Numerical simulations of stress generation and evolution in Volmer–Weber thin films

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Journal of the Mechanics and Physics of Solids 56 (2008) 2727–2747

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Numerical simulations of stress generation and evolution inVolmer–Weber thin films

Juan S. Tello�, Allan F. Bower

Brown University, Providence, RI 02912, USA

Received 25 June 2007; received in revised form 26 February 2008; accepted 27 February 2008

Abstract

We present a detailed model of the stresses and shape changes that occur in polycrystalline thin films during

Volmer–Weber growth. Our model tracks the shape of an array of islands as they grow and coalesce into a continuous film.

The islands change shape as a result of the deposition flux, as well as surface and grain boundary diffusion. Stress is

generated in the film as a result of forces exerted between neighboring islands as they meet to form a grain boundary. The

internal stresses in the islands and the diffusive changes on their surfaces and grain boundaries are computed using a

coupled finite element scheme. Interactions between neighboring islands are modeled using a cohesive zone law. Our model

predicts stress-thickness vs. thickness behavior that is in excellent agreement with experiments. Specifically, we observe a

three-stage growth process consisting of a stress-free pre-coalescence stage, a rapid tensile rise at coalescence, and an

eventual transition to a steady-state. The steady-state stress may be tensile or compressive, depending on the deposition

rate, the grain size, and the properties of the film. Detailed parametric studies are conducted to establish the influence of

material properties and growth conditions on the stress history, and the results are compared with experimental

observations and previous models.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Volmer–Weber growth; Thin film stress; Cohesive zone; Finite element analysis

1. Introduction

Polycrystalline thin films are among the most studied types of thin film systems, partly because of the widerange of technological applications that rely on them. Among these are magnetic storage media, thermalbarrier coatings, piezoelectric sensors and actuators, among others. The successful performance of these filmsdepends critically on the ability to control and faithfully reproduce the manufacturing process. Consequently,understanding the effects of growth conditions and material properties on the resulting film characteristics isof crucial importance.

A characteristic feature of polycrystalline films is that, upon deposition of film material onto a substrate,atoms of the film material bond more strongly with one another than they do with the substrate. As a result,

e front matter r 2008 Elsevier Ltd. All rights reserved.

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ess: [email protected] (J.S. Tello).

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ARTICLE IN PRESSJ.S. Tello, A.F. Bower / J. Mech. Phys. Solids 56 (2008) 2727–27472728

these atoms gather into clusters instead of wetting the substrate uniformly. With further deposition, theseclusters continue to grow and eventually coalesce to form a continuous film.

During this process, called Volmer– Weber growth, polycrystalline films develop internal stresses that varysignificantly over the course of deposition. The stresses can be determined experimentally by measuring thecurvature k of the substrate, which is related to the stress in the film by means of the Stoney (1909) formula

f ¼ 16kMshs (1)

where the ‘film force’ f is the resultant force per unit film thickness, and hs and Ms are the substrate thicknessand biaxial modulus, respectively. Experimental observations are often reported in terms of f, or the‘instantaneous stress’, sins, given by

sins ¼qf

qh(2)

where h is the volume-equivalent film thickness. Most experimental observations share the following features.During the early stages of growth, when the films are in the form of isolated islands (Fig. 1(a)), the film forcetends to have a small compressive value. Then, as islands begin to coalesce (Fig. 1(b)), sins increases rapidly,reaching a peak value when the film becomes continuous. Subsequently, a quasi-steady-state is attained(Fig. 1(c)) in which sins reaches a constant value, sssins, which depends on deposition conditions and materialproperties. Nearly all experiments (Sheldon et al., 2005; Floro et al., 2001; Hearne and Floro, 2005) reportcompressive stresses after coalescence at low growth rates. Increasing the growth rate tends to decrease themagnitude of steady-state compression, and may induce tension at very high rates. In addition, materials withlow surface diffusivities, such as ceramics, tend to exhibit more tensile stresses than say, fcc metals, which havemuch higher surface and grain boundary diffusivities at growth temperature (Sheldon et al., 2005; Hearne andFloro, 2005).

The origin of the small compressive stress exhibited by films before they become fully continuous remains acontroversial issue, and several different explanations have been proposed. Camamarata et al. (2000) explain itin terms of surface stress effects. Friesen and Thompson (2002) have suggested that adatoms moving aroundthe substrate and film surfaces act as effective force dipoles inducing a compressive stress. However, the extentto which these mechanisms are responsible for the observed stresses remains the subject of much debate(Koch et al. 2005a, b; Friesen and Thompson, 2005).

Several quantitative models have been developed to predict the tensile stress that is generated as the islandscoalesce into a continuous film. These models employ an energetic argument in various forms to determine themaximum tensile stress in the islands. The earliest of these models was introduced by Hoffman (1976), whoestimated the maximum tensile stress that can occur in an array of rectangular grains separated by a small gap.He argued that the grains can deform elastically to close this gap if the increase in elastic energy p the

Film

for

ce, f

(ar

bitr

ary

units

)

0

(a)

(b)

(c)σins<0 σins = σinsss

Film thickness, h (arbitrary units)

increasing growth flux

Fig. 1. Schematic of typical experimental observations during Volmber–Weber growth of thin films.

ARTICLE IN PRESSJ.S. Tello, A.F. Bower / J. Mech. Phys. Solids 56 (2008) 2727–2747 2729

reduction in surface energy that occurs when two surfaces are joined together to form a grain boundary. Forgrains with surface energy gs, and interface energy gi, this condition translates into

w2

2LEp2gs � gi (3)

where w is the gap between islands. Noting that the stress follows as s ¼ Ew=L, the upper bound on thestress is

sH

Ep

fEL

� �1=2

(4)

where f ¼ 2gs � gi is the work of adhesion or separation, E ¼ E=ð1� nÞ for biaxial deformation (and thegrains are 2L� 2L), E ¼ E=ð1� n2Þ for plane strain deformation (and the grains are 2L�1). This approachassumes that islands can slide freely on the substrate, and neglects the effects of mass transport and growthflux. Furthermore, gi is commonly understood to represent the energy of a stress-free interface, which is lowerthan that of a tensile one. Also, this model makes no mention what gives rise to the stress, or whether thematerial fracture strength is high enough to support the resulting stress. Nevertheless, under specialcircumstances Eq. (4) provides an acceptable approximation of the stresses that occur during coalescence, asseen in Section 6.3.

A similar model of coalescence stress in elliptical grains was proposed by Nix and Clemens (1999). Theyderived an upper bound for the stress that can occur as a result of the formation of a grain boundary throughelastic deformation of cylindrical islands. Here we summarize their results for the special case of circularislands of radius L. They viewed the cusps between grains as receding cracks with energy release rate given by

G ¼1þ n1� n

� �s2LE

(5)

where s is the volume average stress and n is Poisson’s ratio. They postulated that zipping would take placeuntil G ¼ f, which gives an estimate for the volume average stress of

sNC

E¼

1þ n1� n

� �f

EL

� �1=2(6)

As pointed out by Freund and Chason (2001), this approach ignores the fact that as soon as a finite grainboundary has formed, Eq. (5) no longer represents the energy release rate of the receding crack. Consequently,the Nix–Clemens model significantly overestimates the stress that can result from elastic zipping at grainboundaries. This conclusion was confirmed by Seel et al. (2000), who carried out finite element calculations ofthe zipping process for semicircular islands.

