Comparison between models of insulator and semiconductor thin films islanding

0.17 MB·English

by  F. Lallet

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Comparison between models of insulator and semiconductor
thin films islanding
8
0
0 Franc¸oisLallet∗1,AlainDauger∗∗1etNathalieOlivi-Tran∗∗∗1
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1 SPCTS,UMR-CNRS663847a`73AvenueAlbertThomas,87065Limogescedex,FRANCE

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J

3 Mots-cle´ PVD-CVD,Sol-gelprocessing,MonteCarlosimulation,thinfilmislanding.
2 PACS 04A25
The synthesis of self-organized quantum dots (QD’s) can be achieved through bottom up layer by layer
] depositionprocesses aschemical vapor deposition(CVD)orphysical vapor deposition(PVD).However,
i
c QD’smayalsobesynthesizedviasol-gelroute,whichinvolvesaspontaneousevolutionfromthinfilmsto
s discreteQD’swithoutfurtherdeposition.Theaimofthepaperistodiscussandcomparethephysicalphe-
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nomenainvolvedinQD’sformationwhichinitiatefromthinfilmsurfacerougheningbetweenPVD-CVD
l
r andsol-gelsynthesisapproaches.Weproposetwosimplephysicalmodelswhicharerelevanttoexplainthe
t
m fundamentaldifferencesbetweenthosemethods.
.
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d
n 1 Introduction
o
c Many authors have presented and studied epitaxial QD’s growth through PVD-CVD synthesis pro-
[
cesses.Thephysicalphenomenaassociatedwithsuchapproacheshavebeenwidelystudiedboththeoreti-

