合金薄膜形態變化之穩定分析

行政院國家科學委員會補助專題研究計畫成果報告

合金薄膜形態變化之穩定分析

Morphological Stability of Alloy Thin Films

計畫編號 ︰ NSC 90-2212-E-002-162

執行期限 ︰90 年 8 月 1 日至 91 年 7 月 31 日

主持人︰舒貽忠

國立台灣大學應用力學研究所

中文摘要

本研究計劃乃探討半導體合金薄膜型態變化之穩定分析；此項研究在加強與

擴展半導體領域上是極具有發展潛力的。我們藉由發展材料多重尺度模型，來

建構本研究之理論基礎，並提出一種全新的構想，即藉由探討非均質組成濃度

和薄膜表面粗糙所造成的耦合效應，來研究合金薄膜型態變化之穩定分析。同

時結果顯示對合金薄膜生成中其奇異溫度的大小，乃是藉由不穩定因素諸如彈

性應變能，與穩定因素諸如化學能和表面張力，之間的競爭而得到。

Morphological Stability of Alloy Thin Films Project Number: NSC 90-2212-E-002-162

Project Period: August 1, 2001 {July 31, 2002 Project Investigator: Yi-Chung Shu

Institute of Applied Mechanics, NationalTaiwanUniversity

Abstract

Wehavestudiedthemorphologicalstabilityof semi-conductoralloy lmswith potentialto enhance and enlargethecurrentspectrumofapplicationsthatare of interest to the semiconductor industry. We have developedaframeworkbasedonthemultiscale mod-elingtoperformstabilityanalysisofalloylmgrowth accountingforthejointeectofnonuniform composi-tionand surfaceroughness. Wehaveshownthat the critical temperature is obtained by the competition between the destabilizing in uence of elastic strain energyand the stabilizingin uence of chemicaland surfaceenergies.

1 Introduction

Theinterestinthemorphologyandspinodal decom-positionofanalloylmhasmotivatedmanyeortsto studystrainrelaxationbysurfaceroughnessand com-positionvariation. Itiswellknownthatrelaxationof elasticenergycanhaveastrongin uenceonthe mor-phology of astressed solid. Recent experiment has observed a 3D morphology for a strained epitaxial lm instead of common layer-by-layergrowth. One possibleexplanation forthischangeof growthmode is that a stressed lm can partially relieve its elas-ticenergy bya morphologicalinstability at the free

surfacethroughmasstransport,whichgivesthe for-mationofislands,nonplanarsurfacesorcusplike mor-phologies. Instabilityofthistypearisesasthestrain energy reduction due to morphologicalvariations in surfaceshapesurpassestheincreaseofsurfaceenergy. Strain relaxation via composition modulation isalsoofparticularimportanceinthetheoryof spin-odal decomposition. A homogeneous alloy which is not completely miscible tends to decompose under certain critical temperature [1]. Cahn wasthe rst to demonstrate that a bulk alloy can be stabilized bythepresenceofcoherentcomposition uctuations providedthevolumeofitsunitcellchangeswith com-position. Inthecaseofthin alloylms,thepresence of free surface in an uncapped thin lm allows the partialrelaxationofcoherentstrain. Asaresult,the increase ofelastic energyin athin lm turns outto be smaller than that in its bulk form for the same compositionalstrainmodulation. Thepredicted crit-icaltemperatureT

f c

in thin lmsis therefore higher than T

B c

in bulk alloys. Experimental evidence for strainrelaxationviacomposition uctuationsandfor strain-drivenspinodaldecompositioncanbefoundin [5].

Westudystrainreliefbysurfaceroughnessand composition uctuationin astressedalloylm here. Weproposetoinvestigateitbyderivingarelaxation formula usingenergy bounds. Werst develop

suit-able upper and lower bounds of energy, and then prove the identity of these two bounds if both lm surfaceand compositional strainare smooth. How-ever, ifthe surfaceprole does nothave continuous derivativesashappenedoftenintheshapetransition inthegrowthofstrainedislands[7],theresulting re-laxationformulaturns outto beanupperbound of thestrainenergy.

Another advantage of the current approach is that we do not require any a priori assumption of elastic isotropy or identical material properties be-tweenlmandsubstrateinderivingourresults. Such ageneralizationisimportantasmaterial inhomogene-ity has great in uence on the morphology of het-eroepitaxially growinglms. Freund and Jonsdottir [2] havedemonstrated that the stability criterion is sensitive to the ratio of lm thickness to roughness if both lm and substrate have dissimilar material properties.

