Solid StateCommunications,Vol. 37,pp.271—274.Pergamon PressLtd. 1981.Printed in Great Britain.

VIBRATIONAL PROPERTIES OF CRYSTALLINE AND AMORPHOUS Ge~Si~ALLOYS

Bal. K. Agrawal

Department of Physics,University of Allahabad, Allahabad, India

(Received7 August1980by C. W. McCombie)

A five-atom cluster Bethelatticemethodtreatingtheshort-rangeorderwith proper statisticaleffectshasbeenproposedfor understandingthecrystallineandamorphousalloys. Numerical results for Ge1_~Si~alloysarein excellentagreementwith the experimental data. In the systemstudied the network is randombut the nearest.neighbourcoordinationshouldbetreated exactly.The long-rangecoordination is seento beconcentration-dependentin the crystallinealloysbut concentration-independent in theamorphousalloys.

IN Ge—SiALLOYS theelectronegativity difference networks correspondingto different bond statistics: (i)betweenthe Ge andSi atomsis quite smallandit has segregationnetwork where the referenceatom is alwaysbeenshown[1] that specialfeatures(e.g.an “ionic” surrounded by its ownkind of atoms(ii) chemicallyor asymmetrygap) in the electronicdensityof states ordered network where the referenceatomhas all itsappearin a limited medium-concentration range in the nearest-neighbouringatomsof the other kind in casethechemically orderednetwork wherethe atomsin minority referenceatom is in minority in concentration,and(us)(in concentration)arealwayssurrounded by the majority random network wherenoneof the restrictions includedatoms.On the other hand, asthe massratio of thecon- in the abovetwo networks existsand the occurrenceofstituent atomsQWo./mas= 2.6) is quite large,the effects a kind of atom dependsonly on its concentration.of the short-rangeorder shouldappearmore clearly in For Ge1_~Si~alloyswefirst perform oneatomthe phonondensityof states. cluster—Bethelattice calculation (where the Bethe

The author andhiscollaborators[2] have been lattice is attachedto the referenceatom) throughout thestudyingthe phononsof the GeSisystemusinga Green’s whole concentrationrange.The phonon density for thefunction methodin the low concentrationlimit of Bethe lattice dependson the concentrationx in theimpurities. Localisedmodesdue to singleand pairedSi Ge1_~Si~alloysin the variousnetworks and is thus dif.atomsin Ge matrixwere predictedat 390and 460 ferent for various valuesofx. For illustration wepresentcm

1,respectively,In agreementwith the infrared [3] here the phonondensityfor the Ge0~5Si0.5 alloy for the

andRaman [4] data. In the presentcommunication segregation,chemically orderedandrandomnetworkswe reportthe resultsof astudy of the vibrational prop- in Fig. 2. In the segregationnetwork [Fig.2(a)1 theertiesof the crystalline and amorphousGei...~Si~alloys total density is simply the averagesof the densitiesofusinga five-atom cluster—Bethelattice methodwhere the pure Si and Ge andthereby exhibits the host peaksthe short-range order is treated exactly,both in the at 100,200,290and510cm~.However the peakatnearest-neighbourcoordination andbeyond it with — 400cm

1 due to the localmode of Si in the cageofproper statisticaleffects.Earlier Yndurain [5] hasmade Ge atomsis missing.In the chemically ordered networkasimilar cluster—Bethe lattice calculation in virtual [Fig.2(b)] aseachatom of onekind hasall its neigh.crystal approximation in which all the atomsare equi- boursof a secondkind oneexpectsto observe the fi~odesvalentandthereis no short-range order. The present of Ge in the cageof four Sineighboursandvice versa.calculationconsidersalso thevariation of the inter- We thus observethe acousticmodesand a bandof theatomic force constantsfor Si—Si, Si—GeandGe—Ge optical modesat 400 cm~of the SiGe compound.bonds for interpretatingthe Ramandataof Lannln [4] The optical bandis well separatedfrom the acoustic(seeFig. 1). band. On theother hand,the TO modesof pure Ge and

For detailswerefer to earlier papers [1,5]. The Si alloys are missing.Finally, in the random networkinterpolationschemeusedfor the Bethelattice is similar [Fig.2(c)] oneobtainspeaksnear 100,250,280andto Kittler andFalicov [6]. There are threecovalent 480 cm’ , but againthe Si local modeis missing.Thus___________ the one-atomcluster calculationdoesnot reproduce all* Work supported in partsby ScienceResearchCouncil, the features appearingin the Remandata in anyone

U.K. andby University GrantsCommission,NewDelhi. network and it necessitatesthe considerationof a largercluster which wediscussbelow.