1 cally[1][2][3]andexperimentally[4][5],inparticularthroughthemodelsystemGe/Si[6][7][8][9][10][11][12],
v becauseofitspromisingtechnologicalapplications.HoweverthesynthesisofQD’sepitaxiallygrownon
9
a crystalline substrate can be achieved through a sol-gel approach. Indeed, Bachelet et al [13] have re-
1
cently synthesized and studied the microstructure of zirconia QD’s grown on a c-cut sapphire substrate
6
3 during thermal annealing of a zirconia precursor xerogel thin film deposited on the substrate by sol-gel
. dip-coating.
1
0 Theaimofthisarticleistodiscussthephysicalphenomenainvolvedintheseprocessesthroughtheana-
8 lysisoftwophysicalmodels,basedonenergeticconsiderations,whichhavebeendevelopedfornumerical
0 simulations.Wesuppose,forthesakeofsimplicity,thatQD’ssynthesisisachievedwithoutnucleation.
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2 Physicalmodels
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a
WehavedevelopedaMonteCarlo(MC)algorithmtosimulatetheislandingofathinfilmonaperfect
crystalline subtrate during thermalannealing without further deposition [14]. The numericalthin film is
dividedintomesoscopicdomainswhicharecharacterizedbytheirheightandtheircrystallographicorien-
tationwithregardtotheirnearestneighbors(NN)andthesubstrate.AteachMCstep,adomainiischosen
at randomand the probabilityP to change its height(h ) and/or its crystallographicorientation(c , d )
i i i i
iscalculatedthroughtheclassicalMetropolisscheme[15].Theenergyofthedomainiwithregardtoits
∗ Correspondingauthor:e-mail:[email protected],Phone:[phone],Fax:[phone]
∗∗ Secondauthor:e-mail:[email protected],Phone:[phone],Fax:[phone]
∗∗∗ Thirdauthor:e-mail:[email protected],Phone:[phone],Fax:[phone]
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4Sh.FirstAuthor,Sh.SecondAuthoretSh.ThirdAuthor:Comparisonbetweenmodelsofinsulatorandsemiconductorthinfilmsislanding
nearestneighborsisexpressedas:
ℓ2 NN ℓ2 NN D γ ∆t NN
Ei =γ1(cid:18)h +ℓ(cid:19) (ci−cj)+γ2(cid:18)h +ℓ(cid:19) (di−dj)+Y(1+ν)r ks sT ℓ2 (hi−hj) (1)
i Xj=1 i Xj=1 B Xj=1
wherethefirstandsecondtermoftherighthandsideoftheequalitycorrespondtotheinterfacialenergyof
thedomainwithregardtothesubstrateandtoitsneighborsrespectively.Thethirdtermcorrespondtothe
surfaceenergyrelatedtotheheightsofthedomains.γ1 istheboundarysurfacetension(domain/domain),
γ2theinterfacialsurfacetension(domain/substrate),ℓthedistancebetweendomainianditsnearestneigh-
bors,Y theYoungmodulus,ν thePoisson’sratio,D thesurfacediffusioncoefficient,γ thefreesurface
s s
tension,k theBoltzmannconstantandTtheabsolutetemperature.
B
We presentanotherphysicalmodelwhich allows one to discuss, at least qualitatively,the main para-
metersresponsibleforthemorphologicalevolutionofathinfilmepitaxiallygrownonaperfectcrystalline
substratethroughadepositionprocess(PVDorCVD).Thismodelisinspiredfromthepreviousworksof
Kawamura[16]andRusso[17]whoestablishedMCalgorithmsattheatomicscaletomodelQD’sforma-
tionmodesduringdepositionprocesses.Theenergyofanadatomiiscomputedasthesumofitsbonding
energyandelasticenergy[16][17]:
E =NE −(E −E ) (2)
i B withadatomi withoutadatomi
where N is the number of chemical bonds of the adatom i with its neighbors, E is the energy of a
B
chemicalbondandE thetotalelasticenergy.Fromequation(2),wededuceamodelinwhichanepitaxial
monocrystallinethinfilmisdepositedonaperfectcrystallinesubstrate.Wemodelthisfilmasacubicarray
of mesoscopic domainsi of heighth andwidth ℓ. Each domainis submitted bothto its surface tension
i
γ andtoanelasticstressfieldinducedbythelatticemismatchbetweenthefilmandthesubstrate,ǫ.The
s
volumeenergydensityofathinfilmofinitialthicknessh,duetofreesurfaceenergy,isexpressedasγ /h.
s
Thus, the surface tension of a domain i can be written as (γ h )/h and therefore the surface energy of
s i
domaini(relatedtoNE )inducedbythefreesurfaceenergyis(ℓ2γ h )/hwhereℓ2 isthefreesurface
B s i
areaatthetopofeachdomain.Theelasticstressfieldleadstoanelasticenergyinsideeachdomainofthe
film.Thiselasticenergy(relatedtoE −E )canbeexpressedthroughlinearelasticity
withadatomi withoutadatomi
theory. We suppose for the sake of simplicity that the elastic stress tensor is diagonal (for example for
cubicphasedmaterials).Withthesameassumption,ǫ,whichisthelatticemismatchbetweenthefilmand
the substrate,is constantinthe horizontalplane.Thusthe resultingforceinthe horizontalplanereduces
to 2Yǫℓh with ℓh the area of surfaces of domain i perpendicularto the x and y axis. Futhermore,the
i i
resultingforcesupportedbythedomainiontheverticalaxiszisequaltoνYǫℓ2.Therefore,theresulting
elasticenergyrelatedtodomainiandinducedbythelatticemismatchǫisgivenbytheworkofthiselastic
force,foracharacteristicdisplacementǫℓinthehorizontalplane:
2
E =Yǫ(2ℓh +νℓ )ǫℓ (3)
i
Consequently,theenergyofadomainiwithregardtoitsneighborsj mightbewritten:
NN
γ
E =ℓ2 s −2Yǫ2 (h −h ) (4)
i i j
h
(cid:16) (cid:17)Xj=1
3 Discussion
The formation of QD’s without nucleation is achieved through the evolution of the roughness of a
thin film until nanometer scale islands are clearly identified. In the numerical models, the evolution of
the roughnessof a film is simulated by the evolution of the heights of the discrete mesoscopic domains
(cid:13)c 2003WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
