PHYSICAL REVIEW B 15 OCTOBER 1998-IIVOLUME 58, NUMBER 16

Structure and chemical order of bulk Si12xCx amorphous alloys

Daniele MuraIstituto Nazionale per la Fisica della Materia and Dipartimento di Scienze Fisiche, Universita` degli Studi di Cagliari,

Cittadella Universitaria, S. Prov.le Monserrato-Sestu Km. 0700, 09042 Monserrato, Italy

Luciano ColomboIstituto Nazionale per la Fisica della Materia and Dipartimento di Scienza dei Materiali, Universita` degli Studi di Milano,

via Emanueli 15, I-20126 Milano, Italy

Rita BertonciniCentro Ricerche e Sviluppo Studi Superiori in Sardegna, C.P. 1048, 09123 Cagliari, Italy

Gianni MulaInstituto Nazionale per la Fisica della Materia and Dipartimento di Scienze Fisiche, Universita` degli Studi di Cagliari,

Cittadella Universitaria, S. Prov.le Monserrato-Sestu Km. 0700, 09042 Monserrato, Italy~Received 24 February 1998; revised manuscript received 18 June 1998!

We investigate the structural trends of bulk amorphous Si12xCx alloys in the composition range 0.125,x,0.875 by molecular dynamics. Our simulations, based on the many-body potential by Tersoff, allow for anaccurate analysis of the compound topology, as well as of its chemical order. We also characterize the localatomic coordination and network distortion as a function of the alloy composition.@S0163-1829~98!04640-2#

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I. INTRODUCTION

The amorphous silicon carbide alloy (a-Si12xCx) is a pro-totypical example of a semiconductor binary system formby two atomic species with radically different chemical bhavior. While Si prefers fourfold and tetrahedral coordintion with sp3 orbital hybridization, the C allotropy allows foa much greater diversity of local arrangements and any kof sp1, sp2, andsp3 hybrids. The physical properties of thresulting solid solution are therefore given by a compinterplay between such different behaviors. The possibiof tailoring its structural, mechanical, and electronic propties thus makes silicon carbide a material with very proming technological applications.1–5

The key questions relevant either to applications orbasic physics can be summarized as follows:~i! Can we pre-dict the degree of chemical order as a function of the alstoichiometry?~ii ! Which are the short-range structural fetures of such binary compound?~iii ! Can we characterize thtwo sublattices and bonds formed by Si and C? These plems have attracted quite a few experimental investigatiowhich, unfortunately, did not provide a unique picture.some cases, features of the recorded Raman scattering6 andx-ray diffraction data7 have been interpreted as fingerprinof homonuclear bonds, which, in turn, implies a descriptof the alloy as a chemically disordered structure. This cclusion is consistent with experimental informationvalence-band spectra.8 Conversely, Auger spectra9 have beenread as evidence of the existence of heteronuclear bondschemical order. This conclusion is consistent with a mrecent Raman study on microscopic distribution in Si12xCxalloys by Melendez-Liraet al.2 There it is shown that epitaxial growth most likely produces ordered silicon carbi

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structures. As for theory, a very accurateab initio molecular-dynamics investigation by Finocchiet al.10 has pointed outthat a-Si12xCx ~in x50.5 stoichiometric composition! can-not be classified as a chemically ordered alloy since a sizamount~40–45 %! of homonuclear bonds are containedthe structure. However, remarkably high short-range strtural order was found in the computer-generated samplethat it is incorrect to classify the compound as a randsystem. Besides, Finocchiet al.’s conclusion is conflictingwith a model-potential Monte Carlo simulation11 that pre-dicted a significant degree of chemical order. An attempreconcile the two models has been performed by Terso12

on the basis of thermodynamical arguments. There itbeen shown that both chemical preference for mixed boand difference in atomic size of Si and C favor the alterntion of atomic species in the bond network. Although tamorphous lattice weakens such an effect, still chemicaldering should prevail.

This scenario proves that there is a real need for a msystematic theoretical investigation on the structure achemistry ofa-Si12xCx . It is, in fact, important to establisha rationale for the variation of such properties as a functof the actual compositionx over the full range 0,x,1. Inorder to prevent any bias in the conclusions, this investition should be based on computer-generated samplesenough to provide a reliable statistical model for the madifferent local arrangements available to Si-Si, Si-C, aC-C pairs. To these aims we have undertaken a comptional study of a-Si12xCx based on classical moleculadynamics simulations. A description of our model potentias well as details on the simulation strategy, are given in SII, while the results of our investigations are presentedSec. III.

