Evolution of normal stress and surface roughness in buckled thin filmsG. Palasantzas and J. Th. M. De Hosson Citation: Journal of Applied Physics 93, 893 (2003); doi: 10.1063/1.1528299 View online: http://dx.doi.org/10.1063/1.1528299 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/93/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamic behaviors of controllably buckled thin films Appl. Phys. Lett. 95, 231915 (2009); 10.1063/1.3273385 Contact stress analysis of lightly compressed thin films: Modeling and experimentation J. Appl. Phys. 105, 124907 (2009); 10.1063/1.3143893 Surface effects on buckling of nanowires under uniaxial compression Appl. Phys. Lett. 94, 141913 (2009); 10.1063/1.3117505 Buckling modes of elastic thin films on elastic substrates Appl. Phys. Lett. 90, 151902 (2007); 10.1063/1.2720759 Mesoscale x-ray diffraction measurement of stress relaxation associated with buckling in compressed thin films Appl. Phys. Lett. 83, 51 (2003); 10.1063/1.1591081

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Evolution of normal stress and surface roughness in buckled thin filmsG. Palasantzas and J. Th. M. De Hossona)

Department of Applied Physics, Materials Science Center and Netherlands Institute for Metals Research,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

~Received 17 July 2002; accepted 22 October 2002!

In this work we investigate buckling of compressed elastic thin films, which are bonded onto aviscous layer of finite thickness. It is found that the normal stress exerted by the viscous layer on theelastic film evolves with time showing a minimum at early buckling stages, while it increases at laterstages. The normal stress also shows a minimum as a function of applied compressive stress, whichdepends strongly on the viscosity of the underlying layer and strain values. Furthermore, withdecreasing viscosity the film roughness amplitude also shows a minimum at early buckling stages.The effect of the viscosity becomes more pronounced with increasing strain in the film. Finally,decreasing elastic film thickness and/or increasing viscous layer thickness also enhance bucklingroughness. ©2003 American Institute of Physics.@DOI: 10.1063/1.1528299#

I. INTRODUCTION

Thin films in modern device technology are often in astate of compression. Actually, the mismatch between ther-mal expansion coefficients may produce compressivestresses in thermal barrier coatings and heteroepitaxialgrowth may be accompanied by the evolution of compres-sive stresses. Nevertheless, by adhering a compressed thinfilm to a low viscosity glass, the compressive stresses can berelieved. In particular, this methodology has been explored inthe growth of low dislocation and relaxed heteroepitaxialsemiconductor films.1 Moreover, atomic force microscopyand cross-sectional scanning electron microscopy/transmission electron microscopy have shown buckling ofcompressively strained SiGe films~deposited on borophos-phorosilicate! during annealing.2

In general, any freestanding film, which is subject tocompression, will spontaneously display buckling at laterallength scales that strongly depend on its elastic properties,the thickness, and the magnitude of the applied stress.3 Thefilm expands out of its plane, which leads to buckling, with acharacteristic wavelength that is the result of the competitionof the in-plane strain relaxation and the elastic stress due tobending.4,5

So far, a stability analysis of the buckling problem forthin elastic films with a finite thickness onto a viscous layerpredicted the growth rates of preexisting undulations thatdevelop in time.4 However, the former calculations did notencounter the problem of how the buckling amplitude devel-ops assuming an initial rough profile of the elastic film, aswell as the how the normal stress that the elastic film feelsfrom the viscous films changes with progressing film buck-ling. This will be the topic of the present work where forsimplicity we will consider the case of the initial elastic filmsurface roughness to be self-affine type. The latter assump-tion is based on the fact that the formation of self-affine

roughness has been observed for a wide range of depositedthin films ~i.e., metallic, semiconductor, and organic!.6–9

II. BUCKLING FILM THEORY

We consider an elastic film of thicknesshf , YoungmodulusE, and Poisson ration. The elastic film is assumedto be bonded onto a viscous substrate with viscosityh andthicknesshg , which is also assumed to be bonded onto arigid substrate~Fig. 1!. The elastic film is under compressivestresss, which is related to a misfit straine by s5Ee/(12n). When the film has buckled under compression andbending with the vertical displacementh(r ) (!hf) given byRefs. 3 and 4 for wavelength much larger thanf hf

Ehf3

12~12n2!¹4h1shf¹

2h1sN50. ~1!

