Growth of hexagonal quantum dots under preferential evaporation

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Growth of hexagonal quantum dots under preferential

evaporation

Guido Schifani, Thomas Frisch, Jean-Noël Aqua

To cite this version:

Guido Schifani, Thomas Frisch, Jean-Noël Aqua. Growth of hexagonal quantum dots under

pref-erential evaporation.

Comptes Rendus Mécanique, Elsevier Masson, 2019, 347 (4), pp.376-381.

Growth

of

hexagonal

quantum

dots

under

preferential

evaporation

Croissance

de

boîtes

quantiques

hexagonales

sous

évaporation

préférentielle

Guido

Schifani

a

,

Thomas

Frisch

a

,

Jean-Noël

Aqua

b

,

∗

a_{UniversitéCôted’Azur,CNRS,InstitutdephysiquedeNice,ParcValrose,06108Nice,France}

b_{SorbonneUniversité,CNRS,InstitutdesnanosciencesdeParis,INSP,UMR7588,4,placeJussieu,75005Paris,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received19November2018 Accepted26November2018 Availableonline12April2019

Keywords:

Hexagonalquantumdots Preferentialevaporation Heteroepotaxialgrowth

Mots-clés :

Boîtesquantiqueshexagonales Évaporationpréférentielle Croissancehétéroépotaxiale

WeperformnumericalsimulationsofhexagonalquantumdotsofAlGaNsemiconductors. Weshowthatthecompetitionbetweensurfacemassdiffusionandevaporationrulesthe morphologyofthequantumdots.Thesystemdisplaysthreedifferentbehaviors:presence ofseparatedislandswithoutawettinglayer,islandsdissolvingintothewettinglayer,or islandsthatdonotevolve.Thefirstbehaviorisofspecialinterestbecauseitsoptoelectrical propertiesare significantly improvedincomparison with quantum dots witha wetting layer.

©2019Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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é

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é

Nous effectuons une simulation numérique des boîtes quantiques hexagonales de semiconducteursAlGaN.Nous montrons quelacompétition entre ladiffusion demasse ensurfaceetl’évaporationdétermine lamorphologiedes boîtesquantiques. Lesystème montre trois comportements différents : des îlots séparés sans couche de mouillage, desîlotsse dissolvant dansla couchedemouillage oudesîlots nepouvantévoluer. Le premiercomportementprésenteunintérêtparticulier,car lespropriétésoptoélectriques sontconsidérablementamélioréesparrapportauxboîtesquantiquesavecunecouchede mouillage.

©2019Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

*

Correspondingauthor.

E-mailaddresses:guido.schifani@inphyni.cnrs.fr(G. Schifani),thomas.frisch@inphyni.cnrs.fr(T. Frisch),aqua@insp.jussieu.fr(J.-N. Aqua).

https://doi.org/10.1016/j.crme.2019.03.012

1631-0721/©2019Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Fig. 1. PreferentialevaporationfollowingEq. (2).Themodelinvolvessixpreferentialevaporations,correspondingtothefacetsoftheislands.Forsimplicity, wechoseFev

001=1,p=1/10 ande=4. 1. Introduction

Thegrowthofself-organizedquantumdots(QDs)hasbeenwidelystudiedduetotheir opto-electronicproperties,from screendisplaystosingle-photonemitters[1–3].Thedevelopment ofa modelthat canpredictthesize,theshape,andthe surface densityofstrainedQDsremains achallengingtask, sinceitinvolvesthedynamicinterplayofelasticity,capillarity, wetting,faceting,andalloyingeffects [4–14].

Therearetwoessentialphenomenaforthemorphologicaldevelopmentofself-organizedQDs.Thefirstoneisthe devel-opmentofthespatial instability,whichselectsasize.Thesecondone isa coarseningphenomenonthat broadensthesize distributionoftheislands.Coarseningisageneralphenomenoninwhichthesizeofapatternincreases,whosedescription dependsontransportmechanismsbetweenQDs.