10 357 © 1998 The American Physical Society

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II. COMPUTATIONAL FRAMEWORK

In the present work we made use of the classical inatomic potential by Tersoff.13–15 Although model potentialsare less accurate and transferable thanab initio or semi-empirical tight-binding methods, they are invaluablestudying structural trends in complex systems when the oall computational workload could be remarkably high. Oinvestigation was indeed very computer intensive, sincestudied seven random alloys differing in stoichiometry. Easample was generated by an expensive procedure of queing from the melt.

Tersoff interatomic potential has been successfullytended to multicomponent systems containing Si, C, anGe by the introduction of a single new parameter for eapair of atomic species.15 Following a suggestion contained ithe original paper, we have further improved the potentialthe exploitation of an additional term. This contributiogives a greater flexibility to the model and allows for a mo

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accurate description of the effects of local environmentthe net interatomic forces. Details are given in the followin

A. The interaction potential

The configurational energy of a given assembly of atoi 51,2, . . . ,n is written as15

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n

Ei51

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Vi j , ~1!

where the pair potential Vi j represents a balancbetween a steric repulsionVR(r i j )5Ai j exp@2lij rij# anda covalent environment-dependent attractionVA(r i j )52Bi j exp@2mij rij#:

Vi j 5 f c~r i j !@VR~r i j !1bi j VA~r i j !#. ~2!

The cutoff function

f c~r i j !5H 1 for r i j ,Ri j

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enters either in the definition of the pair potentialVi j and inthe definition of the bond-order contributionz i j to theenvironment-dependent coefficientbi j , according to

bi j 5x i j ~11b iniz i j

ni !21/2ni ~4!

where

z i j 5 (kÞ i , j

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f c~r ik!g~u i jk !v i j ~5!

and

g~u i jk !511ci

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$di21@hi2cos~u i jk !#2%

. ~6!

Equation ~6! contains theu i jk angle formed by the pivoatom i and two its neighborsj ,k. In order to simplify thecalculation of interatomic forces, the termv i j was originallyset equal to 1. Consequently, with respect to the formulaof the Tersoff potential for elemental semiconductors,only new parameter is represented by the termx i j , whichdescribes the relative strength of heteronuclear bonds.

The multicomponent potential contains either atomicrameters (b,n,c,d,h) or pair terms (A,B,R,S,l,m). In theformer case, Tersoff’s prescription is simply to use therameters for a given elemental system even in the multicponent environment. On the other hand, the parameterspending on pairs of atoms are calculated by averaging othe atomic values. The full list of parameters is givelsewhere.15

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The straightforward application of the above potentprovides results conflicting withab initio data. In Fig. 1 wecompare the partial pair correlation functions for the amphous Si0.5C0.5 alloy as obtained by the above potential~fullline! and by Finocchiet al.10 through Car-Parrinello molecular dynamics~dashed line!. It is apparent that in the case oSi-C and Si-Si pairs even the main features~like the positionand relative intensity of the first and second coordinatpeaks! are poorely reproduced. As a consequence, the inmation on the chemical ordering of the alloy is different

FIG. 1. Partial pair correlation functions fora-Si0.5C0.5 com-puted by means of the original Tersoff potential~full line! and byab initio molecular dynamics~dashed line!.

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PRB 58 10 359STRUCTURE AND CHEMICAL ORDER OF BULK . . .

the two models. Following Tersoff,12 we can introduce aparametert5nC-C/nC-Si, defined as the ratio between thaverage number of C neighbors (nC-C) and the average number of Si neighbors (nC-Si) for C atoms. This is a conveniencharacterization of the chemical disorder of the alloy sint51 corresponds to a perfectly random system wherenC-C5nC-Si, while t50 describes the opposite situation of a pfectly ordered sample, like crystalline SiC. Theab initio in-vestigation by Finocchiet al. predicts a valuet50.86 and,therefore, provides a picture of chemical disordered forx50.5 alloy. Conversely, the present Tersoff-based simution ~similar to the previous study by Kelires11! has given avalue t50.5 which represents a remarkably different sitution of ~partial! chemical ordering.

We have therefore decided to improve the present mopotential by fully exploiting thev i j term. We have set15

v i j 5exp@l3ik

3 ~r i j 2r ik!3#, ~7!

where the additional parameterl3ikis now introduced. The

new expression forv i j ~which recovers the original valueonly if r i j 5r ik) provides a more refined description of thenvironment-dependent termbi j . We have also more accurately determined the values ofR andS for the Si-Si and C-Cpairs. This calibration was operated on the structural propties of the corresponding amorphous elemental semicontor. The determination ofl3 for Si-Si, Si-C, and C-C pairswas in turn performed on the partial pair correlation funtions and average coordination numbers ofa-Si0.5C0.5. Thealloy reference sample was that one described by Finoccetal. In Table I we summarize the set of new and recalibraparameters. The remaining potential parameters~not listed inTable I! are set identical to their original value.