2 The Free Energy of an Inho-mogeneous Alloy Film

2.1 General Problem

Consideralm/substratesystemshownin Figure1. The lm surface is described by x

3 =  h(x p ) > 0. LetS betheboundedLipschitzdomain,hbeviewed as the average thickness of the lm and H be the heightofthesubstrate. Thereferencedomainofthe lm/substratesystemisdenotedby

(h) =S H;  h(x p )
: (2.1) Set (h) f =S 0;  h(x p )
; s =S( H;0) (2.2) which are the domains for the lm and substrate, respectively.

The total energy of the lm/substrate system

Ω

(h)

f

Ω

s

S

λ

h

H

Figure 1: Aheterogeneouslm/substratesystemwitha non- atsurface.

perunitareaincludes W (h) tot =W (h) e +W (h) s +W (h) c ; (2.3) whereW (h) e

istheelasticenergyofthelm/substrate system, W

(h) s

thesurfaceenergy of thelm surface, andW

(h) c

thechemicalenergy. Theelasticenergyper unit areaisgivenby

W (h) e = inf u2V 1 jSj Z (h) ' e (E[u];x)dx; (2.4) where' e :IM 33 s IR 3

![0;1)istheelasticenergy density,E2IM

33 s

thestraindependingon displace-mentu: (h) !IR 3 E[u]= 1 2 ru+(ru) T
; (2.5) and thespaceV dened by

V = n u:u2W 1;2 ( (h) ;IR 3 ); uj Sf Hg =0 o :(2.6) Note thattheinhomogeneityofthelmdueto vari-ouscausescanbeseenfromthedependenceof'

e on x.

Thesurfaceenergyperunitareais W (h) s = 1 jSj Z S (n)dS(n); (2.7)

where (n) is the surface tensiondepending on the lm surfacenormaln. If (n)isindependentof ori-entationnand isequalto ,(2.7)becomes

W (h) s = 1 jSj Z S q 1+  h 2 ;1 +  h 2 ;2 dx p : (2.8)

W (h) c = 1 jSj Z (h) f ' c (c(x p ))dx; (2.9) where c(x p

): S !(0;1) is the concentration of the alloylmand '

c

:(0;1)!IR is thevolumedensity ofthe chemical and entropic partof thefree energy inthelm. Usingtheregularsolutionmodelgives '

c

(c)=c(1 c)+R T[clnc+(1 c) ln(1 c) ](2.10); whereistheinteractionparameter,Rthegas con-stant,andT thetemperature.

Considerahomogeneousalloylmwithplanar freesurfaceand denoteitas thereferencestate. We wish to study the thermodynamic instability of the alloylmbyexaminingifanyjointcomposition mod-ulationc(x) andsurfaceundulation

 h(x

p

)maylower thefreeenergyofthereferencestate. Notethatonce theperturbedc(x) and

 h(x

p

)aregiven,theenergies W (h) c andW (h) s

canbecalculatedimmediatelyby in-tegrating(2.8)and(2.9). However,theelasticenergy W

(h) e

isstillunavailableasweneedtosolvethe min-imizationproblem (2.4)which is diÆcultin general. The analysis can be greatlysimplied by assuming the periodicity of the lm surface and composition andwillbeconsidered next.

3 Asymptotical Formulation of Limiting Elastic Energy

We now study an important case for slightly un-dulating surface and uctuating composition; i.e., h<< ;d. Instead of solvinga diÆcultminimizing problem, we show that the asymptotical expansion oftheelasticenergy

 W e

upto therstorder inis givenbythefollowingtheorem.

Theorem1 Let the eective elastic energy density andeigenstress begiven by

 W e and p .  p (x p )2W 1;1 per (X;IM 22 s )\W 2;2 per (X;IM 22 s ); f(x p )2W 1;1 per (X;IR )\W 2;2 per (X;IR ): Then, liminf !0  1   W e W 0
 W 1 +W 2 ; (3.1) where W 0 = Z X U 0 (x p )f(x p )dx p ; W 1 = 1 2 Z Z r p
[ p f(x p )]
u (1) p dx p ;(3.2) W 2 = Z X r p
  p f 2 (x p ) 
b p (x p )dx p : (b). Suppose f(x p )2W 1;1 per (X;IR); b  (x p );  p (x p )2W 1;2 per (X;IR 3 ): Wehave limsup !0  1   W e W 0
 W 1 +W 2 ; (3.3) where W 0 ;W 1 andW 2 are given by(3.2).