Freund and Chason (2001) developed a more rigorous model of island coalescence based on theJohnson–Kendall–Roberts model (Johnson et al., 1971) of adhesive contact. They predicted a volume-averagestress of1

sFC

E� 0:44

fEL

� �2=3

(7)

for the coalescence of two-dimensional cylindrical grains.All of these models neglect the combined effect of deposition flux and mass transport, which must play an

important role because the observed stresses vary with the magnitude of the deposition flux. A possiblemechanism for tensile stress generation that takes into account the competition between growth flux anddiffusion was suggested by Sheldon et al. (2005), who observed that as grain boundaries form, atoms tend todiffuse to sites of high tensile stress, tending to relax it. However, if the growth flux is high compared with therate at which atoms diffuse down the grain boundary, many of these sites will remain vacant. As a result, the

1This result is for plane stress. The plane strain solution, which is not presented in their paper, should have a different coefficient, but the

same power-law behavior.


grain boundary develops a tensile stress that should increase with increasing growth flux and decreasingdiffusivity and should saturate at some value.

The origin of the steady-state stress during the post-coalescence stage of growth is less well understood, butsome analytical models have been developed that estimate the relationship between compressive stress andgrowth flux in rectangular two-dimensional grains. One of these models was presented by Chason et al. (2002),who postulated that the large compressive stresses seen in experiments arise as a result of an increased surfacechemical potential, which is induced by the growth flux and drives excess material into grain boundaries. Thismodel is insightful, and is the first to attempt to include the effects of growth rate on stress. However, itrequires a number of parameters which are not easily estimated, such as the magnitude of the excess chemicalpotential, the coalescence tensile stress, and the adatom concentration on the surface.

A more sophisticated model of compressive stress evolution was developed by Guduru et al. (2003) whotook into account the through-thickness variation of grain boundary stress, and treated the materialincorporated into the grain boundary as an array of dislocations, which give rise the film stress. This modelincludes the effects of grain boundary diffusion, which is driven by gradients in grain boundary stress.

In this paper we present a two-dimensional continuum model of the stresses that develop inside the grains ofa polycrystalline thin film during the process of coalescence and growth. Our model extends previous work byincluding a detailed description of the attractive forces that act between neighboring islands as they coalesce toform a grain boundary. These forces, which originate from atomic-scale interactions, are approximated usinga cohesive zone law developed by Xu and Needleman (1994). In addition, our model accounts for elasticdeformation inside the islands, as well as for the shape changes and stress evolution that result from surfaceand grain boundary diffusion as well as from the deposition flux. The governing equations are derived fromstandard balance laws and are solved using a coupled finite element scheme. The reference configurationcontinuously evolves as a result of growth flux and diffusion, and we keep track of these changes by repeatedlyregenerating the finite element mesh.

While the model lacks level of detail of atomistic formulations, and neglects some potentially importanteffects including the possibility of faceting, varying grain size, and three-dimensional effects, it represents asubstantial improvement upon previous models and reproduces many experimental observations with a highdegree of accuracy including: (i) the general behavior of stress-thickness vs. thickness during growth (Hearneand Floro, 2005; Chason et al., 2002; Sheldon et al., 2005), (ii) the dependence of steady-state stress withgrowth flux, and (iii) the magnitude of observed stresses. In addition, the model makes a number ofpredictions about stress behavior of thin films during growth. Firstly, it suggests that the instantaneous steady-state stress should approach the equilibrium grain boundary stress as the growth rate is reduced to zero.Secondly, the model shows that a tensile rise at the point of coalescence can be present in the absence of elasticdisplacement, and can occur as a result of mass transport and growth flux.

The remainder of this paper is organized as follows. In Section 2 we describe the model, list our assumptionsand derive the governing equation for mass transport. Section 3 includes a description of the numericalmethod used to solve the governing equations and compute the evolution of the islands. In Section 5 wecalculate the island shape evolution as it reaches its equilibrium configuration starting from an arbitrary, non-equilibrium shape. Then we compare the features of this equilibrium configuration with two differentanalytical models, which are discussed in detail in the Appendix. In Section 4 we carry out a dimensionalanalysis of the governing equation for diffusion and suggest a functional relationship for the various stressmeasures in dimensionless form. In Section 6 we show the stress contours and stress histories of an islandduring a typical growth simulation and study the influence of the various system parameters on stress. Finally,in Section 7 we compare the predictions of our model with experimental observation in nickel and aluminumnitride films.

2. Model description

We idealize the process of Volmer–Weber growth by considering an array of two-dimensional elastic islandsattached to an elastic substrate as shown in Fig. 2. The island centers are periodically distributed a distance 2L

apart. Individual islands are bounded by the island/substrate interface, Gi, and island surface, G. Initially, theislands are semicircles of radius R5L so that they are isolated and stress free. At t ¼ 0 the islands begin

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δs

Gra

in b

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tripl

ej

unctio

n

Γ

nγs

Ti = -F(δ) δi1

Region ofcomputation

Symmetric

x1 / L

0.5 1 1.5 2

Symmetric

Fig. 3. Representative geometry of half of an evolving island. The arrows represent the cohesive tractions, given in Eq. (8).

LL

Elastic Islands

ElasticSubstrate

Deposition flux

symmetry

x1

x2

Γ

symmetry

Γi

Fig. 2. Periodic array of two-dimensional islands during early stages of Volmber–Weber growth.

J.S. Tello, A.F. Bower / J. Mech. Phys. Solids 56 (2008) 2727–2747 2731

growing as a result of a deposition flux, and are subject to shape changes due to diffusion along G. We neglectany mass transport along Gi, and enforce traction and displacement continuity across it. As islands come intoclose proximity of each other they interact through atomic scale surface forces. These forces induce a state ofstress sij in the islands and substrate, and also act as a driving force for formation of new grain boundaries.

Due to the symmetry of the system we confine attention to the region 0px1pL, as shown in Fig. 3, whichshows the configuration of an island after the formation of a grain boundary. The curve G, which previouslyconsisted only of the island surface now contains the grain boundary and triple junction as well. Note that theisland shown in Fig. 3 has an unnaturally large triple junction region for illustrative purposes.

