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Acta Materialia 67 (2014) 189–198

Grain growth and complex stress evolution duringVolmer–Weber growth of polycrystalline thin films

Hang Z. Yu, Carl V. Thompson ⇑

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

Received 6 September 2013; received in revised form 16 December 2013; accepted 17 December 2013Available online 25 January 2014

Abstract

Volmer–Weber growth of polycrystalline thin films involves poorly understood kinetic processes that occur far from equilibrium andlead to complex co-evolution of the surface, microstructure and intrinsic stress of the films. Here we present a comprehensive study con-sisting of in situ stress measurements, microstructure characterization and analytical modeling for various metallic films that grow by theVolmer–Weber mechanism. We find that, under conditions of intermediate atomic mobility, stress evolution after coalescence involves aturnaround from a compressive to a tensile stress state. The thickness at which the stress turnaround is observed increases as the sub-strate temperature increases or the deposition rate decreases. We show that this phenomenon is associated with two competing mecha-nisms: grain growth during deposition (at homologous temperatures as low as 0.17) and adatom attachment to surface sites at grainboundaries. Grain growth during film deposition not only causes a tensile component of the intrinsic stress, but also leads to changesin the magnitude of the compressive stresses, from being independent of, to scaling with, the inverse of the grain size. Analyses of thesephenomena lead to insights into stress evolution under conditions of low and high atomic mobility, as well as intermediate mobility, andare general for stress evolution during Volmer–Weber growth of polycrystalline films.� 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Thin film; Grain growth; Intrinsic stress; In situ; Modeling

1. Introduction

Volmer–Weber growth of polycrystalline films involvesnucleation of 3-D islands on a substrate surface, and sub-sequent growth, impingement and coalescence of islandsto form continuous films [1–4]. Complex kinetic processesare involved during Volmer–Weber growth, which is typi-cally carried out far from equilibrium. Minor changes inprocessing conditions can lead to major differences in sur-face morphology, grain structure and residual stress in thefilms. This strongly influences the performance and reliabil-ity of polycrystalline films in a wide variety of applications,including nano- and microelectromechanical devices andsystems [5–7], protective coatings [8,9] and magnetic,

1359-6454/$36.00 � 2013 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2013.12.031

⇑ Corresponding author. Tel.: +1 617 253 7652; fax: +1 617 258 6749.E-mail address: cthomp@mit.edu (C.V. Thompson).

plasmonic and flexible electronic devices [10–12]. It is crit-ical, therefore, to comprehensively understand the underly-ing mechanisms for structure and stress evolution duringpolycrystalline film growth, so that the properties of thesefilms can be tailored to meet specific engineering goals.

Depending on the homologous temperature of the sys-tem (defined as the growth temperature divided by themelting temperature in K: Th = T/Tm) or the atomic mobil-ity, two types of intrinsic stress behaviors have been identi-fied [13,14] in polycrystalline films. In Type I behavior,which occurs under conditions of low atomic mobility(e.g. Pt at room temperature), a tensile stress develops asislands coalesce and is retained during further growth ofcontinuous films. In Type II behavior, which occurs underconditions of high atomic mobility (e.g. Au at room tem-perature), a tensile stress also develops during coalescence,but subsequent growth leads to evolution into a compres-sive state. In Type II behavior, post-coalescence evolution

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190 H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198

toward a compressive state results in a peak tensile stressassociated with the end of the coalescence process.

The development of a tensile coalescence stress isunderstood to be the consequence of grain boundaryformation, as explained by the Hoffman–Nix–Clemensmechanism [15–19]. In Type I behavior, the post-coalescencestress remains highly tensile due to epitaxial inheritance ofthe coalescence stress [15]. The post-coalescence compres-sive stress observed in Type II behavior has been attributedto the Laplace pressure that develops in the very smallislands in the pre-coalescence growth stage [20,21], excessatom incorporation into grain boundaries [22–24],excess atoms trapped between surface ledges [25] and excessdefects on the film surface during deposition [26]. Mostrecently, Gonzalez-Gonzalez and co-workers [27,28]attributed the post-coalescence stress to the interactionsbetween grains involving grain rotation.

