2000_seel_tensile stress evolution during deposition of volmer_weber thin films

JOURNAL OF APPLIED PHYSICS VOLUME 88, NUMBER 12 15 DECEMBER 2000

Tensile stress evolution during deposition of Volmer–Weber thin filmsSteven C. Seel and Carl V. Thompsona)

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139

Sean J. Hearne and Jerrold A. FloroSandia National Laboratories, Albuquerque, New Mexico 87185-1415

~Received 1 August 2000; accepted for publication 21 September 2000!

A simple model is presented that predicts the kinetics of tensile stress evolution during thedeposition of thin films that grow by the Volmer–Weber mechanism. The generation of a tensilestress was attributed to the impingement and coalescence of growing islands, while concurrent stressrelaxation was assumed to occur via a microstructure-dependent diffusive mechanism. To model theprocess of island coalescence, finite element methods were employed and yielded average tensilestresses more consistent with experimental observations than those predicted using previouslyreported analytical models. A computer simulation was developed that models the process of filmgrowth as the continuous nucleation of isolated islands, which grow at a constant rate to impingeand coalesce to form a continuous polycrystalline film. By incorporating the finite element resultsfor stress generation and a microstructure-dependent stress relaxation model, the simulationqualitatively reproduced the complex temperature-dependent trends observed fromin situmeasurements of stress evolution during the deposition of Ag thin films. The agreement includessimulation of the decreasingstress relaxation rate observed during deposition at increasingtemperatures. ©2000 American Institute of Physics.@S0021-8979~01!01501-8#

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I. INTRODUCTION

Stresses are commonly generated during the growtthin films on substrates. The reliability and performancethin film devices depend on the properties and behaviothe deposited films. Understanding the origins of stressecontrol the level of stress is of great importance for applitions in microelectronics, magnetic storage, and microetromechanical systems.

Measurements of stress during the early stage of deption of polycrystalline thin films have been performedmany researchers.1,2 For refractory materials with low adatom mobility, tensile stresses observed during depositioncreased with film thickness, becoming approximately cstant once the film became fully continuous. Tensile stresgreater than 1 GPa were measured in refractory matesuch as W, Ti, and Cr during deposition at room temperatFor fcc materials with high adatom mobility, tensile stresspeaked in the early stages of film growth and decreasebecame compressive for thicker continuous films. The mamum in the tensile stress during deposition of fcc matersuch as Ag, Cu, and Au at room temperature was onorder of 100 MPa.

For films that grow by the Volmer–Weber mechaniscrystallites of critical size nucleate on the surface of the sstrate as isolated islands. As the islands grow in diamethey impinge to form a network of islands, eventually colescing into a continuous polycrystalline film. As two islanimpinge and form a grain boundary at their intersectiopart of the free surface of each island is eliminated, resul

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in a significant energy reduction. Hoffman postulated thaneighboring islands are within close proximity, they wstretch toward each other to form a grain boundary to redthe interfacial energy at the expense of an associated senergy.3 Transmission electron microscopy~TEM! observa-tions of films at various stages of growth, coupled with stremeasurements, supported the idea that tensile stress getion during the early stages of film growth was associawith the process of impingement and coalescence of growislands.4–6

In these works, the maximum tensile stress measuduring deposition of high mobility materials was typically aorder of magnitude lower than that for low mobility materals. However, if low mobility materials were depositedhigher temperatures they behaved much like high mobimaterials.7 Conversely, high mobility materials depositedlower temperatures exhibited large intrinsic stresses simto low mobility materials.8 These observations are consistewith the existence of a diffusive stress relaxation mechanwith a strong dependence on temperature. Since the mistructural scale during deposition can be on the order ofÅ,6,9 diffusion distances along surfaces and grain boundawill be short so that significant stress relaxation may ocduring deposition.

In this study, we present a simple model for tensile strgeneration and relaxation that captures the kinetics of tenstress evolution during deposition of thin films that growthe Volmer–Weber mechanism. For comparison with preous analytical models for tensile stress generation, welized finite element methods~FEM! to model the island coalescence process. By minimizing the sum of the positstrain energy and associated reduction in interfacial ene

9 © 2000 American Institute of Physics

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7080 J. Appl. Phys., Vol. 88, No. 12, 15 December 2000 Seel et al.

the equilibrium configuration resulting from island coalecence was determined as a function of island radius.magnitude of the average tensile stress calculated using Fwere compared to the analytical model for island coalescestress presented by Nix and Clemens.10

We also performedin situ thin film stress-thickness measurements using a wafer-curvature based technique fordeposited on oxidized silicon substrates. Films were depited at different temperatures to study the kinetics of strevolution during deposition. TEM and scanning electron mcroscopy~SEM! were used to correlate the microstructuwith the measured stress-thickness product at a given thness. Qualitative comparisons were made between theperimental results and the model for tensile stress generaand relaxation. In addition, we have developed compusimulations that model the process of film growth from tinitial stages of island nucleation, through island growth aimpingement, to film continuity. Our goal is to reproduce tqualitative trends observed experimentally for tensile strevolution during film deposition, using the FEM results fstress generation and the proposed microstructure-depenstress relaxation model.