When the islands impinge on each other they interact through atomic scale forces, which we approximateusing a cohesive zone law (Xu and Needleman, 1994). The cohesive zone law specifies the traction on theboundary G as a function of the separation 2d as

Ti ¼ �F ðdÞdi1 and F ðdÞ ¼ sm exp 1�dD

� �dD

(8)

Here, d ¼ 0 corresponds to the equilibrium spacing between the two crystals that meet at the grain boundary.Since neighboring crystals will generally have different orientations this does not necessarily represent aseparation of one interatomic lattice parameter. When do0, neighboring grains are understood to be closerthan this equilibrium position, so that they repel each other ðto0Þ; for d40 the traction is tensile, increases to

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0 2 4 6

−0.8

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−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

( δ−wgb)

wtj

ΔδF( )

σm

wtj

Dgb

Ds

Transition

Sample diffusivity distribution

D(δ) = Dgb+(Ds - Dgb) g(δ)

δ

δ / Δ

g

Fig. 4. Left: cohesive zone law (Eq. (8)) and truncation function gðbÞ (Eq. (12)); right: sample diffusivity distribution plotted as normal

arrows proportional to the diffusion coefficient DðdÞ, given in Eq. (16).

J.S. Tello, A.F. Bower / J. Mech. Phys. Solids 56 (2008) 2727–27472732

a maximum value of sm at d ¼ D, and decays to zero as d!1. A plot of F ðdÞ vs. d is shown in Fig. 4. Thework of separation follows as f ¼ 2esmD, and the interface energy is given by gI ¼ 2gs � f. Such approachesare standard in characterizing interplanar potentials in models of fracture (Xu and Needleman, 1994).

The tractions imposed by neighboring islands on each other through the cohesive zone induce a state ofstress sij that must satisfy

qsij

qxj

¼ 0; sijnj ¼ �F ðdÞdi1 (9)

where summation is implied over repeated indices and ni are the components of the normal vector to G,given by

n1 ¼dxG

2

ds; n2 ¼ �

dxG1

ds(10)

where xGi ðsÞ are the coordinates of G as a function of arc length.

The islands grow as a result of attachment of film atoms from a second phase that surrounds the islands.This attachment is quantified by the volumetric flux per unit area, jnðsÞ, which is assumed to be normal to thesurface. In order to avoid depositing material into the grain boundary, we let

jn ¼ j0gðbÞ (11)

where j0 is the magnitude of the growth flux away from the grain boundary, and

gðbÞ ¼

0; bo0;

ð6b5 � 15b4 þ 10b3Þ; 0pbp1;

1; b41;

8><>: b ¼

d� wgb

wtj(12)

is a truncation function that varies smoothly from gð0Þ ¼ 0 to gð1Þ ¼ 1. A plot of Eq. (12) is shown in Fig. 4for wtj ¼ 3D and wgb ¼ 0. The islands change shape as a result of mass transport along G, whose chemicalpotential we approximate as

mðsÞ ¼ �Oðsn þ gskÞ; s 2 G (13)

where sn ¼ sijninj is the normal stress and k is the curvature, given by

k ¼ ni

d2xGi

ds2(14)


gs is the surface energy and O is the atomic volume. In writing Eq. (13) we have neglected small terms such asstrain energy density. In most of our simulations, the grain boundary is essentially flat (k � 0), so that thegrain boundary chemical potential follows from Eq. (13) as �Osn. In contrast, the surface is essentially stressfree (sn � 0), so that its chemical potential is �Ogsk. We call the finite region of width wtj�D where ka0 andsna0 the ‘triple junction’ even though this term commonly refers to a sharp junction between two surfacesthat meet at a grain boundary.

Material flows down the chemical potential gradient according to

js ¼ �DðdÞkT

qmqs; s 2 G (15)

where

DðdÞ ¼ Dgb þ ðDs �DgbÞgðbÞ (16)

is a spatially varying diffusion coefficient and gðbÞ is given in Eq. (12). The coefficients Dgb and Ds have unitsof length3/time, and are given by

Dgb ¼ D0gbdgbe

�Qgb=kT ; Ds ¼ D0sdse

�Qs=kT (17)

where T is absolute temperature, k is Boltzmann’s constant, D0gb and D0

s are the grain boundary and surfacediffusivities, Qgb and Qs are the grain boundary and surface activation energies, and 2dgb and ds are thethicknesses of the respective diffusion layers.

This definition of a separation dependent diffusivity suggests an interpretation of the terms ‘grainboundary’, ‘triple junction’ and ‘surface’ in the context of this framework as follows:

�
Grain boundary is the part of G where dowgb. � Triple junction is the part of G where wgbpdpwgb þ wtj. � Surface is the part of G where d4wgb þ wtj.
This categorization is intended as an approximate guide for analyzing the results of our simulations but it isnot essential to the implementation of the model. However, the stress evolution behavior of the islands ishighly sensitive to the grain boundary width wgb and triple junction width wtj, as evidenced by the parameterstudies shown in Section 6.

During a time step Dt, we solve the diffusion equation for the normal displacement along G in the referenceconfiguration, hðsÞ. Mass conservation requires that

h

Dtþ

qjsqs� jn ¼ 0; s 2 G (18)

Combining Eqs. (13), (15) and (18) gives

h ¼ �DtOkT

qqs

DðdÞqqsðsn þ gskÞ

� �þ Dtjn (19)

3. Numerical method

A finite element approximation to the field quantities ui and h can be obtained by expressing Eqs. (9) and(19) in weak form asZ

GtþDt

F ðdÞdu1 dsþ

ZRtþDt

sijqdui

qxj

dAþ

ZGtþDt

ðh� DtjnÞdhds

þODt

kT

ZGtþDt

ðsn þ gskÞd

dsDðdÞ

ddh

ds

� �ds ¼ 0 (20)

where RtþDt represents the region occupied by the island at time tþ Dt. Here, dui represents a virtualdisplacement field satisfying dui ¼ 0 on qR and dh is a virtual surface displacement field satisfying dh0 ¼ 0 at


both ends of G. Since Eq. (20) must be satisfied at the end of each time step, we approximate the normal vectorni and curvature k as

niðtþ DtÞ � niðtÞ � ti

dh

ds(21)

kðtþ DtÞ � kðtÞ þ kðtÞ2hþd2h

ds2(22)

and compute the normal stress as

sn ¼ �F ðdÞ n1 � t1dh

ds

� �(23)

where

d ¼ x1 þ u1 þ hn1 (24)

is the gap at the end of the time step in the deformed configuration.Eq. (20) is used as a basis for a finite element calculation that tracks the shape and mechanical state of a

periodic array of islands during Volmer–Weber growth. Our goal is to compute the evolution of the boundaryG as well as the stresses and elastic displacements in the island as a function of time. A generic simulationbegins with semicircle of radius R5L so that the entire surface of the island is outside the influence of thecohesive zone.