271

272 VIBRATIONAL PROPERTIESOF Ge1..~Si~ALLOYS Vol. 37, No.3

0.8CRYSTALLP� ALLOYS CRYSTALLINE Ge1~Si~ALLOYS

0.6 (~)Gel ~5’~x- 0.1

0.4

0.2 _z091 0.00177

x .0.30.6

~O.35 0.00.46 0.2

x=0.5m 04zOll

4

0.0 _500 600 700 800 900 iccoiioo 0.2i.~ (cm’) Cd)Fig. 1. Second-orderRamanspectrain Ge1_~Si~at T 07

0.4330K for somecrystallinealloys reproducedfrom [4].The arrowsindicatemajor first-orderpeakpositions 0.2

multiplied by 2. 0.0 .i~~~~i’l

x 090.46Ø5 S’a5

(0)0.4 SEGREGAT~N 0.2

0.00 100 200 300 400 500

________________ Fig. 3. Phonondensityof statesin crystallineGe1_~Si~0.2 w (cm~)

0.0 alloys forx = 0.1,0.3,0.5,0.7and0.9. The dashedandCHEMICALLY ORDERED (b) dottedcurvesshow,respectively,thecontributionsfrom

0.6 theGe andSi atoms.

0.4 /‘ neighboursgivesriseto aresonancemodeat 130cm~.

I Also, theoccurrenceof aSi atomasits neighbourgives~ 0.2rise to apeakat —315 cm~.Onthe otherhand,one

Ui ~

° 0.0 • finds alocalisedmodedueto the Si atomhavingfourRANDOM (C) neighbouringGeatomsat 395 cnf’. A modedueto

0.8the clusterSi—SiSiGeGeappearsat 250 cm’. It may

0.6be notedthatall theabove-mentionedmodesare ob-

0.4servedatall thevaluesofx(0<x < 1).

Thephonondensityatthereferenceatomn(w) is0.2 thendeterminedby takingaweightedaverageof the0.0

0 100 200 3~0 A00 soo densitiesn~(w)atthe centralatomin thevariousclustersas

~ (cm~)

Fig. 2. Phonondensityof statesfor Ge05Si0~alloy in n(c~)= E w~n0(w) (1)Cthe~a)segregationnetwork,(b)chemicallyorderednet-

work,and(c) randomnetwork.Thedashedanddotted wherew0 is the properweightof the clusterc andiscurvesshow the contributionsfrom the Ge andSi atoms, equaltox or (I — x) timesthe factor [4! Ia! b ! )x

0respectively.The solid curvedenotesthe totaldensity. (1 x)b] accordingto whetherthe centralatomis Si or

Ge. Hereaandb denotethenumberof Si andGeneigh-The five-atomclusterconsistsof thereferenceatom boursof the centralatomin the five-atomcluster,

plusits fournearestneighbours.We attachthe Bethe respectively.latticeto thesefour neighboursandcalculatethe phonondensityat thereferenceatomfor everypossiblecon- 1. CRYSTALLINE ALLOYSfiguration of the clusterfor eachconcentration.Thedensitiesof thevariousclustershavingGeasthe central Thecalculateddensityof statesat someconcentra-atom,revealthata Geatom havingall the Si atomsas tionsof x areshownin Fig. 3. A comparisonwith

Vol. 37,No.3 VIBRATIONAL PROPERTIESOF Gej..~Si~ALLOYS 273

0.6

l AMORPHOUS ALLOYS AMORP*O.5 Ge1,Si,, ALLOYS (a)

£ G.1.1Si~ 0.6

~ ~. ~JL...

Z 10 ..—-—.._ x—O.3 ,c.0.3 (b)

50 __-~~__....-_ ~—___ x .05 0.6

~ 7O—~~ .N____..._zO.7 0.4

___________________ :.: tI:::E~E.\....~_.0 100 200 300 400 500 600 ~.O.7 (c)

&, (cnr1) ~ 0.’. A I

Fig. 4. Reducedfirst-orderRamanintensityof amor- 0.2 ~ ~.L=,_~—_t~phousGe

1_~Si~alloys from [7] . Thecurveswere o.onormalisedto thesamearea. 0.4 x.O.