II. TENSILE STRESS GENERATION

Following Nix and Clemens,10 a model for calculatingthe tensile stress generation associated with island impiment and coalescence was examined. Consider an arratwo-dimensional islands with hemispherical shapes that clesce to form a surface with a cycloid shape, as shown in1~a!. The cusps at the surface of the islands were treatedcracks that allowed for a cracklike analysis similar to tGriffith criterion. The resulting expression for the averastress,s&, from this analysis of island impingement and colescence was given by10

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wheren is Poisson’s ratio,E is Young’s modulusgs is thesurface energy of the island,ggb is the grain boundary energy, andr is the radius of the island. An expression for theight of the grain boundary resulting from coalescence, acalled the zipping distance orz0 , for a hemispherical islandwas approximately given by10

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Using typical values for silver given in Table I, a grain rdius of 100 Å results in a zipping distance of 64 Å andaverage stress of 6.8 GPa, while a grain radius of 100gives a zipping distance of 360 Å and an average stres2.2 GPa.

The Nix–Clemens model provides an intuitive undestanding of how tensile stresses can be generated dudeposition, along with simple analytical expressions for cculating stress. However, the model predicts tensile strethat are significantly higher than those observed in expments. While observed stress levels may be mitigatedrelaxation processes, we examined the accuracy of the N


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Clemens analytical approach by performing FEM calcutions using the commercial software ADINA. An island warepresented by a two-dimensional element under plane sconditions and with perfect traction at the island-substrinterface, as shown in Fig. 1~b!. The plain strain conditionimposed in thexy plane implies that impingement is occuring between two infinitely long cylinders with semicirculacross sections. A series of displacements were imposed athe surface to a heightz0 to mimic the zipping process. Forgiven island radius, the positive strain energy from FEmodeling and the associated reduction in interfacial enewere calculated as a function of the zipping distance. Tsum of these two energies represents the change in enerthe system and the negative-valued minimum correspondthe equilibrium value of the zipping distance. For compason with the values calculated from the Nix–Clemens modthe zipping distance and average stress versus island rafrom the FEM modeling are shown in Fig. 2. From the FEmodel with plane strain conditions and traction at the islasubstrate interface, the zipping distance decreases withisland radius raised to the 0.675 power, compared to the 0dependence in the Nix–Clemens model, while the averstress has an exponential dependence of20.814 on the is-land radius, compared to an inverse square root depend

FIG. 1. ~a! Schematic of the island impingement and coalescence proresulting in tensile stress generation. The dashed lines represent thespherical island at impingement, while the solid lines represents the cycsurface of the island after coalescence through grain boundary zippingheight z0 . ~b! FEM model of island coalescence represented by a twdimensional element under plane strain conditions. Thex axis represents theisland-substrate interface where traction was imposed. They axis is an axisof symmetry along which sliding was allowed. The arrowed lines areseries of displacement that represent zipping to a heightz0 of an island ofradiusr.


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7081J. Appl. Phys., Vol. 88, No. 12, 15 December 2000 Seel et al.

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TABLE I. Materials properties for Ag at room temperature.

MaterialYoung’s modulus,a

E ~GPa! Poisson ratio,a nSurface energy,b

gs ~J/m2!Grain boundary

energy,b ggb ~J/m2!

Ag 87.3 0.354 1.5 0.47

aG. Simmons and H. Wang,Single Crystal Elastic Constants and Calculated Aggregate Properties: A Habook, 2nd ed.~M.I.T. Press, Cambridge, MA, 1971!.

bL. E. Murr, Interfacial Phenomena in Metals and Alloys~Addison-Wesley, Reading, MA, 1975!.

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in the Nix–Clemens model. The material properties forused in the calculation of Fig. 2 for both the FEM results athe Nix–Clemens model are shown in Table I.

The Nix–Clemens model for tensile stress generationsulting from island coalescence utilizes an analysis originused to describe surface roughening.11 The geometry of theisland coalescence and surface roughening problems arcompletely analogous which may cause the Nix–Clemmodel to overestimate the stress. Our FEM model of islacoalescence represents island zipping in a straightforwmanner and predicts smaller stress more consistent withperimental observations.