Here we present a systematic study of the microstructureand intrinsic stress evolution in a range of materials undera range of experimental conditions. In particular, a stressturnaround phenomenon that strongly depends on temper-ature and deposition rate has been discovered and investi-gated under conditions of intermediate atomic mobility (orat intermediate homologous temperatures). The fact thatnone of the previously proposed models [20–28] canexplain this phenomenon calls for a better understandingof the underlying mechanisms for stress evolution duringVolmer–Weber growth. Based on microstructure charac-terization and analytical modeling, we conclude that thestress turnaround is associated with an increase in the grainsize during deposition. This grain growth in films adherentto rigid substrates leads to a tensile stress, and therefore acontribution to the tensile component of the stressobserved during film growth. We show that the magnitudeof the compressive component is controlled by a competi-tion between the attachment of adatoms to 2-D islandson the surface of grains and attachment to surface sitesat grain boundaries. The number of grain boundary surfacesites decreases as grains coarsen. Grain growth duringdeposition therefore affects the mechanisms for the devel-opment of both tensile and compressive stresses. We showthat analyses of the effects of grain growth during filmgrowth explain the transition from Type I to the intermedi-ate type and to Type II behavior under conditions ofincreasing atomic mobility.

2. Experimental procedures

Films were deposited in an ultra high vacuum e-beamevaporation system, with a base pressure of 5.0 � 10�9

torr. A residual gas analyzer was used to monitor the com-position and the pressure of the gases inside the chamber.During growth, the chamber pressure was kept below1.0 � 10�8 torr. Films of platinum, palladium, nickel andgold – all face-centered cubic (fcc) metals – were studied.All source materials had purities above 99.99%.

We made in situ stress measurements during film depo-sition using the cantilever deflection method [29,30]. Whena film is deposited on a cantilever, the intrinsic stress in thefilm causes the cantilever to deflect. Based on Stoney’sequation [31], the force per unit width (F/w) exerted atthe film–substrate interface is proportional to the deflectionof the cantilever. The intrinsic stress evolution can thus bedetermined by measuring the magnitude of the deflectionduring deposition. In this work, the cantilever deflectionwas measured using a capacitance-based method in whichwe monitored the capacitance change between the freeend of the cantilever and a fixed electrode during deposi-tion. This method has a spatial resolution of 1 nm for can-tilever deflection and a time resolution of 100 Hz [29,30,32].The cantilevers used in this work had a dimension of50 � 18 � 0.2 mm, and were cleaved from 200 lm thick(10 0) Si wafers coated with 33 nm low-stress silicon nitridelayers.

Depositions and in situ measurements were carried outat 300–473 K. The deposition rate was monitored using aquartz crystal microbalance and calibrated using an atomicforce microscope. The microstructures of the films werecharacterized using a JEOL 2011 transmission electronmicroscope in bright-field and dark-field mode. Samplesfor plan view transmission electron microscopy (TEM)imaging were prepared by growing the films on SiNx mem-brane windows so that they could be characterized immedi-ately upon removal from the deposition system. Samplesfor cross-sectional TEM imaging were thinned and pre-pared by mechanical polishing followed by ion milling.The average grain size in a film was determined by tracingthe boundaries of hundreds of grains on transparencies fol-lowed by automated image analysis of the transparencies.Each data point in the grain size plot was determined usingmultiple TEM images.

3. Stress turnaround at intermediate homologous

temperatures

In situ stress measurements give the force per unit width(F/w) as a function of the film thickness h. Due to a forcebalance at the interface between the film and the substrate[25], F/w =

PjrjDhj, where rj and Dhj are the stress in, and

the thickness of, the jth layer of the film andP

jDhj = h. Wedefine the instantaneous stress rin as the slope of the stress

curve at a given thickness, rin ¼ DðF =wÞDh . If the stress in the

bulk of the film does not change during deposition, theinstantaneous stress rin is equal to the stress in the newlydeposited surface layer. Otherwise, the instantaneous stressis influenced by both the bulk and the surface of the film.This can be quantitatively understood as follows. Assumethat the film thickness is h at time t. During a very smallperiod of time (t! t + Dt), a new layer of thickness Dh isdeposited on the film surface with a stress of r*(t) in thislayer. During the same period, the change in the averagestress in the bulk of the film is D�rðt! t þ DtÞ. The change

Fig. 2. Ni films deposited at 373 K at different deposition rates. Arrowsindicate the stress turnaround thickness.

H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198 191

in (F/w) is then DðF =wÞ ¼ r�ðtÞDhþ D�rðt! t þ DtÞh, andthe instantaneous stress at a given time t is

rinðtÞ ¼DðF =wÞ

Dh¼ r�ðtÞ þ D�rðt! t þ DtÞh

Dhð1Þ

The two terms on the right side of the equation correspondto the contributions from the surface and the bulk of thefilm, respectively.