Above,thedetailsofnotationssuchasf;U 0

;b 

;u (1) and taredescribedin [6].

Remark 1 Ifbothlmsurfacef(x p )andeigenstrain E I (x p

)aresmooth,Theorem 1tellsusthat the lim-itingelasticenergyofalm/substratesystemcanbe approximated asthe sum of three energies W

0 , W

1 and W

2

provided the ratio of the lm thickness to the periods of roughness and composition variation is small;i.e.,  W e =W 0 +(W 1 +W 2 )+o(); where o() 

! 0 as  ! 0. On the other hand, if the surface prole doesnot have continuous deriva-tivesashappenedoftenintheshapetransitioninthe growthofstrainedislands[7],theresultingrelaxation formulaturnsouttobeanupperboundofthestrain energy(3.3).

T

_c

f

(α )

k

α

1

α

2

k

Wave number k

Temperature T

T

c

f(α )

2

1

m

cm

(T, )

α

Stable

Figure 2: Instability with respect to coupled composi-tionalandmorphological modulationsinastressedalloy lmdeposited on athick substrate. Here0 <1 <2 andk =kcm(T;)separates the boundaryof instability inthe k-Tplane. Dashed linesaretheboundariesof in-dependentcompositionalandmorphologicalinstabilities.

4 Stability with Respect to Morphology and Spinodal Decomposition

We now apply Theorem 1 to studying the thermo-dynamical stability of a nonhydrostatically stressed thin lm. Suppose the lm is madeof a binary cu-bicalloyoftheformA

c B

1 c

orternarycubicalloyof theformA

c B

1 c

C,anditsstressfreelattice parame-tera(c)dependingoncompositioncfollowsVegard's law;i.e., = 1 a da dc j c 1 6=0; (4.1) where isthesoluteexpansioncoeÆcient.

As any smooth periodic function can be expanded by multi-dimensional Fourier series, we may assume that the compositional and mor-phological modulations ^ h(x p ) and ^c(x p ) take all possible combinations of sin(2mx

1 )sin(2nx 2 ), sin(2mx 1 )cos(2nx 2 ), cos(2mx 1 )sin(2nx 2 ), and cos(2mx 1 )cos(2nx 2

), and sum all these combinationsweightingbytheirFouriercoeÆcients.

^ h(x p ) = sin(2x 1 )sin(2x 2 ) (4.2) ^ c(x p ) = sin(2x 1 )sin(2x 2 ); (4.3) for simplicity. Wehaveshown that the newcritical wavenumberseparatingthe boundary ofinstability in thek-Tplaneis k cm (T;)=k m + 8 2 c 1 (1 c 1 ) R (T T f c ()) (1+) 2  E 1   2 ; where k m = p 2 (1+) E 1  e m 2 :

The result can also be shown in Figure 2. The 3D critical wave number k

m

coincides with that in [3]. On the otherhand, fora purecompositional modu-lation,thereisaslightdierencebetweenthecritical temperatureT

f c

andthat obtainedby[4]in thecase ofsmall. Thisisbecausecompositionalmodulation used hereis2Dinsteadof1Din [4].

References

[1] J.W. Cahn. On Spinodal Decomposition. Acta Metall., 9:795{801,1961.

[2] L.B. Freund and F.Jonsdottir. Instability of a BiaxiallyStressed Thin Film onaSubstrate due to Material Diusion over its Free Surface. J. Mech. Phys. Solids,41:1245{1264,1993.

[3] H.Gao.SomeGeneralPropertiesofStress-Driven SurfaceEvolutioninaHeteroepitaxialThinFilm Structure. J. Mech. Phys. Solids, 42:741{772, 1994.

[4] F. Glas. Elastic State and Thermodynamical Properties of Inhomogeneous Epitaxial Layers: Applicationto Immiscible III-VAlloys. J. Appl. Phys,62:3201{3208,1987.

[5] F.Peiro,A.Cornet,J.R.Morante,S.Clark,and R. H. Williams. In uence of the Composition Modulation on the Relaxation of In

0:54 Ga

0:46 As StrainedLayers.Appl.Phys.Lett.,59:1957{1959, 1991.

[6] Y. C. Shu. Strain Relaxation in an Alloy Film withaRoughFreeSurface. ToappearinJ. Elas-ticity,2002.

[7] J. Terso and R. M. Tromp. Shape Transition inGrowthofStrainedIslands: Spontaneous For-mation of Quantum Wires. Phys. Rev. Lett., 70:2782{2785,1993.