During the growth of a real film, impingement of a priodic array of islands does not occur as described beforelow substrate coverage, island impingement will typicainvolve an island that has not yet impinged with any othislands. To model the first coalescence of an island usFEM, an island was represented as a plane strain tdimensional element with either traction at the islan

FIG. 2. Comparison of FEM calculations with results from the NixClemens model, showing~a! the zipping distance,z0 , divided by the islandradius,r, and~b! the average stress as a function of the island radius.

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substrate interface@island ~i! in Fig. 3~a!# or with sliding atthe island-substrate interface@island ~iii ! in Fig. 3~a!#. Thezipping due to the single coalescence was representeddisplacements imposed along only one side of the islaThe equilibrium zipping distance@Fig. 3~b!# and averagestress@Fig. 3~c!# versus island radius resulting from a sing

FIG. 3. ~a! Schematic of island coalescence showing~i! the first and~ii ! thesecond coalescence of an island with traction at the island-substrateface, and~iii ! the first and~iv! the second coalescence of an island wsliding at the island-substrate interface.~b! The equilibrium zipping dis-tance,z0 , divided by the island radius,r, and ~c! the average stress asfunction of the island radius from FEM modeling of a two-dimensionelement under plane strain conditions for the cases shown in~a!.


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coalescence are shown for both the cases of tractionsliding at the island-substrate interface. If traction was iposed at the substrate, the coalescing island stretchedwards the forming grain boundary to eliminate part ofsurface. Since island movement was constrained due totion, tensile stresses were generated in the island. If slidwas allowed along the substrate, the island moved its cetowards the forming grain boundary during zipping, resultiin a very small, compressive mean stress. However, withtraction, no load transfer between the island and substcan occur, so the stress will not result in a substrate cuture.

At higher substrate coverage, an impinging island mlikely will have already coalesced with another island foring what will be referred to as an island cluster. The secocoalescence of an island was modeled using a plane selement with one edge pinned and displacements impoalong the opposite edge to mimic zipping to a heightz0 . Forthe case with traction at the island-substrate interface,second coalescence@see island~ii ! in Fig. 3~a!# approxi-mately doubled the average stress in the island, as showFig. 3~c!. If sliding at the island-substrate interface is alowed, the consequence of subsequent impingements isobvious. If the entire island cluster is able to slide, thenland coalescence will result in a slightly compressive streas shown previously. However, sliding of an island maycome inhibited after a few coalescence events, as discuin the following paragraph. If sliding was allowed along thisland-substrate interface but with one edge of the islpinned, the second coalescence@see island~iv! in Fig. 3~a!#generates tensile stresses similar in magnitude to thecoalescence of an island with traction, as shown in Fig. 3~c!.

The mechanism responsible for island sliding mustexamined more closely to understand how load transfertween the film and substrate can occur even with interfsliding. When islands coalesce, large shear stresses areerated near the grain boundary due to the displacementsulting from zipping. These shear stresses can drivemovement of dislocation-like entities along the filmsubstrate interface away from the grain boundary.12 Islandsliding will result if the ‘‘dislocation’’ can travel across thentire island. As the substrate coverage increases, moslands have coalesced with two or more islands. The moment of the dislocation-like entities across the entire islawill be opposed by shear stresses from previous coalescevents. Consequently, we would expect that sliding onisland will become inhibited after the island has coaleswith a few other islands. If island sliding is inhibited, subsquent coalescence of an island will generate tensile strein the film, as shown in Fig. 3~c!. Since traction now existsbetween the island and the substrate, the tensile stresseration will results in a substrate curvature. Ifperfectslidingoccurs at the island-substrate interface, no load transferoccur between the film and substrate and consequentlysubstrate curvature will be measured.

A potentially more accurate representation of the geoetry of island coalescence would use an axially symmecoalescence analogous to contacting two spheres. Howthe axisymmetric representation conflicts with the bound


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condition of perfect traction at the island-substrate interfaThe islands attempt to zip towards each other in the planthe substrate but are prevented from doing so by the tracwith the substrate. Even though the geometries are differthe axisymmetric model with sliding was found by FEMgive quantitatively similar average stresses resulting frisland impingement compared to the plane strain case wsliding. Since both the traction and sliding cases cantreated with the plane strain geometry, the plane strain fielement results were implemented in the modeling of islaimpingement.