Our in situ stress measurement results are shown inFigs. 1 and 2. Fig. 1 shows stress evolution during deposi-tion of a number of polycrystalline materials. At room tem-perature, Pt and Au display typical Type I and Type IIstress behavior, respectively. We have also discovered anintermediate type of behavior, shown here for Ni depositedfrom 300 to 473 K and for Pd deposited at 300 K, all at adeposition rate of 0.5 A s�1. Similar to Type II behavior, inthis intermediate behavior the stress decreases from amaximum tensile value after film coalescence. However,unlike Type II behavior, the instantaneous stress rin

(i.e. the slope of the stress curve at a given thickness;

rin ¼ DðF =wÞDh ; with h being the film thickness) continues to

evolve during film thickening, eventually turning aroundto trend back to an increasingly tensile state. In other words,the force per unit width (F/w) reaches a local minimum. Werefer to the associated film thickness as the turnaroundthickness. It is interesting to note that the melting pointsof Ni and Pd lie between those of materials displaying TypeI (e.g. Ti, Cr, Pt) and Type II (e.g. Au, Ag, Cu) behavior atroom temperature [13,14]. The stress turnaround can thus beviewed as a signature of the intrinsic stress developed insystems of intermediate atomic mobility. The data for Nifilms in Fig. 1 show that the turnaround thickness increaseswith increasing substrate temperature.

Fig. 2 shows the deposition rate dependence of theturnaround behavior in Ni films deposited at 373 K. The

Fig. 1. Intrinsic stress evolution at different homologous temperatures,shown here for Pt, Pd and Au films deposited at 300 K and Ni filmsdeposited at 300, 333, 373, 398, 423 and 473 K. All depositions werecarried out at a deposition rate of 0:5 A s�1. Arrows indicate thethicknesses at which stress turnaround happens.

deposition rates were kept constant at 0:5 A s�1 duringgrowth of the initial 15 nm in all cases, to ensure that thefilms had similar grain structures right after coalescence.From 15 to 100 nm, deposition rates were 2.5, 1.3, 0.8,0.5 and 0:3 A s�1, respectively. Clearly, the turnaroundthickness increases as the deposition rate decreases. Thefindings in Figs. 1 and 2 indicate that at least two mecha-nisms operate during post-coalescence growth: one causesa tensile component of the intrinsic stress and the othercauses a compressive component that is favored at highertemperatures or at lower deposition rates.

4. Grain growth and the tensile component

Evolution of a film’s microstructure during depositionstrongly influences the development and evolution of itsintrinsic stress. However, this effect has not been taken intoaccount in previous models [20–28]. TEM was used tostudy the microstructures of films in this work. Fig. 3(a)–(c) shows plan-view, bright-field TEM images of Ni filmsdeposited at 300 K (a homologous temperature of 0.17),with film thicknesses of 7, 20 and 37 nm, respectively.The grain size is clearly seen to increase with the film thick-ness. For Fig. 3(d) image analysis was used to quantify theaverage grain size in Ni films as a function of the film thick-ness. The range of film thickness matches the plots inFig. 1. It can be seen that the grain size scales with filmthickness, and this linear relationship is very weakly depen-dent on temperature. Cross-sectional TEM imaging con-firms that all the films consist of columnar, transversegrains; we did not find any evidence for renucleation of3-D islands on the grain surfaces during film thickening.

Grain growth is a thermally activated process. For puremetals, its activation energy is approximately equal to halfthe activation energy for self-diffusion [33,34]. However,the grain growth in Ni films shown in Fig. 3 happens at

Fig. 3. Plan-view, bright-field TEM images of (a) 7 nm, (b) 20 nm and (c)37 nm Ni films deposited at 300 K. (d) The grain size-thickness relation-ship in Ni films deposited at 300, 373 and 473 K.

192 H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198

a homologous temperature of 0.17. This can be attributedto the high driving force for grain growth due to the nano-scale grain size in the film. Grain growth during and aftercoalescence of metallic films deposited at Th � 0.2–0.3 iswell known [1]. Grain growth has also been observed inbulk nanocrystalline Cu [35] and Pd [36] at room tempera-ture (Th � 0.16) and Ni near room temperature (Th � 0.2–0.3) [37,38]. In thin films, grain growth at equivalentlysmall grain sizes may occur at still lower homologous tem-peratures than in bulk materials, owing to the relativelylow impurity contents achieved when films are formedunder high vacuum conditions. In addition, experimentaland modeling work [39–44] has shown that grain growthcan occur at room temperature or cryogenic temperaturesunder an influence of stress. During thin film growth, there-fore, the high internal stress (as shown in Figs. 1 and 2)may be another factor influencing grain growth. In the cur-rent experiments, we find that the grain size in Ni depositedat 300–473 K scales with the film thickness at which depo-sition is stopped. This scaling of grain size with film thick-ness is often observed in thin films and sheets in whichnormal grain growth stagnates [45,46]. Mullins [45] hassuggested that this phenomenon is caused by the develop-ment of grain boundary grooves and trapping of theboundaries of most grains. In some cases, abnormal graingrowth occurs when surface or strain energy favors thegrowth of a sub-population of grains with a specific texture[1,45]. This phenomenology is consistent with the behaviorobserved in the Ni films deposited at 473 K. However,given that normal grain growth occurs during deposition

of Ni films at lower temperatures, it is not clear whypost-deposition abnormal grain growth does not also occurat lower temperatures.