Theformationofself-organized QDsresultsfromtheStranski–Krastanow growthmode[15].Athinsemiconductorfilm isdepositedandgrowsasaplanarlayer.Aboveacriticalthickness,QDsemergefromthislayer.Thisformationisexplained byapartialrelaxationoftheelasticstressofthestrainedfilm,whichisalsosubmittedtocapillarityandwettingeffects.For lowmisfit[16,17],theinstabilityisreminiscentoftheAsaro–Tiller–Grinfeld(ATG)instability[18,5].Afteritsinitialgrowth, theassemblyofQDsundergoescoarsening,drivenbytheefficientelasticrelaxationofthelargestislands.Theinitiallyrough isotropicQDs (prepyramids)henceripenand, asthey displaysteep enoughslopes,they transformintoanisotropic QDs of varioussizes,especiallypyramidsanddomes[19].

We are interested inGaN QDs. Itis observed experimentally that, afterdeposition ofthe semiconductor film,a pref-erential evaporationcomesinto play[20–22], whichdependson theslopeof thesystemandinwhich thewetting layer evaporatesfasterthattheQDs.ThiseffectdrivesthesystemtoobtainQDswithoutawettinglayer,bywhichthe optoelec-tricalpropertiesimprovesignificantly.

2. Continuumequation

Wemodelthegrowthandevaporationofasemiconductorfilmbyacontinuumframeworkthatdescribessurfacemass diffusion. The film, lying in between z

=

0 and the free surface z

=

h

(

x

,

y

,

t

)

, evolves following the mass conservation equation:

∂

h

∂

t

=

D



μ

+

F

dep

₋

_Fev



₁

_{+ ∇}

_h2 ₍₁₎

where Fdep, Fev correspondtothedepositionandevaporationfluxes,while

μ

= ∂(

F

el

₊

_F

s

_)/∂

_{h isthechemicalpotential} of the inhomogeneous surface, givenby the functional derivative of the elastic and capillary free energies

F

el and

F

s. Wework withIII–Vsemi-conductors,where Fev_{isnot negligibleanddependsonthelocalorientationsn}

0

= {

0

,

0

,

1

}

and ni

=

1₂

{

cos

(

iπ₃

+

π₄

),

sin

(

iπ₃

+

π₄

),

√

3

}

,i

=

1

,

. . .

6.Weconsiderthefollowingpreferentialevaporationfunction

Fev

(

m

)

=

F₀₀₁ev

⎧

⎨

⎩

1

+

1

−

p

π

6



j=1 1



σ=0

(

−

1

)

σarctan



η



||

m

−

mj

|| −

(

−

1

)

σ



e

||

mj

||

⎫

_⎬

⎭

(2)

where F₀₀₁ev istheevaporationofthesubstrate(001), p

=

F₁₁₃ev

/

F₀₀₁ev theevaporationratioofthefacetsandofthewetting layer,

η

and



e parameterizethe dependence of evaporationon the localfilm orientation relatedto m

= {

hx

,

hy

}

,while

mi

=

2

/

√

3

{

ni,x

,

ni,y

}

.WeplotinFig.1thepreferentialevaporationforan hexagonalanisotropy.The surfaceenergy

F

s

=



γ

(

h

,

n

)

d2S dependsonthesurface energyas

γ

=

γ

f[1

+

γ

a

(

n

)

+

γ

w

(

h

)

]. Here

γ

f istheenergyofa thick-film,

γ

aisthe

anisotropy energy and

γ

w the wetting contribution.We consider again an hexagonal asymmetry for the surface energy anisotropy,describedgenericallyby

Fig. 2. Surface energyanisotropygiveninEq. (3).The preferentialorientations representthefacets ofthe islands.Theparametersfor thefacets are

Aα=0.07,ηα=4,andα=10−3andfortheflatorientationA0=0.2,η0=10,and0=0.1.