III. STRESS RELAXATION

Tensile stress generation resulting from island coacence occurs due to the localized displacements at the gboundary. One possible mechanism by which the tenstress can relax is through transport of matter to the straregion within the grain boundary as shown schematicallyFig. 4. A fast diffusion path for atoms is along the islansurface, which acts as a source of atoms, and down the gboundary. Since both surface and grain boundary diffusare required in series, either diffusive mechanism mayrate limiting. With or without sliding at the film-substratinterface, the stress relaxation rate,s, will take the form13–15

s52C0

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whereC0 is a material-dependent, temperature- and streindependent constant,s is the average stress in the film,h isthe film thickness,k is Boltzmann’s constant,T is tempera-ture in Kelvin, andQ is the activation energy for the ratelimiting diffusive process, either grain boundary or surfadiffusion. Implicit in this expression is that the grain sizscales with the film thickness. Unlike the uniform straiassociated with lattice mismatch or thermal expansion mmatch, the strains created by island coalescence resultlocalized surface displacements due to island zipping. Csequently, matter diffusing to the grain boundaries can reall of the tensile stress generated by island coalesceSince the diffusion distance along the surface and grboundary is very short, this diffusive process may be anportant stress relief mechanism even at low temperature

FIG. 4. Proposed stress relaxation mechanism involving the diffusionatoms across the surface of the island and along grain boundaries tstrained regions near the grain boundary. Since surface diffusion andboundary diffusion occur in series, either process may be rate limiting.


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IV. IN SITU FILM STRESS MEASUREMENTS

Ag films were grown by electron-beam evaporation inultra-high vacuum~base pressure of 1310210Torr, deposi-tion pressure of 531029– 331028 Torr!. The 100-mm-thickSi ~001! substrates had either a native oxide or a thermgrown oxide~similar results were obtained for both typessubstrates!. The substrates were prepared using a solvclean followed by a sulfuric acid-hydrogen peroxide etprior to loading in the vacuum chamber. The substrate woutgassed at 350 °C for 1 hin vacuoat a pressure below 5310210Torr. Prior to deposition, the substrate was allowto equilibrate at the desired growth temperature for at leah. The temperature during deposition was monitored usinthermocouple metal bonded to a witness wafer, anddeposition rate was controlled using a quartz crystal moniWafer curvature was measuredin situ using a sensitivemultibeam optical deflection technique.16 If the thickness ofa film is much less than the substrate thickness, the proof the average film stress and film thickness can be relatethe substrate curvature using Stoney’s equation, whichcommonly used to interpret substrate curvatumeasurements.17,18 The stress-thickness was determinfrom substrate curvature measurements during growth ofat a constant deposition rate of 2 Å/s at various substtemperatures@Fig. 5~a!#. For comparison, we have replottethe data in Fig. 5~b! as the stress-thickness product dividby the nominal film thickness. Since the sensitivity of t

FIG. 5. ~a! Stress-thickness vs nominal film thickness for Ag thin filmdeposited at 2 Å/s on oxidized Si substrates. Inset is an enlargement oinitial 200 Å. ~b! The same data replotted as stress-thickness divided bynominal film thickness.


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curvature measurements is related to the stress-thickproduct, the calculated values in Fig. 5~b! at very small filmthicknesses tend to be noisy.

In order to study the evolution of microstructure, filmhaving a laterally graded thickness were grown by movinshutter across the wafer during deposition.Ex situplan-viewTEM micrographs and SEM images were then used to chacterize the microstructure as a function of the nominal fithickness. Figure 6 shows a typical sequence of TEM micgraphs at different nominal film thicknesses of Ag deposiat 30 °C and at a rate of 2 Å/s. Consistent with previostudies,2,6 the maximum in the tensile stress-thickness fordeposition temperatures coincided with the late channel swhen the fractional substrate coverage was greater than 0

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FIG. 6. Sequence of plan-view TEM images of Ag films at nominal thicnesses of~a! 120, ~b! 160, and~c! 260 Å.


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V. SIMULATION OF THIN FILM GROWTH

The growth of thin films by the Volmer–Weber mechnism involves the nucleation and growth of individual ilands that impinge and coalesce to form a continuous filmshown in a sequence of simulated structures in Fig. 7. Tfilm growth was modeled in a two-dimensional simulatiby tracking the nucleation and growth of circles, which reresent the intersection of islands with the substrate. For splicity, the film-substrate interface energy was assumedequal the substrate surface energy so that the nucleatelands have hemispherical shapes~i.e., the island heightequals the radius of the island!. Nucleation was assumed toccur continuously during deposition at a constant rate

FIG. 7. Sequence of simulated structures of polycrystalline film formaby a Volmer–Weber mechanism, under conditions of continuous nucleaand a constant radial growth velocity.