Grain growth leads to elimination or redistribution ofthe excess free volume associated with grain boundaries,and can lead to a tensile stress in a film adherent to a sub-strate [47]. During deposition, grain growth continuouslychanges the bulk stress in the film and gives rise to a tensilecomponent of the measured instantaneous stress. Considera very small period of deposition (t! t + Dt), duringwhich the average grain size increases as d! d + Dd andthe film thickness increases as h! h + Dh. The increasein average stress in the bulk of the film owing to densifica-tion is [48]:

D�rðt! t þ DtÞ ¼ MDa1

d� 1

d þ Dd

� �

Here, Da is the excess volume per unit area of grain bound-ary and M is the biaxial modulus of the film. The measured(F/w) increases by DðF =wÞðt! t þ DtÞ ¼ D�rðt! t þ DtÞh.The linear relationship in Fig. 3(d) leads to Dd

d ¼ Dhh .

Therefore, the instantaneous stress caused by grain growthis

rggin ðdÞ ¼

DðF =wÞðt! t þ DtÞDhðt! t þ DtÞ ¼ MDa

Dd

d2� h

Dh¼ M

Dad

ð2ÞGrain growth continues to occur during deposition so

that more grain boundary area is eliminated in the bottomlayers of the film than in the top layers during deposition.As a result, this mode of grain growth gives rise to a straingradient through the thickness of the film, with the top lay-ers being less tensile than the bottom layers. Given that theinitial grain size is only a few nanometers (see Fig. 3), thetotal densification stress in the bottom layers can exceedthe yield stress of the film during subsequent thickening.Considering the effect of strain gradient and yielding, theexpression in Eq. (2) can be modified as

rggin ðdÞ ¼ M

Dad

1

1þM Dad

1ry

!ð3Þ

where ry is the yield limit of biaxial tensile stress. The der-ivation of Eq. (3) is shown in Appendix A.

5. A model for the compressive component of the intrinsic

stress: competition between adatom attachments to 2-D

islands and to grain boundaries

Chason and co-workers [22–24] proposed that the com-pressive stress that develops in Type II behavior is associ-ated with adatom incorporation into grain boundariesduring film growth. The model assumes that the surface–grain boundary transition occurs at a much lower rate thansurface diffusion and grain boundary diffusion, andtherefore is the rate-limiting process [22]. Subsequently,the important role of grain boundaries in developing

H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198 193

compressive stress has been shown by a number of experi-ments and simulations (e.g. [29,49]).

Here we develop a model that accounts for effects of grainsize, and the corresponding grain boundary length per areaof surface, on the magnitude and evolution of the compres-sive component of the intrinsic stress observed in the inter-mediate type and Type II behavior. It should be notedthat Chason and co-workers argued that some componentof the compressive stress observed due to adatom trappingat boundaries during deposition can be relieved throughout-diffusion from grain boundaries during interruptionsof growth. Our treatment does not require that this aspectof Chason et al.’s model be correct, as we are focusing onthe instantaneous stress during deposition. We discuss themechanism of “reversible” evolution of the compressivestress during a growth interruption elsewhere [50].

Growth on a single-crystal surface often occurs throughnucleation, growth and coalescence of 2-D islands [51,52].For a continuous polycrystalline film with a flat surface,such processes can occur on each grain’s surface, whereinboth the edges of 2-D islands (monolayers) and grainboundaries (GBs) provide low chemical potential sites foradatom attachment (see Fig. 4(a)). Adatoms attaching tothe perimeter of 2-D islands form a more close-packed con-figuration than adatoms that attach to GB surface sites. As aresult, the adatoms at GB surface sites are in higher energystates and thus have a higher detachment rate. Also, the den-sity of 2-D islands is usually higher than the density of GBsites. Consequently, most adatoms contribute to 2-D islandgrowth (i.e. film thickening), while only a small fraction

Fig. 4. A model for the compressive component of the intrinsic stress. (a) ProEnergy landscape for adatoms deposited between a 2-D island and a GB. (c) T2-D island incorporation and adatom–GB attachment, respectively. (For interpto the web version of this article.)

attach to GBs. The corresponding energy landscape isshown in Fig. 4(b). Here ES is the activation energy for sur-face diffusion. Dl1 and Dl2 are the chemical potential differ-ences between the grain surface and 2-D island edges, andbetween the grain surface and GB surface sites, respectively.

