Intrinsic tensile stress and grain boundary formation during Volmer–Weber film growthAshok Rajamani, Brian W. Sheldon, Eric Chason, and Allan F. Bower Citation: Applied Physics Letters 81, 1204 (2002); doi: 10.1063/1.1494459 View online: http://dx.doi.org/10.1063/1.1494459 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/81/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Competition between tensile and compressive stress mechanisms during Volmer-Weber growth of aluminumnitride films J. Appl. Phys. 98, 043509 (2005); 10.1063/1.1994944 Reversible stress changes at all stages of Volmer–Weber film growth J. Appl. Phys. 95, 1011 (2004); 10.1063/1.1637728 Tensile stress generation during island coalescence for variable island-substrate contact angle J. Appl. Phys. 93, 9038 (2003); 10.1063/1.1571964 Intrinsic stress, island coalescence, and surface roughness during the growth of polycrystalline films J. Appl. Phys. 90, 5097 (2001); 10.1063/1.1412577 Tensile stress evolution during deposition of Volmer–Weber thin films J. Appl. Phys. 88, 7079 (2000); 10.1063/1.1325379

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Intrinsic tensile stress and grain boundary formation during Volmer–Weberfilm growth

Ashok Rajamani, Brian W. Sheldon,a) Eric Chason, and Allan F. BowerDivision of Engineering, Brown University, Providence, Rhode Island 02912

~Received 20 December 2001; accepted for publication 21 May 2002!

The finite element calculations in this letter present a detailed picture of tensile stress evolutionduring island coalescence in polycrystalline films. Earlier models suggest that these stresses areproduced only when islands initially coalesce. In contrast, our model predicts that stress evolutionis an inherent part of the growth process when two neighboring surfaces grow together to form agrain boundary. This view of coalescence stress is also consistent with experimental observations.© 2002 American Institute of Physics.@DOI: 10.1063/1.1494459#

Island coalescence during the growth of polycrystallinefilms can produce significant intrinsic tensile stresses, par-ticularly when there are low surface mobilities duringdeposition.1–5 Figure 1 depicts stress generation during is-land coalescence. As two neighboring islands grow, theyeventually become close enough for adjacent surfaces to ex-perience a force of attraction and snap together. This forms anew grain boundary and produces an elastic distortion of theislands that generates stress.

While earlier work attributes tensile stress to the forma-tion of grain boundaries,1–5 the mechanism by which thisoccurs was not originally addressed in sufficient detail toallow quantitative comparisons with experiments. Hoffmanfirst estimated the stress rather generally asMd/L, whereMis the biaxial modulus of the film,L is the grain size, andd isthe gap between neighboring surfaces just before they snaptogether.1,2 Nix and Clemens recently developed a more so-phisticated model, which accounts rigorously for the nonuni-form strain distribution in the grains after coalescence.6 Thisapproach assumes that stress is generated only by the initialcoalescence event. Thereafter, they assume that the filmgrows coherently with the underlying deformed lattice. Thishas two consequences: First, significant coalescence stressonly develops if the neighboring surfaces of adjacent islandsare nearly parallel~otherwise the islands can only snap to-gether over a negligible distance!. Second, the growth stressremains constant after initial coalescence. This description isnot consistent with experiments where tensile stresses in-crease substantially after initial coalescence occurs.7–9

In this letter, we present a model of tensile stress gen-eration as adjacent islands grow into each other. This modelpredicts that most of the stress evolves after the time whereislands initially impinge (tC). In our description, the materialadded aftertC brings the neighboring surfaces of adjacentgrains closer together. As growth proceeds, grain boundaryformation due to ‘‘zipping’’ becomes more energetically fa-vorable, because the material that is added reduces the elasticstrain associated with bringing adjacent surfaces together.With a periodic array of faceted islands, stress evolution be-gins attC . In the undeformed configuration in Fig. 1, the top

surface of each island is at a distanceh from the substrate~i.e., the film thickness!, and the facets on each side are atangle u to the horizontal. The reference configuration isstress free. Both islands and substrate are idealized as isotro-pic, linear elastic solids with Young’s ModulusE and Pois-son’s ration. The solid is assumed to have infinite out-of-plane thickness and deforms in plane strain.

We begin by evaluating the stress due to initial islandcoalescence, using an approach similar to Nix6 and Freund.10

The excess free energy of an island before coalescence isF052h gS cscu ~the energy of the unstrained bulk solid andthe top facet are not included because they are constant con-tributions for all calculations!. When coalescence first occursat tC , a length of the undeformed surface forms a new grainboundary of initial lengthb1 , such that the free energy isnow approximated as

F152gS~h cscu2b1!1g Ib11E8L2WE~u,b1 /L,h/L !,

~1!

where the last term is the total elastic strain energy per is-land. Here,g I is the surface free energy per area of the newlyformed grain boundary,gS is the precoalescence free energyper area of the side facets in Fig. 1,E85E/(12n2) is theplane strain modulus of the film, andWE is a dimensionlessfunction that depends parametrically on grain-boundary ge-ometry but is independent of the elastic properties or grainsize. The equilibrium grain boundary lengthb1 is determinedby minimizing F1 , which corresponds to the condition

]DF

]b150. ~2!

whereDF5F12F0 .

a!Author to whom correspondence should be addressed; electronic mail:brian–sheldon@brown.edu

FIG. 1. Schematic of the deformation that occurs when islands first impinge.The dashed line shows the undeformed configuration, and the solid linedepicts the deformation associated with creating a grain boundary of heightb1 .