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that the added number of new nuclei per unit time per uexposed substrate area was constant. The nucleated isgrow with a constant radial growth velocity proportionalthe deposition rate.19 At early times, most of the islands havimpinged with less than two other islands and therefore hformed at most one grain boundary. As islands grow larand more impingements occur, triple points are formwhere grain boundaries meet within an ensemble of thislands or grains. Eventually, all grain boundaries terminat triple points when the film is fully continuous. The asumed growth conditions of continuous nucleation at a cstant rate and a constant radial growth velocity result incontinuous film composed of grains with a Johnson–Mstructure.20

In the simulation, once two islands impinged, the eqlibrium zipping distance and stresses within the island wcalculated using the FEM results for the plane strain geoetry with traction at the island-substrate interface. The cwith sliding at the island-substrate interface will be consered later. For each new coalescence, the average stresculated using the FEM results was simply added to anyisting average stress in the island. The principalsuperposition states that two strains may be combineddirect superposition, with the order of application havingeffect on the final strain of the body. However, for subsquent coalescence events, the calculation of the strainergy, used to determine the equilibrium zipping distance athe corresponding average stress, neglected the exisstresses and strains in the island. Once coalescence bettwo islands occurred, further lengthening of the grain bouary due to island growth was assumed to generate no ational stress. However, continued deposition was specifieoccur ‘‘epitaxially’’ so that new material inherited the streof the underlying layer.21

During deposition, islands with different radii will impinge and behave differently from the symmetric cases csidered previously. In the simulation, the energy minimiziz0 was determined for the coalescence of islands with dsimilar sizes using the FEM calculations of strain energydifferent zipping distances as a function of island radiustwo islands with dissimilar sizes impinged, the boundarythe smaller island was found to zip more than if it had colesced with an island of the same size. Consequently,stress in the smaller island impinging on a larger island wgreater than if it had coalesced with an island of the sasize. The opposite trends are true for the larger island.

Stresses generated by island coalescence were assto be relaxed by a microstructure-dependent diffusive strrelaxation mechanism, similar to that described by Eq.~3!.The proposed expression for stress relaxation is the mappropriate for continuous films since the stresses areproximately equibiaxial and the grains have formed bouaries on all sides. However, after a single coalescenceisland has only one grain boundary and the stresses areequibiaxial. Nonetheless, the form of the stress relaxarate given by Eq.~3! captures the origin of the driving forcfor relaxation and the microstructural dependence ofmechanism. The stress relaxation rate given by Eq.~3! wasused in the simulation with the film thickness term replac

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by the island height, and using the average stress in thland. This implementation implies that stress relaxationeach individual island can be treated independently. Thapproximations are not expected to significantly affectresults of the simulation.

The simulation of stress evolution during thin film depsition is based upon the nucleation and growth simulatcoupled with the FEM results for stress generation andanalytical model for the microstructure-dependent stresslaxation mechanism. The experimental inputs to the simution were the deposition rate and temperature, which baffect the rates of stress generation and relaxation.physical dimensions of the simulation are unitless so a sing factor must be defined that relates the microstructudimensions of the real film to the simulated structure. FrSEM images of discontinuous Ag films, the fractional sustrate coverage versus film thickness was measured fodeposition conditions. For each deposition temperaturescaling factor was determined that gave reasonable agment between the simulated microstructure and the meascoverage versus nominal film thickness, as shown in FigThis scaling factor influences both the island-size-dependstress generation model and the microstructure-depenstress relaxation model. In addition, a diffusivity for the ralimiting diffusive process of the stress relaxation mechanmust be supplied. An activation energy of approximatelyeV and a pre-exponential factor of an appropriate magnitwere chosen to best match the simulation to the experimtally measured stress-thickness versus thickness curvedifferent deposition temperatures. For each deposition tperature, results were averaged over approximately 2island/grains from ten simulations using different randostarting seeds. Assuming traction at the island-substrateterface, the average stress-thickness versus thickness anerage stress versus thickness from simulations of Ag fideposited at different temperatures are shown in Fig. 9.

In another set of simulations, the equilibrium zippindistance and average stress were calculated using theresults with sliding at the island-substrate interface. Contrto the case with traction, two different types of coalesce

FIG. 8. Fractional substrate coverage vs film thickness from SEM imaduring various stages of deposition of Ag at different temperatures.lines are from simulations of microstructural evolution. For each depositemperature, a scaling factor was determined that gave reasonable ament between the simulated microstructure and the measured coveranominal film thickness.