At the beginning of each time step the geometry is specified by a set of ‘control points’ that define the islandboundary, G, and substrate perimeter. Then, the surface curvature k and normal vector ni are computed byfitting cubic parametric splines through these control points. The boundary is subdivided into five-noded lineelements whose size is made proportional to the local radius of curvature. Each element contains threeelasticity nodes and two diffusion nodes, with two degrees of freedom per node. The diffusion nodes containthe nodal values of h and qh=qs, while the elasticity nodes contain the nodal values of u1 and u2. To ensure thatthe unknown variables h and qh=qs are continuous across neighboring elements they are interpolated betweennodal values using cubic Hermitian interpolation functions. The displacements on the surface as well as in thebulk are interpolated using quadratic interpolation functions. The interior of the island is meshed withsix-noded triangular quadratic elements following the methodology described by Peraire et al. (1993). Theinitial mesh has a prescribed uniform element size. Subsequent meshes have a spatially varying element sizeaccording to the error estimator developed by Zhu and Zienkiewicz (1988).

Interpolating Eq. (20) on the resulting finite element mesh yields a system of non-linear algebraic equationsfor the nodal values of ui, h and qh=qs, which is solved iteratively. Having obtained a solution for hðsÞ and ui,the boundary is moved to xs

i ðs; tÞ þ hni and new control points are generated and fitted with parametric splines.The new nodes are generated along the resulting boundary at intervals proportional to the local radius ofcurvature.

4. Dimensional analysis

Whenever studying the behavior of a finite element model involving a large number of parameters it is usefulto express the measures of interest in dimensionless form. Since the model is intended to shed light onexperimental observations, we focus on the stress measures that are commonly reported in the experiments.Hence, we will examine the equilibrium grain boundary stress or s�gb, the histories of film force, f, andinstantaneous stress, sins, from which we will extract the peak average stress, smax

ave ¼ ðf =hÞmax, and the steady-state instantaneous stress, sssins. In general, these stresses will depend on all 13 system parameters, i.e.

sL

gs¼Fðgs; sm;D;L;Ds;Dgb; j0;wgb;wtj;E; kT ;O; hÞ (25)

where s stands for either s�gb, sins or sssins. The choice of normalization for the stress measure is arbitrary:stresses can be also normalized as s=sm, sL=f, or s=E, while the film force can be normalized as f =f or f =gs.Since the relationship (25) must be independent of units, we can express the right-hand side as a function of


certain dimensionless groups. In general, any independent combination of parameters is acceptable, but somecombinations may be more appropriate than others. A straightforward dimensional analysis of Eq. (19)suggests a characteristic time for diffusion given by

t0 ¼L4kT

ODsgs(26)

and a dimensionless flux of

jn ¼j0kTL3

OgsDs(27)

With this in mind, the stress measures can then be expressed as a function of seven dimensionless groups

sL

gs¼H

fgs;smE;DL;Dgb

Ds; jn;

wgb

D;wtj

D

� �(28)

while the film force depends on these seven plus the normalized volume-equivalent film thickness, h=L. Here

�
f=gs is the ratio of work of adhesion to surface energy: high values of f=gs tend to induce longer grainboundaries and flatter surfaces; � sm=E is the ratio of cohesive strength to Young’s modulus; � D=L is a dimensionless inverse grain size; � Dgb=Ds compares the grain boundary and surface diffusivities; � jn is given in Eq. (27). It compares the rate of deposition to diffusive displacements on the surface; � wgb specifies the width of the grain boundary. Also, the point s� such that dðs�Þ=wgb ¼ 1 represents the point
at which deposition is truncated, i.e. jn ¼ 0 for dðsÞ=wgbp1;
� wtj measures the width of the region across which jn transitions from 0 to j0.
In the following sections we present representative stress histories and investigate the effect of the variousdimensionless groups on several stress measures. We also compare the predictions of our FEM model with thevarious models of tensile stress generation outlined in the Introduction.

5. Equilibrium results

In order to illustrate some basic features of the model, we first consider a simple example problem, shown inFig. 5. Instead of starting the simulation with widely spaced, small islands, we assume that at time t ¼ 0 thefilm consists of an array of large islands, which just touch, and have an arbitrary non-equilibrium shape, whichconsists of 1

4of an arc of an ellipse, as illustrated in the figure. We allow the islands to evolve through mass

transport, without the deposition flux. The islands are allowed to evolve until they reach their equilibriumconfiguration.

The initial chemical potential distribution, which is far from constant, is shown in Fig. 5(a) together with thesurface energy and normal stress distributions. The region of low chemical potential induced by the tensilestress attracts material towards it and drives the formation of a grain boundary as seen in Fig. 5(b).Subsequently, material continues to flow from the surface (which has negative curvature, and hence highchemical potential) to the grain boundary (which has lower chemical potential). This process continues untilthe grain boundary develops a small compressive stress as shown in Fig. 5(c), which illustrates some features ofthe equilibrium configuration. At equilibrium the surface is stress free, having a constant chemical potential of�Ogsk

�, where k� represents the equilibrium surface curvature. In the grain boundary, where k ¼ 0, thechemical potential is �Os�gb, where s

�gb is the equilibrium grain boundary stress. Since the chemical potential is

constant throughout, it follows that s�gb ¼ gsk�. In the triple junction, which is subjected to a finite tensile

stress and has a large negative curvature, the distributions of sn and gsk exactly balance out to keepm ¼ �Oðsn þ gskÞ ¼ constant. Note that the equilibrium curvature k� is as yet undetermined, although it isoften approximated as k�� 1=L. A simple analysis of the equilibrium configuration can lead to more specificestimates of k� and the resulting equilibrium grain boundary stress, as discussed below.

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L

μΩ μ

ΩμΩ

σn

σn

−κγs

Γ

Equilibriumshape

−κγs

0.8σm

Chemical potential

Normal stress

- curvature x surface energy

Arbitrarynon-equilibrium

shapeTransitional shape

x1 / Lx1 / Lx1/L

Fig. 5. Island with initial non-equilibrium shape reaches equilibrium through mass transport. The normal stress sn is plotted as arrows

that point in the direction of the normal ni, the contribution from curvature is plotted as �gsk (dashed line), and the total chemical

potential is m=O ¼ �ðsn þ gskÞ (solid line). The scale for all quantities plotted is shown in (c). Here, 2esmD=gs ¼ 1:5, E=sm ¼ 300, and

D=L ¼ 140.


In Appendix A we describe two different analytical estimates for the equilibrium grain boundarystress. First we derive the classical ‘force balance’ formula for the triple junction angle, whichstates that 2gs cos y ¼ gi, where gi is the energy of the interface. The curvature follows from the angle ask� ¼ � cos y=L. Within this analysis, the equilibrium grain boundary stress follows from chemical potentialcontinuity as

sCgbL

gs¼ �

gi2gs

(29)

Additionally, we outline a second analytical model of the equilibrium state based on a simplified cohesivezone, known as a Dugdale cohesive zone. In this case a constant tensile traction is prescribed for doD whilefor d4D the surface is stress free. This approach leads to an explicit analytical expression for the shape of thetriple junction region (see Appendix), and predicts an equilibrium grain boundary stress of

sDgbL

gs¼ � exp

gi2gs� 1

� �(30)

in the limit D=L! 0 with smD ¼ constant. The predictions of these two models and those of our FEM modeldescribed in Section 2 are compared in Fig. 6. In Fig. 6(a) the equilibrium grain boundary stress predicted bythe Dugdale model is compared with the predictions of the finite element model for varying cohesive zonelengths. It is evident that the two models agree only for small D=L. This is because two cohesive zones are onlyequivalent in this limit. Fig. 6(b) shows the variation in grain boundary equilibrium stress with Z ¼ f=gs(the ratio of work of adhesion to surface energy) as predicted by the FEM model, the Dugdale model, and theclassical ‘force balance’ model.