Multiple 2-D islands can form on each grain surface.Assuming d is the distance between a GB and its neighbor-ing 2-D island, during deposition there is a 2d wide capturezone (shown in Fig. 4(c)) for which adatoms deposited out-side this capture zone will be captured by 2-D islandsbefore they reach the GB. Adatoms deposited inside thecapture zone will either be captured by 2-D islands or byGB surface sites. We assume that, for the adatoms depos-ited inside the capture zone, the average probabilities ofbeing captured by GBs and by 2-D islands are p and1 � p, respectively. The value of p depends on the differencein the chemical potentials of the sites at GBs and 2-Dislands, as well as on the local adatom concentrations dur-ing growth. For adatoms deposited outside the capturezone, the probabilities of being captured by GBs and by2-D islands are 0 and 1. Therefore, for each grain, the ratiobetween the numbers of adatoms captured by GBs and by2-D islands is

N adatom–GB

Nadatom–2Disland¼ 0� ðd � 2dÞ þ p � 2d

1� ðd � 2dÞ þ ð1� pÞ � 2d� 2pd

d

This process leads to a compressive stress in the surface

layer during deposition (i.e. the r*(t) term in Eq. (1)),and therefore a compressive component of the instanta-neous stress,

cesses for adatom attachment to 2-D islands and to GB surface sites. (b)he capture zone with width 2d. The red and blue arrows refer to adatom–retation of the references to color in this figure legend, the reader is referred

Fig. 5. Descriptive explanation for the stress turnaround phenomenon. (a)A schematic illustration for the post-coalescence mechanisms of tensileand compressive components of intrinsic stress generation. (b) Aschematic plot of the instantaneous stresses caused by grain growth,incorporation of excess atoms at GBs, and the sum of the two, as afunction of 1

d.

194 H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198

rcompin ¼ �M

2pdd

� �

By definition, the value of d cannot exceed the value of theaverage 2-D island spacing Lisland. Assuming a uniform dis-tribution of the value of d ranging from 0 to Lisland on thefilm surface, the compressive component of the instanta-neous stress caused by adatom–GB incorporation will be

rcompin ¼ �M

2pdd

� �¼ �M

pLisland

d

� �; ð4Þ

which is proportional to the inverse of the grain size.An adatom dimer is stable under normal experimental

conditions. In this case, the classical nucleation theory pre-dicts the 2-D island spacing is [52]

Lisland ¼ kg�1=2 D0

Rk

� �e�Es=kBT

� �1=6

where k is the lattice spacing, R is the deposition rate, D0 isthe diffusivity constant and g is a dimensionless prefactorwhose maximum value has been estimated to begmax = 0.25 [53]. Under our experimental conditions, clas-sical nucleation theory gives a value for Lisland for Ni filmson the order of 10 nm, larger than the coalescence grainsize shown in Fig. 3(d). This means that, right after coales-cence, d < Lisland, and there is one island forming on thesurface of each grain. As a result, the entire grain surfaceis covered by the capture zone. In this small grain size re-gime (d < Lisland), the ratio between the numbers of ada-toms attaching to GBs and to 2-D islands is

Nadatom–GB

Nadatom–2Disland¼ p � dð1� pÞ � d

¼ p1� p

and the corresponding instantaneous stress is

rcompin ¼ �M

p1� p

ð5Þ

independent of the grain size.The model articulated in this section shows that, during

growth of polycrystalline films, there is a competitionbetween adatom attachment to the surface sites at GBsand adatom attachment to the periphery of the 2-D islandsthat nucleate on the grain surface. The former process cancause a compressive stress in the surface layer duringgrowth, which leads to a compressive component of theoverall instantaneous stress. The magnitude of the com-pressive component is independent of the grain size whend < Lisland but becomes proportional to the inverse of thegrain size when d > Lisland. The transition of the tworegimes happens at d = Lisland.

