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Ge clusters in Si matrix: structure and dynamics
J. Dalla Torre, N. Barriquand, Mehdi Djafari-Rouhani, Georges Landa
To cite this version:
J. Dalla Torre, N. Barriquand, Mehdi Djafari-Rouhani, Georges Landa. Ge clusters in Si matrix:
structure and dynamics. European Physical Journal B: Condensed Matter and Complex Systems,
Springer-Verlag, 1999, 12 (3), pp.343-346. �10.1007/s100510051014�. �hal-00148776�
Ge lusters in Si matrix: stru ture and dynami s J.DallaTorre 12 ,N. Barriquand 2 ,M.DjafariRouhani 12 andG.Landa 2 1
Laboratoire de Physique des Solides, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse edex-4, Fran e, ESA5477asso ieeauCNRS
2
Laboratoired'Analyseetd'Ar hite turedesSystemes-CNRS,7avenueduColonelRo he,31077Toulouse edex,Fran e
Re eived:date/Revisedversion:date
Abstra t. We have determined by omputer simulations, some stru tural properties of Ge lusters em-beddedinaSi rystallinehostmatrixfor lustersizesvaryingfrom0.5 to1.5nm.Inorderto des ribe inter-atomi for es we have hosena Valen e For e Field(VFF) semi-empiri alpotential. Next we have al ulated the density of vibrationalstatesby diagonalizationof the dynami almatrixde nedwiththe samepotential.Thein uen eofthevolume/interfa eratioofGeonthevibrationalpropertiesisdis ussed.
PACS. 61,46+w Clusters,nanoparti les,andnano rystallinematerials{63, 50+x Vibrationalstatesin disorderedsystems
1Introdu tion
In re ent years, an in reasing e ort has been developed towardstheunderstandingof surprising physi al
proper-ties of lusters. An exampleof potentialappli ations for semi ondu tor lustersistheirnonlinearopti alproperties
thatusuallydonotappearinthebulkmaterials[1℄.Many theoreti alandparti ularly"AbInitio" al ulationshave
been made in order to determine the stru ture of these
lusters[2,3℄. These ab initio al ulations generally deal withonlyfew atoms.Whilesmallsemi ondu tor lusters
fe ts,largeonesseemtoadoptthe rystalstru ture. Oth-erwise,experimentalobservationsof lustersinahost
ma-trix generally show large luster (few nanometers) with rystallinestru ture[4,5℄.Butthestru ture,inparti ular
at the interfa ebetweenthe matrixand the luster, an be strongly modi ed by the hemi al nature of the
ma-trix, the elaboration te hnique and the elaboration
on-ditions, annealingtemperature,et . Ramanspe tros opy an be useful to de ribe stru turaland dynami
proper-ties of su h lusters[1,5℄. Ourpurpose inthis paperisto showthatusingsimpleValen eFor eField(VFF)
poten-tials,one may obtainnotonlythe twoaboveproperties,
but also,thedegreeofthelo alizationofthestatesinside
the lusters. In this study, we have onsidered Ge lus-ters embedded in aSi rystalline hostmatrix ontaining
432atoms. Evenifthe lustersizesdis ussedhere anbe produ ed experimentally[4℄, amore typi al size is about
2 to 5 nm[1,5℄. So, ontrary to the alloy ase, the
num-berofatomsin thesuper ellmust beverylargein order to des ribe orre tly the largedefe ts. With a 432atom
super ell,wehavebeenabletotreat lustersofradiusup to 1.5 nm. This luster sizes are somewherebetween the
sizes used in ab initio models and those in real experi-ments, but ontain the general information, parti ularly
onthelo alizationofthestatesattheinterfa e.We
om-pareourresultstoexperimentaldataobtainedinasimilar ase where theSi matrix ispoly rystalline[4℄.The
inter-fa e betweenthe Ge lusters and the Si matrixis taken tobeperfe twithnodanglingbonds.Theorganizationof
thepaperisasfollows.Inse tionIIwepresentthemodel
usedin thispaperanddes ribetheparametersemployed in our al ulations.Inse tionIII,wepresentanddis uss
theresultsobtained.Finallyinse tionIV,wesummarize ourmain on lusions.