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events can occur. We have assumed that the first few coacence events for a given island will occur with sliding sinthe movement of dislocation-like entities will be relativeunopposed. Increasing the number of coalescence eventfore sliding is inhibited will delay the onset of the rise of thtensile stress until larger thickness. If a coalescing islandslide easily, the average stress resulting from coalescenslightly compressive@see Fig. 3~c!#. However, without anyregions of traction, there is no load transfer from the islato the substrate and no substrate curvature will result. Ssequent coalescence events were assumed to generate tstress since sliding is inhibited within an island, as describin the previous section. These tensile stresses were aassumed to be relaxed by the proposed diffusive mechanOtherwise, the simulations were run under exactly the saconditions as for the case with island-substrate traction.der conditions of island-substrate sliding, simulations ofaverage stress-thickness versus thickness and averageversus thickness were performed for Ag films depositeddifferent temperatures as shown in Fig. 10. For the simutions shown in Fig. 10, islands with fewer than four impingments were assumed to slide, although values of two or thimpingements gave qualitatively similar results.

VI. DISCUSSION

The measured stress-thickness versus thickness cuexhibited four distinct regimes at different nominal film

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FIG. 9. Simulation of Ag thin films with traction at the film-substrate inteface during deposition at 2 Å/s at different temperatures showing,~a! stress-thickness product vs nominal film thickness, and~b! average stress vs nominal film thickness. The simulated microstructure was formed by continunucleation at a constant rate and using a constant radial growth veloproportional to the deposition rate. Stress generation was modeled usinFEM approach, while stress relaxation was assumed to occur vmicrostructure-dependent diffusion mechanism.


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thicknesses. During early deposition times, no appreciastress-thickness was measured up to a critical thicknwhich increased with increasing deposition temperatureseen in Fig. 5~a!. From TEM micrographs, the island densiwas initially very high but decreased with increasing fithickness, indicating the presence of island coarseningsurface diffusion and/or grain growth. If neighboring islanhave different radii then the gradient in chemical potenmay be sufficient to drive island coarsening by surfacefusion, especially at higher temperatures. However, if islaare approximately the same size or once islands are largegradient will be smaller and impinging islands will formgrain boundaries rather than undergoing coarsening.simulation of film growth does not include island coarsenso all island impingements resulted in stress generationthe simulation with traction at the island-substrate interfathe onset of an appreciable stress-thickness occurredmuch smaller film thickness@inset of Fig. 9~a!# than wasobserved in experiments@inset of Fig. 5~a!#.

Even once numerous grain boundaries had formedseen in Fig. 6~a! for a 120 Å Ag film deposited at 30 °C, thmeasured tensile stress-thickness in the film was still vlow. Because the stress relaxation mechanism is stronglypendent on the island size, the tensile stresses generateresult of the initial coalescence are expected to be relavery quickly. However even with a microstructure-dependstress relaxation mechanism, the simulated stress-thick

FIG. 10. Simulation of Ag thin films with sliding at the film-substrate interface during deposition at 2 Å/s at different temperatures showing,~a!stress-thickness product vs nominal film thickness, and~b! average stress vsnominal film thickness. Islands with fewer than four impingements wassumed to slide, while subsequent coalescence events producedstresses. Otherwise, the simulations were performed under the same ctions as those shown in Fig. 9.


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for the case of island-substrate traction was appreciable ebefore a nominal film thickness of 100 Å, as seen in the inof Fig. 9~a!. If sliding along the island substrate was asumed, the first four coalescence events for a given islgenerated no measurable stress, as described in the presection. This had the effect of retarding tensile stress gention in the early stages of film growth, as shown in the inof Fig. 10~a!. Since multiply-impinged islands were assumto support tensile stresses, the average stress increasedmatically as the film approached a percolated structure. FAg film deposited at 30 °C, percolation occurred around 1Å which approximately coincided with the significant increase in the tensile stress-thickness at the same film thness as seen in Fig. 5~a!. In the simulation, the greater thnumber of coalescence events that occurred before islsubstrate sliding was inhibited, the larger the thicknesswhich an appreciable tensile stress-thickness occurred.

The second feature of the stress-thickness versus thness measurements to be considered is the magnitude omaximum tensile stress-thickness and the film thickneswhich the maximum occurred. From Fig. 5~a!, the magnitudeof the maximum tensile stress-thickness decreased~and thefilm thickness coinciding with the maximum increased! withincreasing deposition temperature. For all deposition teperatures, the maximum tensile stress-thickness occuwhen the film was in the late channel stage just before ctinuity, as confirmed using SEM. Since continuity occurrat larger film thicknesses with increasing deposition tempeture, the average grain size at continuity must similarlylarger at higher deposition temperatures. Based on the Fresults in Fig. 3~c!, the magnitude of the maximum tensistress should decrease with increasing grain size. Howepredicting the relative magnitudes of the maximum strethickness with deposition temperature also requires coneration of the influence of stress relaxation.