9 Cd)

Remandata(Fig. 1) showsthat thecalculatedfrequency 00

shiftsof the majorpeaksis invery goodagreementwith 0 100 200 300 400 500

theexperiment.The frequencyof thepureSi (or Ge) . ~cm~’)materialappearingat 515cm~(or 290cm1)de-creaseswith increasein the concentrationof Ge(or Si) Fig. 5. Phonondensityof statesin amorphousGei_~Si~atoms,in away similarto experimentaldata.TheSi alloys for x = 0.1,0.3,0.7and0.9.The curvefor x =

localmodeat 395 cm’ is almostconcentration- 0.5 is similar to Fig. 2(c).Thedashedanddottedcurvesshow respectively the contributionsfrom the Ge andindependentalthoughsomeirregularvariationin fre. Si atomsquencyis seenin the Remandata.Thevariationof theheightsof the peaksin thephonondensitycorrespond-ing to the Si—SiandGe—Gebondswith concentration Bethelatticeis concentration-independent,i.e. theare inagreementwith theintensitiesin the Remandata. phonondensityof the Bethelatticeis the sameatdif-Although theheightof the Si local modepeakdoesnot ferentconcentrationsof Si andGe atoms.increasewith x, in contrastto the Ramandata,the Thusthe probabilityof theoccurrenceof anatomintegratedintensityof theline increasesfor 0 <x <0.5. beyondthe first shell in amorphousmaterialis indepen-A peakcorrespondingto the R.amanpeakat— 340cm~ dentof its concentration.We thus assumeanetwork forappearsin phonondensityat ‘~ 315 cmt atarelatively theBethelattice havingequalprobabilityfor the occur-higherconcentrationof Si atoms.In the low frequency renceof anykind of atom irrespectiveof its concentra-region,abroadpeakappearsnear108cm’ in the tion, i.e. aone-atomclusterBethelattice for aGe

0~5Si05Ramandata.As discussedearlier,wefmd aresonance alloy in arandomnetwork.modedueto heavyGe.atomssurroundedby Si neigh- A five-atomclusterBethelattice calculationsimilarboursat 130 cm_i. to thatfor crystallinealloyswasperformedandthe

densityof statesfor theamorphousalloysat variousconcentrationsare shownin Fig. 5. Oneobservessome2. AMORPHOUSALLOYS . . -istructure,especiallyin the 250—300cm region,arising

TheRemandataof Lannin [7] for the amorphous from theclustersaswell as from theBethelattice. In thealloys (Fig. 4) showthatthe frequenciesof the three calculationwe havenotaddedanyimaginarycomponentmajorpeaksappearingat 290,400and480cnfi are to thefrequencyasincludedearlierby Yndurain [5].concentration-independentin contrastto thoseof the We do not find discretestatessimilar to thosefoundbycrystallinealloys.The concentrationdependenceof the the otherauthor.In the Remanmeasurementsthesefrequenciesof theGe—GeandSi—Sibondpeaksis structuresare smearedout.Thecalculatedphononden-inherentin theconcentration-dependenceof the phonon sity is in goodagreementwith theexperimentalobser-densityof the Bethelattice,whereastheSi localmode vationthroughoutthe wholefrequencyrange.at395 cm

1 arisesfrom thefive-atomclusterandis thusconcentration-independent.In order to haveinvariant Acknowledgement— The authoris gratefulto ProfessorGe—GeandSi—Sibond peaksonehasto assumethatthe J.L. Beebyfor hospitality.

274 VIBRATIONAL PROPERTIESOF Ge1~Si~ALLOYS Vol. 37, No.3

REFERENCES (1971).4. JS. Lannin,Phys.Rev.B16, 1510(1977).

1. Bal K. Agrawal,Phys.Rev. BiS, 1980(in press). 5. F. Yndurain,Phys. Rev.Lett.37, 1062(1976);2. Ba! K. Agrawal& D.N. Talwar, Phys.Rev.B18, Phys.Rev.B18, 2876(1978).

1751,7189(1978);Ba! K. Agrawal, S. Tripathi, 6. R.C. Kittler & L.M. Falicov,J. Phys.C: SolidStateA.K. Misra & D.N. Talwar,Phys.Rev. B19. 5277 9,4259(1976).(1979);D.N. Taiwar & Ba! K. Agrawal, Ciyst. 7. J.S. Lannin,AmorphousandLiquid Semiconduc-LatticeDefects,185 (1980). to’s (Editedby J. Stuke &W. Brenig), p. 1245.

3. A.E. Cosand& W.G. Spitzer,J. Appi.Phys. 42,41 Taylor andFrancis, London (1974).