6. Discussion

6.1. A descriptive explanation

So far, we have discussed two competing mechanismsfor intrinsic stress generation during post-coalescence film

growth: grain growth during deposition and adatom trap-ping at GBs (see Fig. 5(a)). Eq. (3) suggests that the graingrowth component of the instantaneous stress (rgg

in )decreases slightly as the grain size increases. As shown inour model in Section 5, the compressive componentrcomp

inð Þ stays constant in the small grain size regime(d < Lisland) but decreases in the large grain size regime(d > Lisland). These behaviors are schematically plotted inFig. 5(b). As a growing film evolves into the large grain sizeregime, the overall instantaneous stress becomes less com-pressive and eventually turns to tensile, resulting in thestress turnaround phenomenon. At higher substrate tem-peratures or with lower deposition rates, Lisland increasesso that the transition to the large grain size regime occursin thicker films. Under such conditions, therefore, stressturnaround is observed in thicker films. This is consistentwith the experimental results shown in Figs. 1 and 2.

6.2. Quantitative test

The above explanation can be tested quantitativelyusing experimental results for Ni films. First, we obtain

instantaneous stress (rin) values using rin ¼ DðF =wÞDh based

on the data in Fig. 1. rggin is calculated using Eq. (3), so that

the compressive component can be determined as

H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198 195

rcompin ¼ rin � rgg

in . To calculate rggin , we assume that the GB

width Da (excess volume per area) is around 1.0 A for fccmetals [48,54]. The tensile yield stress of nanocrystallineNi has been reported to range from a few hundred MPaup to more than 1 GPa [55–59], depending on the process-ing method, microstructure, film thickness and grain size,as well as the grain size distribution. In this work, the max-imum tensile stress measured in situ was 600 MPa. There-fore, we performed a series of calculations for the tensileand compressive components, with ry ranging from 450to 750 MPa (corresponding to 100 ± 25% of the maximumtensile stress measured in situ), Da ranging from 0.5 to1.0 A and M = 290 GPa [60]. In all cases, two regimesare found in the rcomp

in vs. 1d plot (one example is shown in

Fig. 6(a)). When the grain size is small, rcompin weakly

depends on 1d (flat regime); when the grain size becomes suf-

ficiently large, the magnitude of rcompin decreases linearly

with 1d (linear regime). In addition, the boundary of the

two regimes moves toward larger grain sizes when the filmsare deposited at higher temperatures.

Fig. 6. Quantitative test of the proposed model. (a) Plot of rcompin as a

function of 1d based on data measured in situ and Eq. (3), using Da = 1 A,

ry = 600 MPa and M = 290 GPa for Ni. (b) 2-D island spacing as afunction of temperature determined by the experimental data andmodeling. Inset: an Arrhenius plot.

The plots in Fig. 6(a) are consistent with the model illus-trated in Fig. 4 and the schematic plot shown in Fig. 5(b).First, we see a flat–linear transition of the compressivecomponent when the grain size increases, as predicatedby the model. Here the linear and flat regimes inFig. 6(a) correspond to the large and small grain sizeregimes in the model. Second, when the substrate tempera-ture is higher, this transition occurs in thicker films, whichhave larger grain sizes. According to our model, this isbecause the transition happens when d = Lisland, and Lisland

increases with substrate temperature.By defining the boundaries of the flat regime and linear

regime in Fig. 6(a), we can determine the values of Lisland

during Ni film growth at different temperatures. As shownin Fig. 6(b), Lisland ranges from �12 to 20 nm within thetemperature range of 300–473 K. These values are consis-tent with those calculated based on classical nucleation the-ory. The Arrhenius plot (inset) shows that the effectiveactivation energy associated with Lisland is 0.035 eV. Com-paring with the equation for Lisland (Lisland / e�Es=6kBT ), theactivation energy for adatom diffusion is Es = 0.035 -� 6 = 0.21 eV. Adatom diffusion strongly depends on thecrystallographic orientation of the surface. (200) and(111) X-ray pole figure measurements show that the as-deposited Ni films are (111) textured. Therefore, we con-clude that the activation energy for adatom diffusion on(111) textured Ni film surface is 0.21 eV. This result is con-sistent with values obtained using other techniques. Forexample, using field ion microscopy (FIM), Fu and Tsong[61] concluded that the activation energy for adatom diffu-sion on Ni (111) surface is Es = 0.22 ± 0.02 eV.

Fig. 7 shows the stress evolution in Ni films predictedusing the model (Eqs. (3)–(5)). Here the empirical pre-coa-lescence data obtained from the in situ stress measurementsare shown as continuous lines. The post-coalescence resultsas derived from the model are shown as dashed lines.