pssheaderwillbeprovidedbythepublisher 5
composingthevirtualthinfilm.Theenergyofamesoscopicdomainiofathinfilmsynthesizedviasol-gel
route is calculated with equation (1). One can see that the roughnessof the surface of the film is driven
bysurfacediffusionandsurfacetension.Theenergyofamesoscopicdomainiofathinfilmsynthesized
viaadepositionprocessiscalculatedthroughequation(4).Thisequationallowsonetounderstandthatthe
roughnessofthefilmisdrivenbyacompetitioneffectbetweensurfaceandstressenergies.
The probability of changing the height of a domain with regard to equation (1) follows a monotonic
tendencywithregardtotheintrinsicparametersofthethinfilm.Figure(1)presentstheevolutionoftheis-
landingofnumericalthinfilmsdepositedonasubstratebysol-geldip-coatingafter107MCSasafunction
ofthe initialthicknessofthe film. Theannealingtemperatureis fixed,so alltheparametersofthe equa-
tionarefixed.Onecanseethatastheinitialthicknessofthefilmincreases,thesubstrateislessdewetted.
Indeed,astheinitialthicknessincreases(h = 4nm),theamplitudeoftheroughnesswhichappearsatthe
topofthelayercannotreachthesubstrateasfastasforthinerfilms(h=1nm).
Theprobabilityofchangingtheheightofadomainwithregardtoequation(4)dependsontherelative
values of the parameters γ /h and 2Yǫ2. Indeed, if γ /h > 2Yǫ2 then the energy of the domain i is
s s
expressedasE =K NN(h −h )(whereK =γ /h−2Yǫ2)withK >0,whereasK <0otherwise.
i j=1 i j s
Ontheonehand,ifthePlatticemismatchǫislowand/orwithalowfilmthickness(K >0),thentheeffectof
surfaceenergyispredominant.Thereforetheprobabilityofrougheningislowandthefilmremainsalmost
flatduringthegrowingofthefilmwhichisdescribedthroughtheFrank-van-der-Merke(F-M)deposition
process(orALDAtomicLayerDepositionprocess).Ontheotherhand,ifthenumericalvaluesofǫand/or
filmthicknessarehigh(K <0),thentheprobabilityofrougheningincreases;thereforetheeffectofstress
fieldinsidethefilmispredominantwhichleadstotheformationofislands.Thesurfacerougheningstage
can either initiate from the top of a wetting layer, which is called the Stranski-Krastanov (S-K) growth
mode,ordirectlyfromthetopofthesubstratewithoutpreviousdepositionofawettinglayer(ǫveryhigh)
which is the Volmer-Weber (V-W) growth mode. Figure (2) illustrates the numerical calculations of the
islanding process through the effect of the initial film thickness (top, ǫ=0.04) while the bottom presents
thisevolutionwithregardtolatticemismatch(bottom,h=2nm)after107MCS.Wefocusonthetransition
ofKfrompositivetonegativevaluesastheislandsareformedwhenK <0.Onecannotethattheincrease
ofhorǫleadstoahighernumberofQD’s,whichiscontrarytotheevolutionofathinfilmsynthesizedvia
sol-gelprocessingwherethehigherthicknessofthefilmleadstoalowernumberofQD’s.Moreover,itis
clearthatQD’sformationismoresensitivetothevariationofthelatticemismatchthantothevariationof
the initialthicknessofthe film asE isa functionofǫ2 and1/h.Thusthe modeldescribesqualitatively
i
theevolutionfromaF-MorS-Kgrowthmode(K > 0andK ≈ 0respectively)toaV-Wgrowthmode
K < 0asthenumberofQD’ssynthesizedgrowswiththeincreaseofǫand/orh.Thosetendencieshave
alreadybeenreportedfortheGe/Sisystem[18].
4 Conclusion
OursimpleenergeticmodelsallowustodescribethemaintendencyofQD’sformationfromthinfilm
synthesizedeitherviadepositionprocesses(PVD-CVD)orsol-geldip-coating.
The main difference between those approaches is that thin films synthesized via sol-gel route lead
to a greater number of QD’s when they are thin while thin films synthesized via deposition processes
demonstrate the opposite behaviour. This result is explained through equations (1) and (4) which point
outthat surface rougheningof thin films synthesized via sol-gelroute is drivenby surface diffusionand
surfacetension,whereasitis theconsequenceofa competitivemechanismbetweenelastic stress energy
andsurfacetensionforthinfilmssynthesizedlayerbylayerinPVDorCVDprocesses.
References
[1] L.Nurminen,A.KuronenandK.KaskiPhys.Rev.B63,035407(2000).
(cid:13)c 2003WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
6Sh.FirstAuthor,Sh.SecondAuthoretSh.ThirdAuthor:Comparisonbetweenmodelsofinsulatorandsemiconductorthinfilmsislanding
Fig.1 Simulationoftheislandingprocessofnumericalthinfilmswithoutdeposition(sol-gel)asafunctionofthe
initialthicknessofthefilmafter107MCS.
Fig.2 Simulationoftheislandingprocessofnumericalthinfilmswithdeposition(PVD-CVD)asafunctionofthe
initialthicknessofthefilm(top)andofthelatticemismatch(bottom)after107MCS.
[2] M.Kalke,D.V.Baxter,Surf.Sci.477,(2001)95-101.
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[13] R. Bachelet, A. Boulle, B. Soulestin, F. Rossignol, R. Guinebretie`re and A. Dauger, submitted to Thin Solid
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[15] N.Metropolis,A.W.Rosenbluth,M.N.Rosenbluth,A.T.Teller,E.J.Teller,Chem.Phys.21(1953)1087.
[16] T.Kawamura,T.Natori,Surf.Sci.438,(1999)148-154.
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(cid:13)c 2003WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
Greek symbols – w-greek.sty
α \alpha θ \theta o o τ \tau
β \beta ϑ \vartheta π \pi υ \upsilon
γ \gamma ι \iota ̟ \varpi φ \phi
δ \delta κ \kappa ρ \rho ϕ \varphi
8
ǫ \epsilon λ \lambda ̺ \varrho χ \chi
0
0 ε \varepsilon µ \mu σ \sigma ψ \psi
2 ζ \zeta ν \nu ς \varsigma ω \omega