γ

a

= −

6



j=0 Ajexp



−

η

j



1

−

n

·

nj2

+



j

(3) withthe localnormaln

=

 1

1+∇h2

{−

hx

,

−

hy

,

1

}

.The functional(3) leads witha smallregularization



α ,evenwithsmall anisotropyamplitudesAα (Fig.2).Thewettingeffectappearsduetotheinteractionbetweenthefilm/substrateinterfaceand thefilmsurface.Insemiconductors,itgenerallyhassmallvariations,whichwegenericallymodelas

γ

w

(

h

)

=

cwexp

(

−

h

/δ

w

)

.

Finally,elasticstrainariseshereduetotheepitaxialmisfitm

= (

af

₋

_as

_)/

_as_{betweenthelatticeparametersa}f/s_ofthefilm (f) andthe substrate(s). We will usethe elastic energyup to second order computedassuming the small-surface-slope approximationproposedin[19].Itsamplitudeisgivenbytheelasticenergydensity

E

0

=

Y m2

/(

1

−

ν

)

,whereY and

ν

are Young’smodulusandPoisson’sratio.

The characteristic length andtime scales associated withthe evolutionequation (1) arel0

=

γ

f

/

2

(

1

+

ν

)

E

0 andt0

=

l4₀

/

D

γ

f [19].Forexample,inaGaN system,

γ

f

=

1

.

89 J/m2,andwithits typicalelasticparameters,1 onefindsl0

=

5

.

4 nm foraGaN filmontopofan Al0.5Ga0.5N substrate.The determinationoft0 isdifficult,astheeffectivediffusioncoefficient

D isdelicatetomeasure.Wecomparetheexperimentalresultswithournumericalsimulationinordertosetatimescale t0



1 s.

3. Asaro–Tiller–Grinfeldinstability

The systemofathinfilmsemiconductordeposited onasemiconductor substratepresentsa surface morphological in-stability.ThisinstabilitywasfirststudiedwithintheframeworkofliquidepitaxybyAsaroandTiller[18] andre-derivedby Grinfeld[23].Thefilmdiffusesinordertoshapean undulatedsurface.Thismayonlyhappenforfilmthicknessesabovea characteristicvalue that wewill derivesubsequently.We emphasizethat thisinstability developsthanksto surface diffu-siondrivenbythestrainresultingfromthedifferenceofthelatticesizesbetweenthefilmandthesubstrate.Theinstability starts withanundulationthatdevelopswithtime.Thisundulationtakesplaceatthefilmfree surface.Ifthereisenough matter,theundulationtransformsinbell-shapedislands.

WecansolvetheevolutionEq. (1) byconsideringalinearanalysiswherethefilm-freesurfaceisdecomposedintoFourier modesalongx and y.Thesolutionisanexponentialgrowthh

(

x

,

y

,

t

)

=

h0

+

Aeσt+i kxx+i kyy.

σ

reads:

σ

= −

1

γ

f

∂

2

γ

(

h

,

hx

,

hy

)

∂

h2







h=hc k2

+ |

k

|

3

−

γ

˜

(

hc

,

hx

,

hy

)

γ

f k4 (4)

indimensionlessunits.Here,thesurfacestiffnessisdefinedas

γ

˜

=

γ

+

∂2_γ ∂hi∂hi.

Forh0 largerthansome wetting criticalthicknesshc,the growthratepresentspositive valuesinafinite wavenumber intervalandtheinstabilitydevelopsasshowninFig.3.Thecriticalthicknesshc ischaracterizedby

σ

=

0 and

∂

σ

/∂

k

=

0:

hc

= δ

wln

(

4 cw

/δ

w

)

(5)

4. Hexagonalquantumdots

We presentherenumerical simulationsof asystem withoutpreferential evaporation, inwhich theinitial condition is a constant height h0

>

hc plusa smallrandom perturbation.In orderto test ourmodelwithan hexagonal symmetry of thesurfaceenergyanisotropy,westudytheeffectofdifferentparametersinordertohavethebestones.Weconsidertwo initial heightsh0

=

0

.