The simulation, which accounted for both stress genetion and relaxation, reproduced the experimental trendsdecreasing maximum tensile stress-thickness that occurrelarger film thicknesses with increasing deposition tempeture. In addition to the experimental variables of deposittemperature and rate, two other parameters had to beplied to simulate the stress evolution during thin film depsition. Since the dimensions of the simulation are unitlethe island or grain size scaling of the simulation mustdetermined by matching the fractional substrate coverversus film thickness from the simulation to results froSEM images of Ag films at different nominal thickness,shown in Fig. 8. The island/grain size scaling influences bthe stress generation@see Fig. 3~c!# and the stress relaxatiorate@see Eq.~3!#, and roughly determines the film thicknesat which the maximum in the tensile stress-thickness occsince the film thickness at continuity generally scales wthe grain size. The second parameter required for the silation was the diffusivity for the rate-limiting process fostress relaxation. With an activation energy of 0.3 eV andappropriate pre-exponential factor, the simulation couldused to predict stress-thicknesses within a factor of 2 ofexperimental measurements for all deposition temperatuIf an activation energy of 0.6 eV was used, as reported

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Kobrinsky and Thompson22 for diffusive stress relaxation incontinuous Ag films, the simulation produced the saqualitative trends with deposition temperature, but usinvalue of 0.3 eV gave better agreement with the experimeresults.

The third feature of the stress-thickness versus thickncurves to be considered is the observed temperature dedence of the negative slope region immediately followifilm continuity. Since the zipping mechanism for tensstress generation becomes inactive once the films achcontinuity, the subsequent decrease in the stress-thicknepresumably due in part to stress relaxation. As can be sfrom Fig. 5, the slope of the curves just after continusuggests that the rate of stress relaxationdecreasedwith in-creasing deposition temperature. Even though diffusioncreases rapidly with increasing temperature, the propostress relaxation mechanism in Eq.~3! is also strongly de-pendent on the microstructural scale which is strongly teperature dependent as well. As shown in Fig. 8, the fithickness coinciding with continuity increased with increaing deposition temperature. Therefore, films depositedhigher temperatures had slower relaxation rates after conuity because of the strong thickness dependence ofstress relaxation rate. As shown in Figs. 9 and 10, the silation reproduces the lower postcontinuity stress relaxarates observed at higher deposition temperatures usingsame activation energy~0.3 eV!, which also correctly repro-duces the maximum tensile stress-thickness dependencedeposition temperature.

One last feature of the experimentally measured curthat has not been addressed here is the compressive strobserved at large film thicknesses. Compressive stressesing island-type growth are typically attributed to the Laplapressure induced by surface stresses.23,24 In support of thecompressive stress model, reduced lattice parametersvery small islands have been measured for variousmetals.25 As an island grows in diameter, the lattice paraeter attempts to change according to the size-depenLaplace pressure. If island-substrate sliding occurs easillow substrate coverage, the compressive strain in the isis not imposed on the substrate and will not result in sstrate curvature, which may explain why compressstresses were not measured at very small nominal thickneof Ag. At larger thicknesses once the film reaches a perlated structure, interfacial shear is no longer effectiveisland sliding. As the island grows larger, the substrate exforces on the island which prevent it from adjusting its lattparameter thereby creating stress in the island. Whiletensile stresses resulting from island coalescence are relby diffusion of matter to the grain boundaries, the samefusive mechanism cannot fully relax the compressive strimposed by the substrate. Consequently, once the testresses near the grain boundaries have relaxed, the comsive stress will still be present and may explain the compsive stress observed in thicker continuous films.

VII. SUMMARY AND CONCLUSIONS

We have presented a simple model for tensile stresseration and relaxation that predicts the kinetics of the intr


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sic tensile stresses evolution during deposition of polycrtalline thin films that grow by the Volmer–Webemechanism. For comparison with previous analytical modof tensile stress generation, we utilized FEM to modelisland impingement and coalescence process. By minimizthe sum of the positive strain energy and the associatedduction in interfacial energy, we determined the equilibriuconfiguration resulting from island coalescence as a funcof island radius. The magnitude of the average tensile stcalculated using FEM is more consistent with experimenmeasurements than the stresses calculated using theClemens crack-closure model.

