undergo tensile stress, indicating that thefilm contains a large number of vacancies, regard-less of the overall stress state. It is well known that,


Fig. 3. Atomic structures of grown films (red atoms are originally in substrate, green atoms are deposited ones). (a) 1 eV; (b) 20eV; (c) 60 eV; (d) 100 eV.

upon energetic particle irradiation, interstitials andvacancies are often formed in pairs in crystals, adefect called Frenkel pairs [22]. It should be men-tioned that the amorphous structure and the smallsp3 fraction of the grown films indicates that the

films are not crystalline. Thus, the vacancy and theinterstitial mentioned here are not defined in theconventional sense. Instead, one can regard a vac-ancy as a free-volume-excess, and an interstitial asfree-volume-shortage in such amorphous structures.


Fig. 4. (a) Depth profile of mass density; (b) mass density ver-sus incident energy.

4. An analytical model

The above simulation results demonstrate that:(1) penetration occurs when the kinetic energy ofincident ions is beyond a critical value; (2) thegrown films are under-dense compared with thesubstrate; and (3) local atomic stress in the grownfilms is not uniformly in compression. The firstobservation contradicts with the observation fromthe two-dimensional simulation, and suggests thatthe resulting stress may be caused by differentmechanisms in different ion energy ranges. Thesecond and third observations indicate that the

Fig. 5. (a) Change in the number of sp3-bonded atoms duringdeposition; (b) sp3 fraction as a function of ion energy.

grown film contains not only interstitials, but alsovacancies. The existence of significant fraction ofvacancies in the films makes the subplantationmodel inapplicable. Motivated by these discrep-ancies between MD simulations and the restrictionof the subplantation model, in what follows, wepropose a new model to account for the stress gen-eration mechanism.

When striking the substrate, the energetic ionsdisplace atoms from their lattice sites, forming adisplacement spike in which defects, such as inter-stitials and vacancies, are produced. According tothe simple model of Kinchin and Pease [23], the


Fig. 6. Atomic stress (red atoms indicate neutral and positive stresses, blue atoms indicate negative stress). (a) 1 eV; (b) 60 eV.

number of displaced atoms produced per ion isgiven by Nd = Q /2Ed, where Q is the kineticenergy of incident ions, Ed is the threshold energyfor displacing an atom from its lattice site. Thethreshold energy depends not only on materialproperties, but also on the crystallographic direc-tion in which ions impact [23]. The produceddefects are primarily interstitial–vacancy pairs[11,21]. The rate with which the interstitial–vac-ancy pairs are created per unit area on the path ofions is related to the number of displaced atoms,Nd. Simultaneously, a significant fraction of thekinetic energy is transferred to concentrated latticevibration, forming thermal spikes during whichmembers of the defect pairs recombine. The num-ber of recombinations is related to the number ofjumps that a defect can make during the lifetimeof a thermal spike. According to the calculationdone by Seitz and Koehler [24], the number ofjumps a defect can make is proportional to(Q /E0)5 /3, where E0 is the activation energy of aninterstitial–vacancy pair, and is typically severaltimes less than Ed. The production and recombi-

nation of interstitial–vacancy pairs reach a steadystate at which the density of defects remains con-stant. The steady-state density of interstitial–vac-ancy pairs, rF, can be determined by the follow-ing equation:

rF

rA

fa � aNdfi�brF

rA�QE0�5/3

fi (5)

The left-hand side of Eq. (5) is the net incrementrate per unit area of interstitial–defect pairs. It isproportional to the flux of forming atoms, fa, andthe relative density of the defects. The variablerA is the density of the film. The first term on theright-hand side represents the rate per unit areawith which defects are produced, which is pro-portional to the ion flux, fi(fi � fa), and the num-ber of displaced atoms produced per incident ion,Nd. The second term is the rate per unit area withwhich the defects are recombined, which is pro-portional to the ion flux, the number of atoms thatreceive more than the recombination activationenergy, and the relative density per unit area of the


defects. The variables a and b are material-depen-dent coefficients. Solving Eq. (5), one has

rF

rA

�aQ /2Ed

fa /fi � b(Q /E0)5/3 (6)

The volumetric strain, e, is proportional to the den-sity of interstitial–vacancy pairs, and can be writ-ten as

e �VV

� krF

rA

(7)

where k is the ratio of the relaxation volume ofan interstitial–vacancy pair and the atomic volume.Within the framework of elasticity, the biaxialstress, s, in the film is given by

s �Y

2(1�2n)e (8)

where Y is the Young’s modulus of the film, n isthe Poisson’s ratio. Here positive stress representscompression. A simple analysis of Eqs. (6)–(8)shows that the steady-state stress linearly increasesat low incident energy (Q /E0�(fa /bfi)3 /5). Thecompressive stress reaches a peak at Q /E0 =(3fa /2bfi)3/5 with a peak value sP =

kY

2(1�2n)aQ /Ed

5fa /fi

, beyond which it decreases

according to a power-law (Q�2 /3) for high inci-dent energy. This prediction qualitatively agreeswith the results of the MD simulations presentedin the previous section. With proper choice ofmaterial parameters, the model prediction of thecompressive stress as a function of kinetic energyof the incident ions can be calculated using Eqs.(6)–(8), as shown in Fig. 7.