1 andh0

=

0

.

35 and,foreach initialheight, wevarythestrength oftheflatorientation n0 givenin Eq. (3).

1 _{WeusedY=300 GPa,ν=0.20,a}

Fig. 3. Growthrateσ asafunctionofthewavenumberk giveninEq. (4).Fromtoptobottom,thebluecurverepresentsthegrowthrateforh0>hc,the

orangecurvethatforh0=hc,andthegreencurvethatforh0<hc.

Fig. 4. Roughnessasafunctionoftime.Thebluecurve,theorangecurve,andthegreencurverepresentthenumericalresultsforA0=0.2,A0=0.35,and

A0=0.5,respectively.(a)Initialheighth0=0.1 and(b)initialheighth0=0.35.

Fig. 5. Islandprofilesfordifferentframetimes.Thecolorsrangefrombluetoreddependingonthevalueoftheheight,fromthewettinglayerheightto thetopoftheisland.(a)t=500 and(b)t=1200.Theinitialheightish0=0.1.

We plot in Fig.4 the roughness w

=

√

<

h2

_>

_{− <}

_h

_>

2 _{as a function of time. We notice that the instability with a} biggerinitialheighth0developsfasterthantheonewithasmallinitialheight.Thisfeatureisexplainedbythedecreaseof wettinginteractionsforlargeheightsh.Thewettingeffectflattensthesystemandcompeteswiththeelasticitythatdrives thegrowthofislands. So,ifthewettinginteractions arestronger,theinstabilitywilltakemoretime todevelop.Asimilar behaviorhappenswiththestrengthoftheflatorientation (001)A0.Forsystemsinwhichtheflatorientationisfavorable, thedevelopmentoftheislandstakesmoretime.Astheflatorientationbecomesadeeperlocalminimum,thesystemtakes moretimetogofromsmallslopestothefacets’preferentialorientation.

WeplotinFig.5and6QDsprofilesfordifferenttimeswheretheanisotropystrength A0

=

0

.

1 andtheheightish0

=

0

.

1 andh0

=

0

.

35,respectively.We canseethatduring coarsening,differentshapesare encountered(truncated-elongated pyramids),butitdependsonthetotalmass.

5. Preferentialevaporation

Wenowstudytheevolutionofthesysteminthepresenceofevaporation.Indeed,onecharacteristicsoftheIII–Vsystems understudyisthelowerboundingonthesurface,whichleadstoasignificant evaporationduringannealing.Wetakethis extra-effectintoaccount and,in addition,weconsider anisotropicevaporation, asoneknowsthat thefacetsevaporateat differentspeeds.

Fig. 6. Islandprofilesfordifferentframetimes.Thecolorsrangefrombluetoreddependingonthevalueoftheheight,fromthewettinglayerheightto thetopoftheisland.(a)t=500 and(b)t=1200.Theinitialheightish0=0.35.

Fig. 7. Roughnessasafunctionofthetime,forthreedifferentvaluesoftheevaporationfluxFev001.Thegreenrhombusarefor F ev

001=0.04,theorange

squaresareforFev

001=0.022,andthebluedisksareforF ev

001=0.018.Theinitialheightish0=1.75.

Our investigationconcerns thenthe long-time dynamicsofthesystemduring long enoughannealing/evaporation.We performnumericalsimulationsofEq. (1) forthepreferentialevaporationgivenbyEq. (2):itwillcompetewiththesurface diffusioninthecoarseningdynamics.Whenevaporationishighenough,theislandsdonothavetime toappearbeforethe entiresystemisevaporated,becausesurfacediffusionisslowerthanevaporation.Whenevaporationismoderate/low,two behaviorscan bemet.The firstbehavior iswhenislands develop but,asthemass evaporates,theisland dissolvesin the wetting layeras,whentheinitialheightislowerthanhc,theislandscannotdevelop.Thesecondpossiblebehavioroccurs when evaporation ispreferential, in which casethe wetting layer vanishes, whileislands remain in the system, keeping theirshape.Asthemorphologyisaresultofthesystem’sdynamics,nopreferentialinsightcanbegivenapriori.