From additional FEM modeling, the traction imposedthe island-substrate interface was found to strongly affboth the stress in the film and the measurable substratevature resulting from island coalescence. When islasubstrate sliding was allowed, two types of island coalcence behavior were considered. At low substrate coverisland sliding is relatively unconstrained, and coalesceproduces very small, slightly compressive stresses. Furtmore, load transfer between the film and substrate canoccur if perfect sliding is assumed, and no curvature evotion will result. At higher substrate coverage, the movemof the dislocation-like entities may be opposed by shstresses from previous coalescence events. If sliding oisland is inhibited, tensile stresses will results from subquent coalescence events.

We have also performedin situ wafer curvature mea-surements during deposition of Ag thin films depositedoxidized silicon substrates. Films were deposited at differtemperatures to study the kinetics of stress evolution dudeposition. TEM and SEM were used to correlate the evoing microstructure with features of the stress-thicknecurves.

Computer simulations were developed to model the pcess of film growth through continuous nucleation of isolaislands that grow to impingement, and eventually formcontinuous polycrystalline film. Using the FEM results fstress generation and a microstructure-dependent streslaxation model, the simulation reproduced the qualitattrends observed experimentally. Comparisons of measments and the simulation suggest that the delayed onsetmeasurable stress-thickness until a relatively large nomfilm thickness is reached can be attributed, at least in parsliding at the island-substrate interface. The magnitude ofmeasured maximum stress-thickness decreased, and occat a larger film thickness, with increasing deposition teperature, consistent with the stress generation model fograin size that increases with increasing deposition tempture, as observed in TEM micrographs. The measurementhe slopes of the measured stress-thickness curves at dent temperatures show that once a film is continuous,stress relaxation ratedecreasedwith increasing depositiontemperature. Although the proposed stress relaxation menism is thermally activated, the slower relaxation rate afcontinuity reflects the strong film-thickness dependencethe stress relaxation mechanism. While a compressive scomponent could be added to the simulation to accountthe compressive stresses measured at large thicknesse


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model for tensile stress generation and relaxation, in its crent form, captures the general characteristics of the expmentally measured tensile stress evolution during deposof Ag thin films.

ACKNOWLEDGMENTS

The authors would like to thank M. J. Kobrinsky fouseful insights and Paul Kotula for TEM. This work wasupported by the NSF through Contract No. DMR-97101Sandia is a multiprogram laboratory operated by Sandia Cporation, a Lockheed Martin Company, for the United StaDepartment of Energy under Contract No. DE-ACO94AL85000.

1K. Kinosita, Thin Solid Films12, 17 ~1972!.2R. Koch, J. Phys.: Condens. Matter6, 9519~1994!.3R. W. Hoffman, Thin Solid Films34, 185 ~1976!.4K. Maki, Y. Nakjima, and K. Kinosita, J. Vac. Sci. Technol.6, 622~1969!.

5J. D. Wilcock, D. S. Cambell, and J. C. Anderson, Thin Solid Films3, 13~1969!.


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9D. W. Pashley, and M. J. Stowell, J. Vac. Sci. Technol.3, 156 ~1966!.10W. D. Nix and B. M. Clemens, J. Mater. Res.14, 3467~1999!.11C.-H. Chiu and H. Gao, Int. J. Solids Struct.30, 2983~1993!.12H. Gao, L. Zhang, W. D. Nix, C. V. Thompson, and E. Arzt, Acta Mate

47, 2865~1999!.13J. F. Turlo, Ph.D. thesis, Stanford University, Stanford, CA, 1992.14M. D. Thouless, Acta Metall. Mater.41, 1057~1993!.15M. J. Kobrinsky and C. V. Thompson~unpublished!.16J. A. Floro, E. Chason, and S. R. Lee, Mater. Res. Soc. Symp. Proc.406,

491 ~1996!.17G. G. Stoney, Proc. R. Soc. London, Ser. A82, 172 ~1909!.18P. A. Flinn, D. S. Gardner, and W. D. Nix, IEEE Trans. Electron Devic

ED–34, 689 ~1987!.19C. V. Thompson, J. Mater. Res.14, 3164~1999!.20H. J. Frost and C. V. Thompson, Acta Metall.35, 529 ~1987!.21F. A. Doljack and R. W. Hoffman, Thin Solid Films12, 71 ~1972!.22M. J. Kobrinsky and C. V. Thompson, Appl. Phys. Lett.73, 2429~1998!.23R. Abermann and R. Koch, Thin Solid Films129, 71 ~1985!.24R. C. Cammarata, Prog. Surf. Sci.46, 1 ~1994!.25C. R. Henry, Cryst. Res. Technol.33, 1119~1998!.


2000_seel_tensile stress evolution during deposition of volmer_weber thin films

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