Fig. 6(b) suggests that the FEM model and the Dugdale model are in excellent agreement with respect tovariations in the energetic parameter Z, while the classical force balance approach diverges significantly fromboth models at larger Z’s.

6. Transient results

We next turn to investigate the history of stress that develops during growth and coalescence of the islandsunder the application of a constant deposition flux. A typical simulation starts with a semicircular island ofradius R5L so that it is stress free and isolated, as shown in Fig. 7. At t ¼ 0 we apply a growth flux

ARTICLE IN PRESS

0 0.05 0.1 0.15 0.2 0.25

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

σ gb

L /

γ s

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

σ gb

L /

γ s

0.5 1 1.5 2 2.5 3

Dugdale modelClassical modelFEM

λ = Δ / L η = φ / γs

λ = 0.025

η = 2

Fig. 6. Variation in equilibrium grain boundary stress as predicted by the FEM model compared with Eqs. (29) and (30). The triple

junction angle follows as cos y ¼ sgbLðl� 1Þ=gs, where l ¼ 0 for the classical model, and Z ¼ f=gs.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0

0.5

1

σ11 / σm

0.90.80.70.60.50.40.30.2

-0.0

1

0.1

x 2

x1

L

0

0.5

1

x 2 L

0

0.5

1

x 2 L

Lx1L

x1L

t / t0 = 0 t / t0 = 0.34 t / t0 = 0.9

Fig. 7. Representative evolution of island morphology and stress during coalescence and growth. Z ¼ f=gs ¼ 0:68, l ¼ 0:05, Dgb=Ds ¼ 1,

jn ¼ 1:25, wgb=D ¼ 0, wtj=D ¼ 1, E=sm !1. The characteristic time is given in Eq. (26).


jn ¼ j0gðbÞ, where gðbÞ is given in Eq. (12). The island begins to grow self-similarly and stress free as long as itsleftmost boundary point remains far from the cohesive zone (Fig. 7(a)). Then, when this point comes within afew D’s of the symmetry boundary x1 ¼ 0, the tensile traction on the surface lowers the local chemicalpotential, tending to drive mass transport in that direction. The color code in Fig. 7 represents the stresscomponent s11 as it evolves throughout the coalescence and growth phases of deposition. The cohesive zoneexerts a compressive traction in the grain boundary (Fig. 7(c)) and a tensile one on the triple junction region(red zone in Fig. 7(b) and (c)). Although it may appear that Fig. 7(b) shows an elastically stretched island, thematerial is essentially rigid, and ui � 0 everywhere. All the shape changes occur as a result of mass transport.This simulation has been run with a relatively large value of D=L for purposes of illustration. This wouldcorrespond to very small islands, since in real systems D�1 A and L ranges from about 80 nm up to severalmicrons. More realistic values are considered in Section 7, where we compare the predictions of the FEMmodel with experimental observations.


6.1. Stress histories

Fig. 8 shows the history of film force, f, and instantaneous stress, sins, as a function of deposited filmthickness h for the simulation shown in Fig. 7. These results have been reported earlier (Tello et al., 2007) butare included here for the sake of completeness.

When the island is isolated (Fig. 7(a)) it is free of traction and internal stress, so that f ¼ sins ¼ 0. Duringcoalescence (Fig. 7(b)) f increases due to the cohesive tractions, and sins reaches a peak value. With furtherdeposition, sins approaches a steady-state value, sssins, henceforth referred to as steady-state stress. Thevariation sssins with growth rate and diffusivity ratio Dgb=Ds is explored in Section 6.2.

6.2. Effect of growth flux and diffusivity ratio on steady-state stress

One of the features that has been most closely observed experimentally is the variation of steady-state stresswith growth flux in various materials (Sheldon et al., 2005; Hearne and Floro, 2005). Fig. 9 shows thisvariation for several diffusivity ratios, representing different material systems. At low growth ratesðjnDs=Dgb51Þ, sins is essentially the equilibrium grain boundary stress, given approximately by Eq. (30). Asthe growth rate increases, for materials with Ds=Dgb�1, the stress becomes more compressive, until it reaches aminimum near jn ¼ 103, and then begins increasing rapidly until it saturates at a tensile stress that depends onthe flux cutoff, wgb, and the cohesive zone strength, sm. Material systems with Dgb5Ds do not go through aminimum in their transition from the equilibrium value at low jn to the saturation value at high jn. This isbecause the low Dgb precludes the flow of additional material into the grain boundary even in the presence of alarge chemical potential gradient.

The reason for the existence of a minimum in the sins vs. jn curves with high Dgb’s is not immediatelyobvious. Compressive stress is developed when extra material is incorporated into the grain boundary, whichcan occur only if there is a strong chemical potential gradient driving material into the grain boundary, sinceno direct deposition takes place inside the grain boundary. The mechanism for generation of this high

0.6 0.8 1.2 1.4 1.6 1.8

0

0.6

f / γ

s

0

2

4

6

σ ins

L /

γ s

(a)

(b)

(c)

Steady-stateσss L / γs

σss <00.4

0.2

−0.2

−0.4

0.4 1

h / L

0.6 0.8 1.2 1.4 1.6 1.80.4 1

h / L

ins

ins

Fig. 8. Evolution of film force, f =gs (top) and instantaneous stress, sinsL=gs (bottom) for simulation shown in Fig. 7. The points labeled

(a)–(c) correspond to the likewise labeled frames in Fig. 7.

ARTICLE IN PRESS

0 1 2 3 4 5

−2

−1

0

log10 jn

σss L

/ γ s

σss L / γs Dgb / Ds = 1Dgb / Ds = 0.1Dgb / Ds = 0.01

^1 2 3 4

−3−2−1

01234567

Nor

mal

ized

film

for

ce,

f / γ

s

jn = 105^

jn = 104^

jn = 103^

Dgb / Ds = 1

−0.5

−1.5

−2.5

Normalized film thickness, h / L

0.5 1.5 2.5 3.5 4.5

ins

ins

Fig. 9. Representative behavior of normalized instantaneous stress (Eq. (2)) with film thickness (left) and the variation in steady-state

stress with growth flux and diffusivity ratio (right). Other parameters are D=L ¼ 0:02, Z ¼ 1:5, E=sm � 1, wgb ¼ 0 and wtj ¼ D. All curves

approach a value of sssins � sDgb ¼ �0:47gs=L (Eq. (30)) as jn ! 0.