The initial condition forthe systemunder studyiscomposed ofa constant height h0 plus a randomperturbation. In comparison withthe simulations presented in the previous section, herewe choose an initial height h0 of theorder of magnitudeofthetypicalislandheights,sincetheevaporationwilldecreasethemassofthesystemuntilthemassvanishes, and in order to let the instability develop, the system will evaporate a considerable amount of mass before instability develops.Wealsochoseaweaksurfacestiffnessforthe(001)orientation,sincethenumericalsimulationsarefaster,aswe haveshownpreviously.

We expect that, for big evaporationfluxes, the instability cannot develop, because evaporationwill win against mass diffusion,andtheinstability wouldnot evolve.Onthecontrary,iftheevaporationfluxisslow,massdiffusionwouldwin againstevaporation,andthesystemwillvanish.

We plot in Fig. 7 the temporal evolution of the roughness w

=

√

<

h2

_>

_{− <}

_h

_>

2 _{as a function of the time, for a} system with three values of preferential evaporation Fev₀₀₁. We observe that roughnessfirst increases, before it starts to decreaseuntilbecomingnegative. Oncetheislandshavetheirpyramidalshape,thepreferentialevaporationeffectismore quantitativeandmakestheroughnessdecrease.

5.1. Vanishingofthewettinglayer

Wenowpresenttheresultsconcerningthevanishingofthewettinglayer.WeperformnumericalsimulationsofEq. (1) foraninitialheighth0

=

1

.

75,varyingtheevaporationfluxofthewettinglayer F001ev .Weletthesystemevolve,andwhen thewettinglayerpresentsnegativevalues,westopthesimulation.WeplotinFig.8thelayerprofileatthemomentwhen thewettinglayerisnegative.Wenoticethattheregionwherethewettinglayervanishesexhibitasmallperiodicbehavior, yetunsettled.

Fig. 8. ProfileobtainedbynumericalsimulationofEq. (1),forthetimewhenthewettinglayergoestonegativevalues.Thegray spotsrepresentthenegative valuesofthewettinglayer.TheevaporationfluxischosentobeFev

001=0.02 andtheinitialheightish0=1.75.Sincethesimulationisnotrealisticfor

negativevaluesoftheheight,westopthesimulationhere.

6. Conclusion

Westudynumericallyathree-dimensionalsystem,wherethesurfaceenergyanisotropypresentsahexagonalsymmetry, inordertofavorhexagonalislands.Weanalyzetheeffectsoftheflatorientationandtheeffectsoftheinitialheightonthe system.

– Wefindthatasystemwithmoremasspresentsfastercoarsening.Thisbehaviorisrationalizedbythewettingpotential effect,whichtriestoflattenthesystem, anddecaysexponentiallywiththe height,sothat, whentheinitial heightis bigger,theeffectisweaker,andtheislandsevolvefaster.

– Asimilarbehavior occurswithaflat orientationinthesurface energyanisotropy.Ifitisweak, awettinglayer isnot preferential,sothesystempresentsahighdensityofislands,andalso,asthisorientation isweak,coarseningwillbe faster.

We extendourmodelwithapreferential evaporation. Theexperiments show thatisland facetsevaporateslowerthan thewettinglayer.Duetothat,weaddtoourmodelapreferentialevaporation.Weshowthat,dependingontheevaporation fluxofthewettinglayer,islandshavetimetodevelopornot.Ifislandsdevelop,weshowthat,forcertaintimes,thewetting layervanishes.

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