In Eq. ~1! sN is the normal stress exerted on the film by theviscous substrate~at the elastic/viscous film interface!, andr5(x,y) the in-plane position vector. Assuming the Fouriertransformh(r )5*h(k,t)e2 ik"rd2r , Eq. ~1! yields the normalstresssN

sN5E Fshfk22

Ehf3

12~12n2!k4Gh~k,t !e2 ik"rd2k, ~2!

with k25kx21ky

2 . The problem of an infinite viscous layer(hg→1`) was solved in the past by Mullins.5 For the linearboundary value problem the velocity of the elastic film sur-face]h/]t is proportional to a strain ratesN(k,t)/h for eachFourier modeh(k,t) of the form4

]h~k,t !

]t5g

sN~k,t !

2hk, g5

sinh~2hgk!22hgk

11cosh~2hgk!12~hgk!2,

~3!

with sN(k,t)5$shfk22@Ehf

3/12(12n2)#k4%h(k,t). Inte-gration of Eq.~3! yields4

h~k,t !5h~k,0!ea(k)ta!Author to whom correspondence should be addressed; electronic mail:hossonj@phys.rug.nl

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 2 15 JANUARY 2003

8930021-8979/2003/93(2)/893/5/$20.00 © 2003 American Institute of Physics

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a~k!5E

24h~12n2!F sinh~2hgk!22hgk

11cosh~2hgk!12~hgk!2G3@b~hfk!2~hfk!3#, ~4!

with b512e(11n) ande represents the misfit strain.

III. ROUGHNESS MODELS AND RELATEDPARAMETERS

In the following we will assume translation invariantfilm surface roughness or̂ h(k,t)h(k8,t)&5@(2p)4/A#3^uh(k,t)u2&d2(k1k8). A is the average macroscopic flatsurface area, and̂...& means ensemble average over possibleroughness configurations. Therefore, since Eq.~4! yields theroughness spectrum̂uh(k,t)u2&, we can obtain the rmsroughness amplitude of the buckled film surface and the av-erage normal stress

w~ t !5F ~2p!5

A E0<k<Qc

^uh~k,0!u2&e2a(k)tk dkG1/2

, ~5!

savN5H ~2p!5

A E0

QcFshfk22

Ehf3

12~12n2!k4G 2

3^uh~k,0!u2&e2a(k)tk dkJ 1/2

~6!

under the constraint thatw(t)!hf because the present theoryrequires weak buckling.4

Furthermore, in order to calculate the roughness relatedparametersw(t) and sav

N , the knowledge of the initialroughness spectrum̂uh(k,0)u2& is necessary in Eqs.~5!–~6!.Indeed, a wide variety of surfaces/interfaces in thin films arewell described by self-affine fractal scaling6–9 where^uh(k,0)u2& follows the power law scaling relations^uh(k,0)u2&}k2222H if kj@1, and ^uh(k,0)u2&}const if kj!1. j is the in-plane roughness correlationlength of the initial film surface. We also definew0 as theinitial roughness amplitude of the film surface att50. Theroughness exponentH is a measure of the degree of surfaceirregularity, such that small values ofH characterize moreirregular surfaces at short roughness wavelengths~,j!. Thisscaling behavior can be described by the simple Lorentzianmodel10

^uh~k,0!u2&5A

~2p!5

wo2j2

~11ak2j2!11H, ~7!

with a5(1/2H)@12(11aQc2j2)2H# if 0 ,H,1, and Qc

5p/a0 with a0 on the order of the atomic spacing. It shouldbe realized thatQc is a rather large number and Eq.~1!breaks down earlier. For other self-affine roughness modelssee also Ref. 9.

IV. RESULTS AND DISCUSSION

Our calculations were performed for film Young’s modu-lus E570 GPa, Poisson’s ratiov50.35, viscosityh5c(1.331011 Pa s) (c.0; variation of the parameterc alters theviscosity!, initial rms roughness amplitudew050.5 nm(!hf), and initial roughness correlation lengthj510 nm. Inthe following we will investigate the evolution of the normalstresssav

N and the surface roughness amplitudew(t). Fi-nally, we will assume in all cases a constant applied stress,ignoring any stress relaxation at the interfaces which couldreduce the applied stress in the film.