−γ κs

μ/Ω

σn

−0.1 0.1

Equilibrium

configuration

Small compression

−γsκ

μ/Ω

Gradient

responsible

for increased

compression

−0.1 0.1

σn

μ = −Ω (σn + γsκ)

Non-equilibrium

configuration

Large compression

x1 / L

−0.05

0.8σm

0 0.05

x1 / L

−0.05 0 0.05

Fig. 10. Schematic representing variation of chemical potential (m=O, solid arrows) near the triple junction. Also shown are the

contributions to the chemical potential from the normal stress (sn, dashed lines) and curvature (shown as �gsk, dotted lines). Case (a)

shows the equilibrium configuration of the triple junction. Case (b) corresponds to an identical system in out-of-equilibrium conditions

with jn�103. Both cases have Ds=Dgb ¼ 1. As shown, m=O is the difference between �gsk and sn.


compressive stress is illustrated in Fig. 10, which shows schematics of the triple junction under two differentgrowth conditions. Fig. 10(a) shows the variation of stress, surface curvature and chemical potential in nearequilibrium conditions (jn51). The free surface outside the triple junction has curvature k � �1=L, so that thesurface chemical potential is m � Ogs=L. In the triple junction region, the two adjacent crystals are subjected toa large tensile (attractive) stress, whose contribution to the chemical potential is balanced by a large convexcurvature. Inside the grain boundary, the curvature vanishes, so that a small compressive stress develops givenby Eq. (30). Larger growth fluxes ðjn\103Þ drive the islands away from equilibrium, as illustrated in Fig. 10(b).Under these conditions a chemical potential gradient develops just inside the tensile region of the triplejunction, which drives diffusion into the grain boundary. The amount of material that is driven into the grainboundary depends on the deposition flux and the ratio Dgb=Ds. It is greatest for Dgb=Ds ¼ 1 (assumingDgbpDs), in which case the compressive stress can reach between 5 and 10 times the equilibrium value s�gb.


6.3. Coalescence stresses

We now turn to investigate the magnitude of the peak tensile stress during coalescence. We focushere on the limit as Ds and Dgb! 0, resembling the behavior of films made of ceramics, such as diamondand AlN. In such case, the peak tensile stress becomes a function of just two independent dimensionlessgroups:

smaxave

E¼F

fEL

;smE

� �(31)

where f ¼ 2esmD is the work of adhesion of the interface, which is related to the surface and interface energiesby f ¼ 2gs � gi. The effect of varying the parameters wgb=D and wtj=D will be considered in Section 6.4; fornow they are set to 0 and 1 and held constant.

Fig. 11 shows the variation in peak coalescence stress with the parameters f=EL and sm=E. Alsoshown are power-law fits of the curves in order to compare these results with the models (4), (6) and (7). Thesemodels predict that the stress will vary like smax

ave =E�ðf=ELÞn where n ¼ 12 in the Hoffman and Nix–Clemens

models, and n ¼ 23in the Freund–Chason model, and should be independent of sm=E. It must be born

in mind that these models neglect the effects of deposition and diffusion, so they cannot be preciselyinterpreted as modeling the coalescence process considered in our calculations. Instead, theymodel the ‘contact stress’ that arises as a result of grain boundary formation through elastic deformation.However, they are often used as predictors of tensile stress at coalescence during growth of polycrystallinefilms. With this in mind we can compare their predictions with the results shown in Fig. 11. It is evidentfrom Fig. 11(right) that the average stress depends on at least two dimensionless groups, not just onf=EL as predicted by the models. Surprisingly, for all values of sm=E the stress varies approximately asðf=ELÞ1=2, in agreement with the Hoffman and Nix–Clemens models, but differing slightly from theFreund–Chason model.

This result should not be interpreted as an indication that the Nix–Clemens model is a more accurate modelof contact stresses than the Freund–Chason model. In fact, separate simulations of the stresses generated inelastic islands that interact through a cohesive zone, but do not change shape by surface diffusion ordeposition, give results that are in excellent agreement with the Freund–Chason model. In the simulationsshown here, the islands change shape as a result of the deposition flux, rather than by elastic deformation. Thegood agreement with Hoffman and Nix–Clemens is fortuitous. Nevertheless, all three analytical modelspredict the trends observed in our simulations, despite the difference in the underlying mechanism forgenerating stress.

−2.8

−2 A=0.132, n=0.44

log 1

0 σ a

ve

/ E

A=0.223, n=0.54

A=0.086, n=0.46σmax/Eave

σmax/E = A(φ/EL)n

1 2

1

2

3

4

5

6

7

8x 10−3

Ave

rage

str

ess,

σav

e / E

φ/EL = 0.0002φ/EL = 0.0006φ/EL = 0.0012φ/EL = 0.0018

E/σm=50

E/σm=200E/σm=100


0.5 2.51.5

log10 φ / EL

−3.6 −3.4 −3.2 −3 −2.6 −2.4

E / σm = 50

−2.1

−2.2

−2.3

−2.4

−2.5

−2.6

−2.7

max

ave

Fig. 11. Variation in peak average tensile stress with the parameter f=EL, which appears in the models of tensile stress generation

described in the Introduction.


6.4. Modeling a finite triple junction and material attachment

One of the most unique features of our model is the fact that the grain boundary and surface are not treatedseparately as in most previous models. Instead, the perimeter of the islands smoothly transitions from grainboundary to free surface over the region we call a finite triple junction. The size and character of this region isdetermined by three dimensionless groups: D=L, wgb=D and wtj=D. Fig. 12 shows our model’s sensitivity to thedimensionless group D=L at various jn with all other groups in Eq. (28) held fixed.

Both at very low ðjn ¼ 13:8Þ and very high ðjn ¼ 138; 000Þ growth rates, the steady-state stress is independentof D=L, indicating that the magnitude of the stress scales like L�1. At low rates, the comparatively highdiffusivity allows the material to flow to regions of low chemical potential so as to maintain the island’sequilibrium shape. As a result, the normalized steady-state stress is equal to the equilibrium grain boundarystress, given approximately by Eq. (30). At very high rates the shape is essentially dictated by the growth flux,with negligible diffusive changes. In this situation, sssins will depend strongly on wgb=D and possible on otherparameters as shown in Fig. 14.

In contrast, at intermediate growth rates, where the shape is determined by a competition betweendeposition and diffusion, the normalized steady-state stress is compressive and increases in magnitude withdecreasing D=L. This is because the chemical potential gradient responsible for the large compression(see Fig. 10) increases with decreasing D=L, leading to higher normalized compressive stresses at lower D=L, orlarger grain sizes. In this case there is no simple scaling of the dimensional stress with grain size.

Other stress measures also exhibit a D=L dependence that is worth exploring. Fig. 13 shows the variationpeak average (tensile) stress ðsmax

ave L=gs) with D=L for different growth fluxes.As seen in Fig. 13(right), smax

ave L=gs exhibits an approximate power-law behavior with L=D with highergrowth rates inducing higher stresses.