n η \eta ξ \xi
a
J

3 Γ \itGamma Λ \itLambda Σ \itSigma Ψ \itPsi
2
∆ \itDelta Ξ \itXi Υ \itUpsilon Ω \itOmega

] Θ \itTheta Π \itPi Φ \itPhi
i
c
s Table 1: Slanted greek letters
-
l
r
t
m
a \upalpha q \uptheta o \upo t \uptau
.
at b \upbeta J \upvartheta p \uppi u \upupsilon
m g \upgamma i \upiota v \upvarpi f \upphi
- d \updelta k \upkappa r \uprho j \upvarphi
d
n e \upepsilon l \uplambda ̺ \varrho c \upchi
o ε \varepsilon m \upmu s \upsigma y \uppsi
c
[ z \upzeta n \upnu V \upvarsigma w \upomega

1 h \upeta x \upxi
v
9
1 Γ \Gamma Λ \Lambda Σ \Sigma Ψ \Psi
6 ∆ \Delta Ξ \Xi Υ \Upsilon Ω \Omega
3
Θ \Theta Π \Pi Φ \Phi
.
1
0
Table 2: Upright greek letters
8
0
:
v
i
X
r
a
1
α \bm{\alpha} θ \bm{\theta} o \bm{o} τ \bm{\tau}
β \bm{\beta} ϑ \bm{\vartheta} π \bm{\pi} υ \bm{\upsilon}
γ \bm{\gamma} ι \bm{\iota} ̟ \bm{\varpi} φ \bm{\phi}
δ \bm{\delta} κ \bm{\kappa} ρ \bm{\rho} ϕ \bm{\varphi}
ǫ \bm{\epsilon} λ \bm{\lambda} ̺ \bm{\varrho} χ \bm{\chi}
ε \bm{\varepsilon} µ \bm{\mu} σ \bm{\sigma} ψ \bm{\psi}
ζ \bm{\zeta} ν \bm{\nu} ς \bm{\varsigma} ω \bm{\omega}
η \bm{\eta} ξ \bm{\xi}
Γ \bm{\itGamma} Λ \bm{\itLambda} Σ \bm{\itSigma} Ψ \bm{\itPsi}
∆ \bm{\itDelta} Ξ \bm{\itXi} Υ \bm{\itUpsilon} Ω \bm{\itOmega}
Θ \bm{\itTheta} Π \bm{\itPi} Φ \bm{\itPhi}
Table 3: Boldface variants of slanted greek letters
aaa \pmb{\upalpha} qqq \pmb{\uptheta} ooo \pmb{\upo} ttt \pmb{\uptau}
bbb \pmb{\upbeta} JJJ \pmb{\upvartheta} ppp \pmb{\uppi} uuu \pmb{\upupsilon}
ggg \pmb{\upgamma} iii \pmb{\upiota} vvv \pmb{\upvarpi} fff \pmb{\upphi}
ddd \pmb{\updelta} kkk \pmb{\upkappa} rrr \pmb{\uprho} jjj \pmb{\upvarphi}
eee \pmb{\upepsilon} lll \pmb{\uplambda} ̺̺̺ \pmb{\varrho} ccc \pmb{\upchi}
εεε \pmb{\varepsilon} mmm \pmb{\upmu} sss \pmb{\upsigma} yyy \pmb{\uppsi}
zzz \pmb{\upzeta} nnn \pmb{\upnu} VVV \pmb{\upvarsigma} www \pmb{\upomega}
hhh \pmb{\upeta} xxx \pmb{\upxi}
Γ \bm{\Gamma} Λ \bm{\Lambda} Σ \bm{\Sigma} Ψ \bm{\Psi}
∆ \bm{\Delta} Ξ \bm{\Xi} Υ \bm{\Upsilon} Ω \bm{\Omega}
Θ \bm{\Theta} Π \bm{\Pi} Φ \bm{\Phi}
Table 4: Boldface variants of upright greek letters
2