Fig. 7. Stress curves predicted for Ni using Eqs. (3)–(5). Inset: theaverage stress and the thickness at the stress turnaround point. Theparameters used were Es = 0.21 eV, according to the data in Fig. 6(b),D0 = 1 � 10�9.2 m2 s�1 (based on FIM experiments [61]), g = 0.19 [47]and p = 0.002.

Fig. 8. A stress evolution map as a function of the homologous substratetemperature and the deposition rate, with experimental data for Pt, Ti, Cr,Ni, Pd, Au, Ag, Cu and Al at various temperatures and deposition rates.

Fig. A1. Illustration of how grain growth affects the strain in the film. (a)At time t, the film thickness is h = Rt and the grain size is d = kh = kRt.(b) The time when the layer of thickness z was first deposited. The in-planesize was d0 = kz.

196 H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198

Clearly, the simulated curves capture all the important fea-tures of the experimental results in Figs. 1 and 2, includingthe stress turnaround phenomenon, the temperature andrate dependencies of the turnaround thickness, and theshapes of the stress curves. In the inset, we plot the averagestress at the turnaround point as a function of the turn-around thickness under various conditions. The modelingresults overlap with the experimental data in all the cases.

6.3. Explanations for Type I and Type II stress behavior

For Type I stress behavior, Nix and Clemens [15] havesuggested that the epitaxial inheritance of the coalescencestress can lead to a constant tensile stress during film thick-ening. This mechanism only weakly affects the shapes ofthe stress curves under conditions of high and intermediateatomic mobility. Under conditions of low atomic mobility,however, it becomes the dominant mechanism becauseboth grain growth and adatom diffusion are suppressed.As a result, a constant tensile stress is observed after coales-cence [13–15,30]. This is true even if the adatom GB trap-ping process is not completely turned off. In that case, thecompressive component is constant but small compared tothe tensile coalescence stress.

In Type II systems, such as Au, Ag, Cu and Al filmsdeposited at room temperature, the grain size is alwayscomparable to the film thickness [62]. Nevertheless, thepost-coalescence stress evolves to a compressive state. Thisindicates that the adatom–GB attachment mechanism isdominant compared to the grain growth mechanism.Because of the high atomic mobility and the large 2-Disland spacing, the transition from the flat regime to the lin-ear regime occurs at very large film thicknesses. However,in situ stress measurements are usually limited to film thick-nesses of �100–200 nm. As a result, the stress turnaroundphenomenon is generally not observed. We note that inrecent experiments we have found that, for Au films depos-ited at 300 K with a deposition rate of 1 nm s�1, stressturnaround is observed when the film thickness exceeds1.5 lm.

Consequently, it can be seen that there are three types ofintrinsic stress evolution behavior during Volmer–Webergrowth: Type I, the intermediate type and Type II, withthe post-coalescence stress being tensile, compressive totensile, and compressive, respectively. The transition fromType I to the intermediate type and then to Type II behav-ior occurs as the atomic mobility increases and when therate of deposition is decreased. This behavior can be sum-marized using the mode of stress evolution map shown inFig. 8, in which the three stress behaviors are distinguishedfor different deposition rates and homologous temperatures(Th = the substrate temperature divided by the meltingtemperature, both in K). The transitions between modesare affected by grain growth and surface diffusion, bothof which are thermally activated. The surface diffusiondistance is affected by adatom interactions and thereforethe deposition rate. Recognition of these dependencies

motivated the plotting of the data in an Arrhenius plot.The positions of the boundaries were empirically deter-mined. Note that the stress turnaround phenomenon mayalso occur in Type II materials if the film is deposited tohigh thicknesses. As a working definition, therefore, wehave categorized stress evolution in which the stress turnsaround in the compressive regime as Type II behavior.The intermediate type is restricted to stress evolution inwhich the turnaround occurs before evolution into thecompressive regime. Based on this definition, Fig. 8 plotsthe experimental data for Pt, Cr [63], Ti [64], Ni, Pd, Au,Ag [22] Cu [26] and Al [65]. This map should be semi-quan-titatively applicable to fcc metals in general, due to correla-tions of diffusion coefficients for a given crystal structureand bond type [66]. It is interesting to compare the stresscurves for Ni at 398 K, 0.5 A s�1 and Au at 300 K,0.5 A s�1. Here Au has a slightly lower homologous tem-perature but develops a more compressive stress than Ni.This might be attributable to Ni’s relatively high reactivity

H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198 197

and interactions with residual gases in the chamber, such asoxygen. However, other more fundamental origins of thisdifference cannot be ruled out.