Finally, it is important to study the effects of varying the parameters that dictate the location of the growthflux cutoff, and the width of the transition region (wgb and wtj in Eq. (12)). Note that we have assumed that thediffusivity transitions from Dgb to Ds according to the same parameters. Since these parametersoverwhelmingly determine the details of attachment in and around the cohesive zone, they can be expectedto have a strong influence on the stress during growth.

The general effect that wgb and wtj have on stress behavior is shown in Fig. 14, which shows the variation insteady-state stress, sssins, with wgb=D and wtj=D at various growth rates. At a very high growth rateðjn ¼ 1:4� 105Þ, varying the grain boundary width amounts to following the shape of the cohesive zone sincethe grain boundary is being ‘forced’ to grow at a particular stress value. As the growth rate is reduced, thisstress is partially relaxed through mass transport and the effect of increasing the grain boundary width ismitigated. A similar scenario occurs when varying the triple junction width, wtj=D. For the range of valuesexplored, the steady-state stress varied linearly with wgb=D, with a lower slope occurring at lower growth rates,again due to diffusion-induced relaxation. The fact that such a wide range of behavior can be obtained by

0−4

−3

−2

−1

00.5

1

σss L

/ γ s

jn =138,000^jn =1,380^jn =13.8^

2 4 6 8

−10

−8

−6

−4

−2

0

2

4

jn = 1,380


Nor

mal

ized

film

for

ce,

f / γ

s

Δ/L=0.005Δ/L=0.03Δ/L=0.05Δ/L=0.08

^

σss L / γs

0.040.02 0.06 0.08

−0.5

−1.5

−2.5

−3.5

Δ / L

ins

ins

Fig. 12. Normalized steady-state instantaneous stress vs. cohesive zone size. Other parameters in the simulations are Z ¼ 2esmD=gs ¼ 1:5,sm=E�1� 10�11, Dgb=Ds ¼ 1, wgb ¼ 0, and wtj ¼ D. The dimensionless flux jn follows from the analysis in Section 4.

ARTICLE IN PRESS

0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4L/Δ = 100

L/Δ = 33.33

L/Δ = 12.5

jn =138,000^

2

−1

A=1.442, n=−0.94

log10 L / Δlo

g 10

σmax

L /

γ s

A=0.768, n=−0.64

A=1.378, n=−0.88

A=0.655, n=−0.58

σave L / γs

σave L

γs

max

jn = 13.8^

jn = 138^

jn = 1,380^

jn = 138,000^

= ( )ALΔ

n


Ave

rage

str

ess,

σav

e L

/ γ s

−0.8

−0.9

−1.1

−1.2

−1.3

−1.4

−1.5

−1.6

−1.7

−1.8

1.81.61.41.2 2.2 2.4

L/Δ = 20

max

ave

Fig. 13. Left: normalized average stresses vs. normalized film thickness and L=D at jn ¼ 138; 000. Right: normalized maximum average

stress vs. L=D and jn. Other parameters are f=gs ¼ 1:5, E=sm�1� 1011, Dgb=Ds ¼ 1, wgb=D ¼ 0, and wtj=D ¼ 1. The average stress follows

from the film force as save ¼ f =h.

2 3 3.5 4.5 5−2

02468

1012141618

wtj / Δ

wtj / Δ= 3

0 0.2 0.4 0.6 0.8 1 1.2 1.4−1

0

1

2

3

4

5

6

7

8

wgb / Δ

wgb=0

σss L

/ γ s

σss L

/ γ s

jn = 1.4x105^

jn = 1.4x103^

jn = 1.4x101^

2.5 4

ins in

s

jn = 1.4x105^

jn = 1.4x103^

jn = 1.4x101^

Fig. 14. Effect of wgb and wtj on normalized steady-state for islands grown at various growth rates. Other parameters are: D=L ¼ 140, Z ¼ 1

3

and E=sm ¼ 300, (a) effect of wtj=D, (b) effect of wgb=D.


varying these parameters suggests that the details of attachment at the triple junction during the formation ofgrain boundaries has an overwhelming influence on the observed histories of stress. From the continuumperspective of this model it is impossible to determine realistic values for wgb and wtj from first principles, sincethey would be determined by complex atomic-scale behavior, with atoms hopping around a highly imperfectsurface. Although these parameters were first introduced within the context of the present model, theyrepresent real lengths that describe real characteristics of the attachment process during the formation of newgrain boundaries. This issue should be the subject of extensive computational studies in the future.

7. Comparison with experiments

Finally, in this section we compare the predictions of our model with two sets of experimental observations.We have modeled the deposition process in aluminum nitride (AlN) and nickel (Ni) films using the parametersshown in Table 1. The comparisons have been carried out as follows. In both sets of data, the average grain

ARTICLE IN PRESS

Table 1

Simulations parameters for comparison of model with experiments

Material Grain size 2L (nm) CZ length, D (A) CZ strength, sm (GPa) Diffusivity, Dgb ¼ Ds (m3=s) Surface energy, gs (J/m

2)

AlN 80 1 1.74 9:6� 10�31 2.84

Ni 150 1 4.3 1:77 � 10�26 7

10−2 100 102 104

−600

−400

−200

0

200

400

600

800

1000

Stea

dy−

stat

e st

ress

(M

Pa)

γi=4.73 J/m2, γs=2.84 J/m2

Ds=Dgb=9.6 x 10−31m3/s

FEMAlN Films, Sheldon et al

−600

−400

−200

0

200

400

Stea

dy−

stat

e st

ress

(M

Pa)

FEMNi Films, Hearne et al

101100 102

γi=11.6 J/m γs=7 J/m2Ds=Dgb=1.77 10−26m3/s

Growth Flux (nm / h) Growth Flux (A/s)

Fig. 15. Comparisons between the predictions of the finite element model and those of two different experimental observations: (a) AlN

films (Sheldon et al., 2005) and (b) Ni films (Hearne and Floro, 2005).


size is known (see table). We take the cohesive zone length to be 1 A. In the simulations we vary the growthflux over several orders of magnitude so as to capture the entire range of behavior. Then, we assume that themaximum observed tensile stress in the experiments corresponded to the peak tensile stress in the simulations.Doing so fixes the value of the cohesive strength, sm. In both sets of simulations we assumed that the interfaceenergy is 5

3of the surface energy, which allows us to relate the surface energy to the already inferred cohesive

strength.The growth flux in the simulations is related to that in the experiments by ensuring that the dimensionless

fluxes match, i.e.

j0kTL3

ODsgs

� �experiment

¼j0kTL3

ODsgs

� �simulation

(32)

which amounts to extracting the diffusivity from the stress data. The resulting fits are shown in Fig. 15. Notethat the computational results span a range of growth fluxes of seven orders of magnitude, whereas theexperiments cover less than two. The model reproduces these experimental observations with a high degree ofaccuracy, despite the fact that the inferred surface energy for the case of the nickel films is larger thanexpected. Furthermore, the inferred diffusivity in nickel is about five orders of magnitude higher than that inAlN, which is consistent with the fact that AlN is a ceramic and its diffusivity is much slower than that of Ni.Furthermore, these simulations are the first attempt to model the growth stress dependence on deposition fluxbased on a detailed description of shape changes, surface diffusion and atomic interactions.