APPLIED PHYSICS LETTERS VOLUME 81, NUMBER 7 12 AUGUST 2002

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This criterion was evaluated with the finite-elementmethod, using a representative example of an Ag film withgrain sizeL5120 nm; thickness at initial coalescence ofh0

560 nm; u567.5°, (g I22gS)50.5 J/m2, and E85100GPa. These values giveb1 /L55.9(10)25, which corre-sponds to a predicted grain-boundary height less than oneatomic spacing. These computed results show a significantstress concentration near the contact point between grains,but the average stress in the film is only 0.04 MPa. Largervalues ofu lead to somewhat larger values for bothb1 /L andthe initial stress, however, the calculated stresses for initialcoalescence are substantially less than those observed inexperiments.3–5,11This discrepancy may be resolved by rec-ognizing that the stress continues to increase during subse-quent island growth. As additional material is deposited onthe islands, the adjacent surfaces approach one another. Thisallows an additional portion of the surface to snap together toform a segment of grain boundary. Our principal objective inthis letter is to estimate the stress induced by this mecha-nism, using a continuum model that neglects grain-boundarydiffusion. The inclined facets on each side of the island areassumed to grow at a uniform normal velocityuE ~in theundeformed configuration!. Similarly, the top facet has uni-form growth velocityuT5buE . Growth on both facets isassumed to be coherent with the underlying crystal lattice.We now consider the film at two successive timest and t1Dt. During the time intervalDt, a layer of material withthicknessDhE5uE Dt grows on the sides of the island,while the island height increases byDh5buE Dt. At thesame time, a lengthDb coalesces with the neighboring islandto form new grain boundary.

The ratioDb/Dh is determined by the local thermody-namic force balance at the tip of the grain boundary, con-ducted as follows. For the configuration at some timet, in-troduce a virtual separation of the grain boundary over alengthj near the triple junction. The change in free energy ofthe solid due to this virtual separation can be expressed in theform

DF5jDg2E8L2DWE~u,j/L,h/L,b,t !. ~3!

In Eq. ~3!, the first term represents the increase in freeenergy due to splitting the boundary into two new surfaces,while the second term accounts for the elastic strain energydecrease associated with relaxing stress over a portion of thegrain boundary. As before,DWE is a dimensionless functionof the island geometry at timet, and is independent of ma-terial properties. Configurational force balance requires that(]DF/]j)j5050 be satisfied throughout the process, whichis consistent with Eq.~2!. The corresponding equation thatmust be satisfied for a small time incrementDt is

DjDg5E8L2DWE~u,Dj/L,Dh/L,b,t1Dt !. ~4!

For eachDt, the lengthDj5Db is fixed. During thistime interval, the thickness of material deposited on the edgeof the islandsDhE was calculated by enforcing the thermo-dynamic force balance at timet1Dt. The strain energychange was computed with the finite-element method~FEM!and Eq.~4! was then solved forDb/Dh by iteration. Thisprocedure was repeated to simulate the growth process.

Calculated biaxial stress distributions during postcoales-cence growth are shown in Figs. 2~a!–2~d!. The configura-tion for these calculations is similar to the deformed sche-matic in Fig. 1, except that the grain-boundary height is nowlonger. These results demonstrate that most of the stressevolves during the grain-boundary formation that occurs af-ter the initial coalescence event. This differs considerablyfrom the idea that postcoalescence tensile stress is producedby templated growth onto the strained surfaces formed byinitial coalescence.6 The distributions in Fig. 2 also contra-dict this templated growth concept by showing that the strainat the growth surface is much lower than the average strainin the film. As noted previously, these low strains at thegrowth surface are a logical consequence of the surfaceroughness that is present in both the model geometry and inreal polycrystalline films.8

The volume-averaged results from Fig. 2 are plotted inFig. 3. The stress increases significantly after the initial coa-lescence event, and reaches its limiting value when the frac-tional boundary height reaches 1. The stress value obtained

FIG. 2. In-plane stress distributions based the FEM, with representativevalues for Ag~given in text!. Four times aftertC are shown, with averagefilm thicknesses of~a! 55, ~b! 58, ~c! 63, and~d! 66 nm.

FIG. 3. Volume averaged stress,sV , versus grain boundary height,b, basedon the case in Fig. 2. The stress is normalized to the maximum stress,sMAX , and the height is normalized to the total island thickness,h. For theseconditions,sMAX 58.0 GPa. The isolated point shows the boundary heightand stress according to the Nix–Clemens analysis~See Ref. 6!.

1205Appl. Phys. Lett., Vol. 81, No. 7, 12 August 2002 Rajamani et al.