Figure 2~a! shows the calculation ofsNav versus evolution

time t for various values of applied compressive stresss.Indeed,sN

av initially decreases reaching a minimum, and fur-ther increases and becomes larger than the applied stresss.This can be understood from Eq.~6! if it is written as

FIG. 1. Schematic of the system elastic film/viscous layer/rigid substrate.

FIG. 2. ~a! Average normal stresssNav vs evolution timet for viscosity h

51.331010 Pa s,H50.5, e50.12, hf530 nm, hg550 nm, and variousvalues of the compressive stresss/E. ~b! sN

av vs s/E for various roughnessexponentsH of the initial starting film morphology,h51.331010 Pa s,e50.12,hf530 nm, andhg550 nm.

894 J. Appl. Phys., Vol. 93, No. 2, 15 January 2003 G. Palasantzas and J. De Hosson

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sNav55

~2p!5

A E0

hf21AbFshfk

22Ehf

3

12~12n2!k4G 2

^uh~k,0!u2&e2a(k)tk dk

1~2p!5

A Ehf

21Ab

Qc Fshfk22

Ehf3

12~12n2!k4G 2

^uh~k,0!u2&e2a(k)tk dk61/2

~8!

with the first integral incorporating the contribution of un-stable wave vectors such thatk,hf

21Ab or equivalentlya(k).0 which makes the film unstable to buckling. More-over, the initial film morphology appears to have a moderateimpact onsN

av @Fig. 2~b!#, where it is more pronounced forsmall roughness exponentsH(,0.5). As Fig. 2~b! indicates,

the magnitude of the normal stresssNav will be lower for a

rougher initial film morphology~lower roughness exponentsH and/or smaller correlation length which leads to largerroughness ratiow0 /j for w0 fixed!.

Furthermore, as Figs. 2~b! and 3~a! indicate, with in-creasing applied compressive stresss the normal stresssN

av

decreases up to minimum which is shifted for later bucklingtimes (t@g1) toward lower values ofs(;0.1E), while sN

av

increases and reaches saturation fors.E. The minimum ofsN

av as a function ofs/E becomes rather sharp as the viscos-ity h of the layer underneath the buckled film decreases@Fig.3~b!#. Notably with decreasing misfit straine, the normalstresssN

av decreases in magnitude, as well as its minimumshifts to lower values ofs/E @Fig. 3~c!#. In addition withincreasing elastic film thickness, the normal stresssN

av de-creases in magnitude with the minimum position beingshifted to larger values of the ratios/E as Fig. 4~a! indicates.This is due to the fact that the range of unstable wave vectors(k,hf

21Ab) in Eq. ~8! becomes larger. However, the oppo-

FIG. 3. ~a! Average normal stresssNav vs s/E for h51.331010 Pa s,e

50.12,H50.5, hf530 nm,hg550 nm, and various evolution timest. ~b!sN

av vs s/E for t510 s, e50.12, H50.5, hf530 nm, hg550 nm, andvarious viscosities h5c(1.331011) Pa s. ~c! sN

av vs s/E forn51.331010 Pa s,t510 s, H50.5, hf530 nm, hg550 nm, and variousstrainse.

FIG. 4. ~a! Average normal stresssNav vs s/E for h51.331010 Pa s,e

50.12,H50.5,hg550 nm,t510 min, and various film thicknesseshf . ~b!sN

av vs s/E for h51.331010 Pa s,t510 s, e50.12, H50.5, hf530 nm,and various layer thicknesseshg .

895J. Appl. Phys., Vol. 93, No. 2, 15 January 2003 G. Palasantzas and J. De Hosson

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site behavior is observed forsNav as the thickness of the vis-

cous layer increases@Fig. 4~b!#.The calculations indicate that the norma1 stress exerted

by the viscous layer on the buckled film depends strongly onthe film characteristic, which influence its time evolutionduring buckling. Such behavior of the normal average stresswill also have strong implications on characteristic bucklingroughness parameters~i.e., roughness amplitudew! as willbe shown in the following paragraphs.