It should be mentioned that the analytical modelapplies only in the regime where ion penetrationoccurs (incident ion energy � 10 eV for thepresent case), and thus cannot be used to accountfor the tensile stress generated by low energy ions.In addition, the current model considers that thedamage of the film is composed of only interstitial–vacancy pairs, and ignores unpaired, isolated inter-stitials. Nevertheless, an important improvement ofour model over the subplanation model is that itcan explain the coexistence of compressive stressand vacancies in films.

Fig. 7. Compressive stress as a function of kinetic energy ofincident ion predicted by Eqs. (6)–(8). Here Ed = 8.0 eV, E0

= 3.0 eV, a = 1, b = 0.016, n = 0.2, k = 1, fa /fi = 0.8.

5. Concluding remarks

In the present work, MD simulations are perfor-med to investigate the stress generation mech-anisms during film growth by ion-beam deposition.The simulations show that the steady-state filmstress is found to be strongly dependent on the kin-etic energy of the incident ions. Although thisfinding is consistent with the two-dimensional MDsimulations by Marks et al. [13], the depositiondynamics are substantially different. The mixing ofthe substrate atoms and the newly deposited atomsindicates that energetic ions penetrate below thefilm surface when the kinetic energy of incidentions is sufficiently high. Detailed examinations ofthe atomic stress of the grown films provide strongevidence that the film, though in compression, con-tains not only interstitials, but also a significantfraction of vacancies. This observation contradictswith the presumption of the subplantation model.Based on these analyses, a new model is proposed,where stress generation is attributed to a competingmechanism of generation and recombination ofinterstitial–vacancy pairs. The theoretical predic-tion agrees with the MD simulations.

The deposition rate in our simulation is approxi-mately 10 orders of magnitude larger than that ina real experiment in which annealing plays an


important role in determining the final stress stateof the films. Within the MD simulation frameworkdescribed here, it may not be possible to fullyaccount for annealing during such a short elapsedtime between ion depositions. In fact, annealing isa thermally activated diffusion processes over along period of time during which defects heal. MDsimulation of such a long-period process would becomputationally prohibitive. In the present study,the annealing effect is incorporated into the ther-mostating process that is mimicked by imposingfriction among Langevin atoms. Though artificial,this method accelerates the annealing process, asconvinced by the observation of a similar defect-annihilation effect in the present MD simulations.While the simulations are carried out over onlyvery short time scales, they nevertheless demon-strate that a steady state has been reached. Thebasic mechanisms are believed to be valid overmuch longer times. These simulation results showthat damage due to energetic ions impacting on asurface may be used to modify the residual stressin thin films.

Due to the small time-step and the large numberof depositions, the present MD simulation isrestricted to a small computational domain. How-ever, the effective range of a displacement andthermal spikes generated by ions with energy lessthan 200 eV is typically on the order of 1 nm. Thesize of the computational domain chosen in oursimulations ensures that the physical process ofinterest, defect production and recombination,takes place within the domain. Analysis shows thatthe compressive stress generation mechanism isrelatively insensitive to the size of the compu-tational domain. We conducted a separate simul-ation with the substrate enlarged by a factor of 1.5in all the three dimensions, the stress in the filmgrown at 40 eV agrees with that obtained from thesmaller computational domain to within 5%. Tomore completely address the domain size effect,however, a multi-scale method bridging MD simul-ations and the continuum scale computationswould be necessary.

Besides presenting a mechanism for the forma-tion of compressive stress in the ion-beam depo-sition process, the results of the present study mayhave important implications in a novel approach of

tailoring thin film shape in MEMS [10], where finecontrol of the thin film curvature is required. Dueto residual stress associated with the micro-fabri-cation process, free-standing thin films are oftencurved with the radius of curvature in the range of10–100 mm. Based on the present study, impactingthe concave side of the curved thin film with ener-getic ions generates compressive stress near thesurface of the thin film. According to Bifano etal. [10], this can be used to effectively reduce thecurvature of the MEMS thin films. To develop amore precise method of controlling the curvatureof such free-standing thin film structures using ion-beam bombardment, work is presently underwayto establish relationships between the impact depth,compressive stress and kinetic energy of impactions.

Acknowledgements

S. Zhang gratefully acknowledges helpful dis-cussions with T. Zhu at MIT; H.T. Johnsonacknowledges the support of NSF grant number0223821; W.K. Liu acknowledges support fromNSF and NSF-IGERT; K.J. Hsia acknowledges thepartial financial support by NSF grant numberCMS 98-72306.

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Stress generation mechanisms in carbon thin ﬁlms grown by