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from the Nix–Clemens analysis is comparable to the FEMresults at the same relative boundary height, even though theisland geometries for the two models differ. However, theFEM predicts increasing stress as deposition proceeds, a phe-nomena that was not addressed previously.6,10 The calculatedmaximum stress in Fig. 3 is 8.0 GPa, which is considerablylarger than measured values in metal films.4,5,11 Similar dis-agreement has been noted with earlier models, where it hasbeen suggested that relaxation mechanisms may reduce thetensile coalescence stresses.6,10 This hypothesis is consistentwith data for many metal films, where compressive intrinsicstresses are dominant for times significantly beyond initialcoalescence.3,11,12 Intrinsic tensile stresses are often domi-nant in films with lower mobilities, but they are generallymuch lower than 8.0 GPa.4,5An exception to this is polycrys-talline diamond films, where relaxation mechanisms are be-lieved to be largely inoperative.8 When reasonable values forpolycrystalline diamond are inserted into our model, it pre-dicts stresses of several GPa that are consistent with experi-mental observations.

At the atomic scale, the process described by our modelcorresponds to attractive forces between growth ledges thatare just ahead of the grain boundary. A simplified view ofthis was described previously, where each side facet isviewed as a set of atomic steps.8 The constant value ofuE

used in our current calculations corresponds to the casewhere adatom attachment at the steps is rate limiting. Otherformulations for the growth kinetics are also possible if othermechanisms are fully or partially rate limiting~e.g., surfacediffusion, etc.!.13 The short-range interatomic force that al-lows two neighboring facets to coalesce is only likely to belarge enough when the growth surfaces ‘‘zip’’ together atseparations that are on the order of an atomic dimension orless. This limitation is not inherently built into Eq.~3!. Forthe results in Fig. 2, a value ofDhE50.2 nm ~roughly oneatomic layer! corresponds to a coalescence ‘‘gap’’ that is al-ways less than 0.2 nm. Thus, the calculations presented hereare nominally consistent with attractive forces that act overatomic scale distances. Significant changes in the parametersin Eq. ~3! can create cases where an atomic-scale value forDhE leads to a coalescence gap that is larger than atomic-scale dimensions. Ultimately, these effects can only be as-sessed by extending our model to treat the short-range forcesthat operate at atomic length scales.

The geometry considered in this letter is essentially afaceted analog of Nix and Clemens’ two-dimensionalmodel.6 Extending the FEM approach to three-dimensionalislands gives biaxial stress distributions that are a more ac-curate representation of real films. However, our growthmodel requires finite-element solutions for a large number oftime steps and the full three-dimensional model is thus com-putationally intensive. Several three-dimensional calcula-tions were performed, and the results are generally consistentwith the conclusions obtained from Figs. 2 and 3. Anotherstep toward a more realistic model would consider islandsthat are randomly distributed spatially, and that start growingat different points in time. Seelet al. have presented resultsusing this type of approach, however, their work does notincorporate the tensile stress mechanism that is proposedhere.14

In summary, the FEM results provide a detailed pictureof stress evolution due to island coalescence. The mechanismdescribed here is fundamentally different from the templatedgrowth process that has been described previously.1,6 Thismodel demonstrates that most of the intrinsic tensile stress islikely to evolve during the growth process that occurs afterthe initial coalescence event.

This research was supported by the National ScienceFoundation under Award DMR-0075207 and also by theBrown University MRSEC program under Award DMR-0079964.

1F. A. Doljack and R. W. Hoffman, Thin Solid Films12, 71 ~1972!.2R. W. Hoffman, Thin Solid Films34, 185 ~1976!.3R. Abermann and R. Koch, Thin Solid Films129, 71 ~1985!.4G. Thurner and R. Abermann, Vacuum41, 1300~1990!.5R. Koch, J. Phys.: Condens. Matter6, 9519~1994!.6W. D. Nix and B. M. Clemens, J. Mater. Res.14, 3467~1999!.7S. Nijhawan, S. M. Jankovsky, B. W. Sheldon, and B. L. Walden, J. Mater.Res.14, 1046~1999!.

8B. W. Sheldon, K. H. A. Lau, and A. Rajamani, J. Appl. Phys.90, 5097~2001!.

9K. H. A. Lau, thesis, Brown University, 2001.10L. B. Freund and E. Chason, J. Appl. Phys.89, 4866~2001!.11J. A. Floro, S. J. Hearne, J. A. Hunter, P. Kotula, E. Chason, S. C. Seel,

and C. V. Thompson, J. Appl. Phys.89, 4886~2001!.12R. C. Cammarata, T. M. Trimble, and D. J. Srolovitz, J. Mater. Res.15,

2468 ~2000!.13A. Rajamani and B. W. Sheldon~unpublished!.14S. C. Seel, C. V. Thompson, S. J. Hearne, and J. A. Floro, J. Appl. Phys.

88, 7079~2000!.

1206 Appl. Phys. Lett., Vol. 81, No. 7, 12 August 2002 Rajamani et al.

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