7. Conclusions

Using in situ stress measurement, microstructure evolu-tion and analytical modeling, we have studied the funda-mental mechanisms for intrinsic stress generation andevolution in Volmer–Weber films of low, intermediateand high atomic mobility. The most important results fromthis work include the following:

� The intrinsic stresses in polycrystalline films can be cat-egorized into three types: Type I, the intermediate typeand Type II. The intermediate type of stress evolutionfeatures a post-coalescence transition from a compres-sive to a tensile stress state. This stress turnaroundphenomenon is strongly dependent on processingparameters, with the turnaround thickness increasingas the substrate temperature is increased or the deposi-tion rate is decreased.� Grain growth occurs during deposition at intermediate

homologous temperatures, shown here for Ni. Graingrowth makes the bulk of the film more tensile or lesscompressive, and leads to a tensile component of theinstantaneous stress. During film thickening, adatomattachment to the surface sites of the grain boundariescauses a compressive stress in the surface layer, leadingto a compressive component of the overall instanta-neous stress. The magnitude of the compressive compo-nent is shown to be a function of the grain size. As thegrain size increases, the compressive component changesfrom being independent of the film thickness (flatregime) to scaling with the inverse of the grain size(linear regime).� Grain growth and adatom attachment to grain bound-

aries are competing mechanisms during Volmer–Webergrowth. As the grain size increases, the tensile compo-nent becomes dominant and the overall instantaneousstress changes from compressive to tensile. This is theorigin of the stress turnaround phenomenon. At highertemperatures, adatom diffusion distances are larger sothat the compressive component becomes more impor-tant. In the case of Ni, however, the grain size–filmthickness relationship is weakly influenced by tempera-ture. As a result, at higher temperatures the stress turn-around occurs at higher film thicknesses, if at all.� Grain growth and adatom diffusion are suppressed

under conditions of low atomic mobility (Type I), sothe post-coalescence tensile stress is caused by epitaxialgrowth on the grains formed at coalescence. The transi-tion from Type I to the intermediate type of behaviorand then to Type II behavior is continuous, and canbe achieved by adjusting the homologous temperatureof the system and the deposition rate.

Appendix A. Derivation of Eq. (3) in the text

Grain size scales with the film thickness during deposi-tion, i.e. d/h = k, where k is a dimensionless constant.Assuming that the deposition starts at t = 0, at time t thefilm thickness is h = Rt and the grain size is d = kh, whereR is the deposition rate. The in-plane size in the layerpositioned at z (z is the distance from this layer to the sub-strate-film interface) is also d = kh = kRt (see Fig. A1 forillustration). When this layer was deposited to a thicknessz, its in-plane grain size was d0 = kz (Fig. A1(b)). At a latertime, when the film thickness h is reached, the grain size hasincreased to d = kRt (Fig. A1(a)). Therefore, the total den-sification strain in this layer at time t is

eðz; tÞ ¼ Dak

1

z� 1

Rt

� �ðA1Þ

Eq. (A1) describes how the strain distribution changes withtime. Here, z is the position of a certain layer. If all thestrains are accommodated elastically, the stress distributionevolves as rðz; tÞ ¼ M Da

k ð1z � 1RtÞ. However, given that the

initial grain size is only a few nanometers under normalexperimental conditions, the total densification stress inthe bottom layers can exceed the yield limit of the film dur-ing subsequent thickening. If the layer at zy has reached theyield limit (i.e. layers at z < zy have all reached the yieldlimit),

ry ¼ MDak

1

zy� 1

Rt

� �ðA2Þ

where ry represents the yield stress. Therefore,

zyðtÞ ¼ Rt1

1þ ry ðkRtÞMDa

ðA3aÞ

or

zyðhÞ ¼ h1

1þ ry dMDa

ðA3bÞ

The stress distribution for any time t during depositionis

rðz; tÞ ¼ ry when z 6 zy ðA4aÞand

rðz; tÞ ¼ MDak

1

z� 1

Rt

� �when z > zy ðA4bÞ

Because the stress level in the layers from z = 0 to z = zy

has reached the yield limit, the stresses in these layers willnot increase due to grain growth during subsequent depo-sition. At any time t during deposition, only the layers fromz = zy to z = Rt become more tensile and contribute to thechange in the measured (F/w). As a result,

DðF =wÞðt! t þ DtÞ ¼ Drðt! t þ DtÞðh� zyÞ ðA5ÞThe instantaneous stress caused by grain growth during

deposition is therefore

198 H.Z. Yu, C.V. Thompson / Acta Materialia 67 (2014) 189–198

rggin ðdÞ ¼

Drðt! t þ DtÞðh� zyÞDh

¼ MDad

1

1þ MDary d

!ðA6Þ

which is given as Eq. (3) in the text.

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