2Model
The frame of our model has been presented before[6℄ in
the ase of amorphous Si and we sket h here only the ne essaryba kground to understand the pro edure.Our
Table 1. Parameters for the di erent distributions under study.Numberof Ge atomsina luster,size of the lusters, numberof lusters and resulting proportion of Ge in the Si matrix.
Numberofatoms/ luster 5 17 29 47 87 Clustersdiameter(nm) 0.49 0.80 0.94 1.23 1.47 Numberof lusters 18 5 3 2 1 %ofGe 20.1 19.7 20.1 21.8 20.1
(a)The rstoneisthe onstru tionofthesystem un-der study. The super ell approa h is hosen in order to
des ribethe stru tural and the hemi al disorder.
A tu-ally, in disordered materials the rystal symmetries are broken due to the introdu tion of non periodi and
ex-tended heterogeneitiesin thepure rystal. Asaresult, a disorderedmaterialshouldbemodeledbyaunique
ma ro-s opi ell,but be auseofthe omputationallimitations,
itisapproximatedbyasuper ellwith periodi boundary onditionskeepingin mind that thesize andthe density
of the heterogeneities are limited by this super ell size. So we have de ned a 6[110℄x6[1-10℄x12[001℄=432atoms
rystalline super ell and we have introdu ed at random positions,Gespheri al lusterswithsizesvarying from
0.5to1.5nm(seetableI)withadiamond-like rystalline
stru ture( g.1a)).
(b)As the Ge latti eparameter islarger thanthe Si
one,atomi positionsmustberelaxed. Forthis, thetotal elasti energyisminimizedandthusthesuper ellvolume
a
b
[110]
Fig. 1. a) stru turerepresentation ofa 87atoms luster.b) (001) plane ofthe 1024 atoms super ell, passingthroughthe enterofthe luster,darkestandbiggestspheresareGeatoms.
tentialintheMartins'harmoni formulation[7℄:
E s = X bonds k r (r r bond ) 2 + X bondangles k ( 0 ) 2 (1)
whereristhea tualbondlength,r bond
istheequilibrium bond length orresponding to thepure materialsand we
have taken r Si Ge bond = (r Si bond +r Ge bond
)=2 for the heteroge-neousSi-Gebonds.isthea tualanglebetweentwo
adja- entbondsand 0
istheanglebetweentwoadja entbonds
in thediamondperfe tstru ture. k r
andk
arethefor e parametersrelatedtotheelasti onstantsofthepure
ma-takenformixedparameters,k Si Ge r =(k Si r +k Ge r )=2and k d X Y Z =k b Y (X;Y;Z=Sior Ge).
( )Afterthe minimumtotalstrainenergyis rea hed, the vibrational eigenstates or normal mode frequen ies
are determined by diagonalization of the dynami al ma-trixobtained withthe sameVFFinter-atomi potential.
Finally, an approximation of the Density of Vibrational
States(DVS) isobtainedbysampling thevibrational fre-quen iesspe trum.
3Stru tural and dynami al al ulations
In this paper,wehave fo used our omputations onthe
ase of a 20% parti ular on entration of Ge atoms in
the Si matrix. In this ase, Si hara ter of the matrixis almost preserved and the number of Ge atoms inside is
suÆ ienttointrodu edi erentratiosofdisorder depend-ing on their relative arrangement. This allowsthe study
ofdistributions ofdi erentsize andnumberofsu h lus-ters(see table I). The enter of the luster is randomly
determined ataSisite,aradiusisde ned withase ond
Si atomand all theatoms in this sphereare substituted byGeatoms.
First, we are interested in the stru ture al ulations toevaluatethestrainnearthe luster boundaries.Thisis
performed throughtherelaxationof thestru ture. How-ever,itshould bepointedoutthat the relaxedstru ture
only orrespondsto ametastable on guration,i.e. a
lo- al minimum of theelasti energy. This resultsfrom the de nite positivenature of thedynami al matrixin VFF
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Ge clusters size (nm)
2.34
2.36
2.38
2.40
2.42
2.44
Bond length (
Å)
Ge−Ge
Si−Ge
average
Si−Si
Fig.2.AveragebonglengthsofSi-Si,Si-GeandGe-Getypes asafun tionofthe lusterssizewhilethetotalGe omposition is kept onstant. The dark diamondsare their valuesfor the al ulated Si
0:8 Ge
0:2 alloy.
lows relatively small atomi displa ements, but not for atomi migrationsalongafewinteratomi distan es.