Figure 5~a! shows calculations of the rms roughness am-plitude w(t) versus timet for various viscosities of the un-derlying viscous film~see Fig. 1!. In all cases we havew(t)!hf so that the linear theory is applicable. With de-creasing viscosityh, w(t) decreases for short time scales~stable regime!, and increases for later times by becominglarger than the amplitude of the initial film morphologyw0 ,indicating unstable roughness growth due to film buckling.The viscosity influence is related to the fact that ash de-creases, the factora(k) increases and consequently the con-tribution of unstable wave vectorsk,hf

21Ab. Indeed, Eq.~5! can be written as

w~ t !5F ~2p!5

A H E0

hf21Ab

^uh~k,0!u2&e2a(k)tk dk

1Ehf

21Ab

Qc

^uh~k,0!u2&e2a(k)tk dkJ G1/2

. ~9!

The first integral in Eq.~9! yields the contribution of un-stable buckling to the film roughness amplitudew(t). Fur-thermore, with increasing misfit straine @Fig. 5~b!# theroughness becomes unstable which is characterized by arapid increment ofw(t)/w0 . A change ofe by 1 order ofmagnitude leads to a fast transition from a damped bucklingbehavior@decreasing ratiow(t)/w0] to fast unstable growthof the surface roughness amplitude. This is due to the factthat the range of unstable wave vectors (k,hf

21Ab) in Eq.~9! becomes larger. It leads to positive values ofa(k) andtherefore to a faster increase of the roughness amplitudew(t). The effect of the straine is more pronounced than thatof the viscosityh since the former has direct control on thesign of the factora(k).

Finally, Fig. 6 indicates the influence of the thicknesshf

and hg , respectively, for elastic film and the viscous layer.With decreasing thicknesshf of the elastic film, the mini-mum orw(t) becomes shallower and a more rapid growth ofunstable behavior develops. This is because the range of un-stable wave vectorsk,hf

21Ab increases and consequentlythe contribution of the first integral in Eq.~9! for w(t). Be-sides the initial transit regime~up to the minimum!, the op-posite behavior develops as a function of the viscous filmthicknesshg where w(t) increases with increasing viscousthicknesshg .

From the above calculations it becomes clear that in or-der to determine the limits of film stability as a function ofevolution time of buckling, the direct calculation of the sur-face roughness amplitude is necessary. Although by itself

FIG. 5. ~a! Amplitude ratiow(t)/w0 vs evolution timet for various viscosi-ties n, e50.012,H50.5, hf530 nm, andhg550 nm.~b! w(t)/w0 vs evo-lution time t for various strainse, h51.331010 Pa s,H50.5, hf530 nm,hg550 nm.

FIG. 6. ~a! Amplitude ratiow(t)/w0 vs evolution timet for various filmthicknesshf , e50.12, H50.5, andhg550 nm. ~b! w(t)/w0 vs evolutiontime t for various thicknesshg , h51.331010Pa s,H50.5,hg550 nm, ande50.12.

896 J. Appl. Phys., Vol. 93, No. 2, 15 January 2003 G. Palasantzas and J. De Hosson

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linear stability analysis4 gives an indication of the stable andunstable wavelengths, it does not provide full knowledge ofhow these modes contribute as a whole to the bucklingroughness amplitude. Note that the film buckling amplitudeis a roughness parameter which in real thin film systems canbe directly measured, i.e., in terms of scanning probemicroscopy8,9 and x-ray scattering reflectivity.8,10,11

V. CONCLUSIONS

We investigated aspects of buckling of compressed elas-tic thin films, which are bonded onto viscous films of finitethickness. The calculations were limited within the frame-work of linear elastic plate theory, such that the bucklingamplitude remained small in comparison with the film thick-ness. It was found that the normal stress exerted by the vis-cous layer on the elastic film evolves with time. It shows aminimum at early buckling stages, while it increases at laterbuckling stages. Moreover, with decreasing viscosity of theunderlying viscous film, the temporal evolution of the filmbuckling amplitude also shows a minimum. The effect of theviscosity becomes more pronounced with increasing strain inthe film. The normal stress also shows a minimum as a func-tion of applied compressive stress with position and shape,which strongly depends on viscosity and strain values. Fi-nally, the unstable growth of buckling roughness is enhancedwith decreasing elastic film thickness and/or increasing vis-cous layer thickness.

ACKNOWLEDGMENTS

The authors would like to acknowledge support from the‘‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek~NWO!’’. They would like also to acknowledge fruitful dis-cussions with Professor E. van der Giessen on the topic ofmechanical buckling.

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897J. Appl. Phys., Vol. 93, No. 2, 15 January 2003 G. Palasantzas and J. De Hosson

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