8. Conclusions

We have presented and analyzed a model of stress generation and evolution during coalescence and growthof polycrystalline thin films. The model computes the elastic displacements inside the islands and tracks theevolution of the island shape as a result of mass transport and deposition. We have shown that the model is


successful in reproducing several experimental observations for which previous models only providedrudimentary approximations. Specifically, we have shown that the films in our model exhibit the nearlyuniversally observed history of stress-thickness vs. thickness. We have presented several parametric studiesthat reveal the dependence of various stress measures on the relevant dimensionless groups of the model. Wehave compared the predictions of our model with those of previous models of tensile and compressive stressgeneration. Furthermore, the post-coalescence steady-state stress has been shown to depend on growth fluxand this variation has been quantitatively compared with experiments in Ni (Hearne and Floro, 2005) andAlN (Sheldon et al., 2005) films, albeit for unphysically large values of gs and Dgb=Ds. Additionally, the modelexhibits a large degree of compressive stress at intermediate growth fluxes, with this large compressiondiminishing as the growth flux is increased. The largest compressive stresses were observed for the highestvalues of grain boundary diffusivities ðDgb=Ds�1Þ, with smaller compressions occurring for less mobile grainboundaries ðDgb=Ds�0:01Þ.

It must be noted that some important effects have been neglected in this initial version of our model. Theseinclude any potential effects related to the three-dimensional nature of the islands, the non-periodicity of mostsystems, the varying island sizes and shapes, faceting, grain growth and plasticity. All of these effects can bereadily incorporated into the present model with the exception of the possibility of faceting, which requires anew formulation of surface/grain boundary diffusion. These extensions represent promising new directions ofresearch in the area of stress generation and evolution in thin films.

Acknowledgments

This work has been supported by the MRSEC program of the National Science Foundation under Grantno. GMR-0520651.

Appendix A

A.1. Estimate of the equilibrium grain boundary stress

A simple analytical approximation for the equilibrium grain boundary stress can be obtained as follows.Consider a two-dimensional array of grains of periodicity 2L, which form grain boundaries of height z. Thefree surfaces have curvature k ¼ �1=R, and they meet at the triple junction at an angle 2y. The surfaces havean isotropic energy per unit area gs, and the grain boundary energy is (Fig. A1)

gi ¼ 2gs � f (A.1)

where f is the work of separation. The aim of this analysis is to find the equilibrium shape of the grain,consisting of k, z, and y, and from these deduce the equilibrium grain boundary stress. For a given volume of

θ

θR

L

γI

γs

z

Fig. A1. Schematic of periodic unit cell in an array of two-dimensional grains.


material, or area in two dimensions, the shape is determined by any two of these parameters. The totalpotential energy of a unit cell of this array is

UðR; yÞ ¼ 2gsp2� y

� Rþ zðR; yÞgi

where zðR; yÞ follows from volume conservation. In equilibrium, U is stationary with respect to variations in R

and y. This condition yields a relationship between y and the surface and interface energies gi and gs of the form

cos y ¼gi2gs

(A.2)

The grain boundary stress can be determined as follows. The grain half-width is L ¼ R cos y and the surface andgrain boundary chemical potentials are, respectively,

ms ¼ �Okgs; mgb ¼ �Osn

where sn is the normal stress in the grain boundary. In equilibrium these must be equal, so that

s�gb ¼ kgs (A.3)

where s�gb is the equilibrium grain boundary stress. Combining (A.1)–(A.3) gives an expression for the equilibriumgrain boundary stress as

s�gb ¼ �gsL

1�f2gs

� �(A.4)

One important consequence of Eq. (A.3) is that as long as the surface is convex (i.e. ko0), the equilibrium grainboundary stress is always compressive.

A.2. Equilibrium under a Dugdale cohesive zone

A more elaborate description grain boundary stress can be formulated by assuming that the grain boundaryand surface interact through a simple Dugdale cohesive zone law, as shown schematically in Fig. A2. Withinthis characterization, the horizontal force acting between two points separated by a distance 2d is given by

F ðdÞ ¼

�1 for do0 ðgrain boundaryÞ

s for 0pdpD ðtriple junctionÞ

0 for d4D ðfree surfaceÞ

8><>: (A.5)

s

θ (s)γs

2L

e2

e1

n

2δ

Δ

ti

Fig. A2. Periodic cell of an array of grains modeled with a Dugdale cohesive zone. Tractions act on both sides of the surface

symmetrically, but are only shown on one side for visual clarity.


In the triple junction, the chemical potential at a point s along the surface consists of a contribution from thecohesive zone as well as one from the curvature kðsÞ, which now varies with position. Specifically,

m ¼ �Oðsn þ gskÞ

where sn ¼ F ðdÞ cos y is the normal stress, k ¼ �dy=ds is the curvature, defined such that concave surfaceshave positive curvature, and m0 is the unknown but constant chemical potential of the surface and grainboundary at equilibrium. The surface curvature kðsÞ varies with position for 0psps0, where dðs0Þ ¼ D, and itis constant ðk0Þ for s4s0. Simple geometric considerations require that

k0 ¼ �1

R0¼ �

cos yðsþ0 ÞL� D

(A.6)

In equilibrium, the chemical potential is constant and equal to that of the surface, so that in order to find theequilibrium shape of the triple junction, yðsÞ must satisfy

m0Oþ s cos yðsÞ þ gs

dyds¼ 0; yð0Þ ¼ 0; 0pypy0

where y0 is the (unknown) angle of the triple junction at the point where d ¼ D. Once a general solution foryðsÞ is found, y0 follows by noting that

D ¼Z s0

0

sin yds

Within this formulation the triple junction angle can be interpreted as the surface angle at d ¼ D. This angle isgeometrically tied to the curvature of the surface, and hence to the equilibrium chemical potential m0. Aftersome manipulation,

cos y0 ¼Zðl� 1Þ

lþ eZ½Zðl� 1Þ � l�(A.7)

where l ¼ D=L and Z ¼ f=gs. The grain boundary stress follows as

s�gbL

gs¼ �

cos y01� l

The grain boundary stress predicted by this model is plotted in Fig. 6 as a function of l and Z. In the limit asl! 0,

s�gbL

gs

D!0

¼ �e�Z (A.8)

Although there is reason to expect that this analysis would reduce to the classical one presented inAppendix A.1, Eq. (A.8) only agrees with Eq. (A.4) to first order in Z; a parameter that is not in general smallcompared to 1. The reason for this apparent discrepancy is not immediately clear.

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Numerical simulations of stress generation and evolution in Volmer–Weber thin films