Theresultisinagreementwithexperimental on gu-rationswhi harealsoin metastablestates.Thequestion
ofwhetherin lusionsarethermodynami allystableornot anonlybeadressedthroughathermalannealing
pro e-dure, bothin experimental andsimulationgrounds.This
questionisbeyondthes opeofthispaper.
Asa rst he kofour al ulationswehave al ulated the average bond length(notedaverage in g. 2) overall
kindofbonds.
As one an observe, the result is in total agreement
withtheexpe tedvaluefromtheVegard'slaw(L average = 0:2(L Ge L Si )+L Si 2:37
A). Thisaveragevalueis
al-mostthe samewhateveristhe distributionofGe atoms,
a lot of small lusters or a few big ones. Moreover we have reported in gure 2 the average bond length over
1
2
3
4
5
6
7
8
Atom number
−5
−4
−3
−2
−1
0
1
2
3
4
Strain (%)
Fig. 3. Strain al ulation along the [110℄ dire tion passing throughthe enterofa87Ge lusterasafun tionoftheatom numberalongthisdire tion( g.1b)).
of the luster diameter. The al ulated bond lengths of
a Si 0:8
Ge 0:2
alloy where Ge are randomly distributed in the matrixare representedbydarkdiamonds and are in
agreementwithprevioussimilarresults[8,9℄.The interest-ingpointisthatasa rstapproximation,bondlengthsare
onstantandindependentofthe lustersizesforthethree kind of bonds. This means that the relaxation for ea h
kindofbond isthesamefromthealloy aseto that ofa
matrixwitha87atoms luster.Thisindi atethatdueto their ovalentnature,ea hkindofbondkeepsitsspe i
hara terindependentlyofthedistribution.
Figure 3presents the bond strain with respe t to Si,
alongthe[110℄dire tion ofthesuper ellpassingthrough the enter ofthe Ge luster.Be ause ofthe dis reteand
non ompa tnatureofthematerialatthiss alethestrain
is al ulated over a very few number of atoms and the (001) se tion passing through the enter of the 87
lus-Wehavede ned thisstrainatanisiteby" i =(Æx i Æx i+1 )=d Si ,whereÆx i
isthevariationoftheatomi
posi-tionalong the[110℄axis( g. 1b)) andd Si
is thedistan e betweentwoatomsinthisdire tionintheSipure rystal.
Inorder toevaluatethelo alstraindue tothe luster in theSimatrixoveralargerrange,wehaveenlargedthe
su-per ellbyfewmonolayersinea hdire tion.Thisin rease
hasbeenlimitedtokeepalowtime ostofour al ulation. Inthis al ulationtheGe proportionislowerthanin the
previous asesasthisisa87Geatomsina8x8x16=1024 atomsmatrix(8:5%).We anobservearapidvariation
of thestrainfrom approximately3.5% to -3.5%whi h is made overtwoatoms. The strainrea hes 3.5%in the
Ge luster, whi h is 0.5%less than thelatti emismat h
between Ge and Si. This orrespondsto aresidual om-pressionoftheGe lusterbytheSimatrix.Otherwise,the
strainfor theSi matrix rea hes-3.5% at the neighbor-hoodofthe lusterwhi h orrespondstoa ompressionof
theSiatoms.TheabsolutevalueofstrainforSiappearsto
de reasewiththedistan etothe luster.Theseresultsare inquantitativegoodagreementwith al ulationsmadeby
ontinuumtheory[10℄.
Next,wearegoingtodis ussthevibrationalstatesof
theprevioussystems.Asweareinthesuper ellmethod,it wouldbe onvenienttode ne anewwaveve tor on ept
relatedtothenewperiodi ity,leadingsotoanewBrillouin zone.
In order to obtain a smooth Density of Vibrational States (DVS), we aneither average theDVS overaset
0
100
200
300
400
500
DVS (arb. u.)
Wave number (cm
−1
)
5
17
29
47
87
C
GeGe
SiGe
SiSi
Fig.4.Cal ulatedDVSfordi erent lusterssize,from5to87 atoms.TheGe ompositioniskept onstantto 20%,sothe numberof lusters varies (see table I). Thenoted "C" DVS is a linear ombination of the pure Ge and Si DVS. Opti al bands asso iatedto theGe-Ge, Si-Geand Si-Siare indi ated byarrows.
fewwaveve torsinthesuper ellBrillouinzone[6℄.Wehave
adopted this se ond s heme and have used 10 values of the waveve torrandomly distributed inside thesuper ell
Brillouin zone. In Fig. 4,we havereported the DVS for di erent lusters ofvarioussizes, ontainingfrom5to87
atoms.TheGe ontentiskept onstant,andequalto20of lusters presentin thesuper ell. To ompare our results
fordi erentdistributionsofGe lusters,wehavealso
pre-sentedin g.4thelinear ombinationoftheDVSofpureSi andGe.ItsintensityisgivenbyI
ombi =0:8:I Si +0:2:I Ge ,
DVSofahugheGe lusterinSiandisusedasareferen e
to observethe size e e ts, preferably to apure Ge orSi
DVSreferen e.Con erningthegeneralaspe toftheDVS, weobserveanimportantdi eren ebetweenthis
ombina-tionandtheDVSforsmall lusters.Thisisa onsequen e ofthedisorderindu edbythedispersionoftheGeatoms
in theSi matrix.Thisla kofsymmetriessplito the
de-generate eigenstates and as a result smoothes the DVS. So, when the luster sizes be ome larger,the DVS ome
loser to the ombination DVS,showinga "phase segre-gation"ofGeandSispe ies.We anparti ularlyobserve
inFig.4thattheGe-Gemodeshiftstohigherfrequen ies when lustersizes in rease.A learshiftisseenwhen
go-ing fromthe 87atom luster tothe referen e urve.The
results arein good agreement withexperimental Raman spe traof Ge lusters inSi matrixwhi hshowapeakat
270 m-1attributed to theamorphousGe-Gemode[1,5℄. Moreover,we anseethetypi al"3modes"[12℄behavior,
ofSiGe ompoundsfortheopti albands( g.4).Ea h
op-ti al band is asso iated with one kind of bond: Si-Si ( 520 m 1 ),Si-Ge(380 m 1 )andGe-Ge(270 m 1 ).
Thee e tsoftheinterfa e/volumeratioofGeare parti -ularlyvisibleinthedi erentintensitiesofDVSandshould
beobservedinthe orrespondingRamanspe tra.As the
numberof lusters de reases i.e. their size in reases, the number of Ge-Ge bonds in reases whi h is dire tly
ob-served in theintensityof theGe-Geband. Butwhile in- reasing the luster size, the number of Si-Ge bonds at
the interfa esde reasesand as a result,theSi-Ge signa-turevanishes.
4Con lusion
In on lusion,wehave al ulatedsomestru turaland
dy-nami al properties of Ge lusters in an Si matrix by a
semi-empiri al VFF potential. Inthis study the number and the size of the lusters are varying in order to keep
theGeproportionapproximativelyequalto20%.Despite the small size of the super ell used in the al ulations,
we have shown that even for large lusters, the average
bondsareroughlythesameasalloyvalueswiththesame Ge ompositionandingoodagreementwithexperimental
observations.Wehaveobservedarapidvariationofstrain at the luster interfa e and we have shown that in the
1.47 nm lusters, the ore has a latti e parameter lose to theGepure rystal.Finally,wehavedis ussedthe
Si-Ge interfa ee e tsin theDVS.A tually,wehaveshown
the de rease of the Si-Geand thein rease of the Ge-Ge band intensitiesfor large lusters asaresultofthe
inter-fa e/volumeratioof lusters. Ashiftof theGe-Geband, losetoexperimental results,hasbeenobtained.
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