We still need to clarify how to defineR and S for like-atom pairs in Si12xCx with xÞ0 or 1. While R was keptfixed, we found it convenient to varyS according tox. Byassuming a Vegard-like variation of the alloy bulk lattiparametera(x) with stoichiometryx,

a~x!5~12x!aSi1xaC, ~8!

we defined

Sa~x!5Ra1Fa~x!

aaG3

~Sa2Ra!, ~9!

where a5Si, C andRa ,Sa are given in Table I. Finally,within the spirit of Tersoff’s original work, theS parameterfor unlike-atom pairs is a given by a simple average:

SSi-C~x!5@SSi~x!SC~x!#1/2. ~10!

The effect of this modification is shown in Fig. 2 where tpartial correlation functions obtained with the modfied T

TABLE I. Set of new or recalibrated parameters used inpresent paper. All parameters express distances measured in

R S l3,Si-Si l3,Si-C l3,C-C

Si-Si 2.531 2.564 1.581Si-C 1.653C-C 1.654 2.370 1.470

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soff potential~full line! are compared toab initio data.10 Theagreement obtained for the Si-C and C-C pairs is much benow. The resulting chemical order parameter was foundbe t50.6. We remark that the modification of the inteatomic potential~aimed to reproduce anab initio model ofhighly disordered two-component lattice! confirmed the pre-vious findings on~partial! chemical ordering.

B. Simulation strategy

Our computer-generated amorphous Si12xCx sampleshave been studied by means of fixed-volume, cubic, andriodically repeated cells containing as many as 512 atoWe investigated the following stoichiometries:x50.125, 0.25, 0.375, 0.5, 0.625, 0.75, and 0.875. The acdensity of a given sample was determined according to~8!. The use of Vegard’s law was similarly adopted in oththeoretical investigations, although a little negative deviatfrom a purely linear trend has been recently described bsecond-order polynomial:16

a~x!5aSi22.4239x10.5705x2. ~11!

This result has been confirmed by a following experimeninvestigation by Bertiet al.17 We have carefully checked thvalidity of our assumption by directly comparing the strutural properties of three alloys (x50.375, 0.625, 0.750) athe density given by Eqs.~8! and ~11!, respectively. Theresults are shown in Fig. 3 where the total pair correlatfunctions are displayed. As expected, the overall alloy strture is nearly unaffected by the little bowing of the latticparameter versus C-content curve. This conclusion is sstantiated by the observation that even atomic coordinanumbers, as well as the chemical order parameter, werefected by less than 10%.

The amorphous structures have been obtained by queing from the melt. Well-equilibrated liquid samples havbeen first generated at each value ofx by heating up to20 000 K a cubic Si12xCx solid with random distribution ofSi and C atoms on the lattice sites and then coolingsystem down to 8000 K. The liquid samples have been c

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FIG. 2. Partial pair correlation functions fora-Si0.5C0.5 com-puted by means of the modified Tersoff potential~full line! and byab initio molecular dynamics~dashed line!.

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10 360 PRB 58MURA, COLOMBO, BERTONCINI, AND MULA

fully equilibrated at that temperature and then quencdown to 300 K by means of simulations as long as;105

time steps. Finally, the systems have been aged and obsefor 43104 and 1.53104 steps, respectively. Constantemperature simulations have been implemented by velorescaling. Throughout all the simulations we adoptedvelocity-Verlet algorithm and a time step as small as 0.5for an optimal integration of the equations of motion.

III. RESULTS AND DISCUSSION

The overall structural properties of thea-Si12xCx alloysat room temperature are summarized in Fig. 4 whereshow the total particle-particle correlation function. If wanalyze the short-range coordination~i.e., the structures fall-ing at distances below 2.5 Å!, we can observe a nice threemode behavior. The C-C, Si-C, and Si-Si peaks are inclearly detectable at distances 1.52, 1.85, and 2.33 Å, restively. Their relative intensities reflect the actual stoichioetry of the sample. Complementary information on the shrange structure is provided by the total bond-andistributiong(u). In Fig. 5 we report such a distribution fothe stoichiometric alloyx50.5. Theg(u) function has beencomputed by selecting different bond cutoffsr i j

bond for thethree possible pairs (i , j 5Si, C), according to the position othe first minimum of the corresponding partial pair corretion function ~see Fig. 6!. In particular, we adopted:r Si-Si

bond

52.56,r Si-Cbond52.30, andr C-C

bond51.80. The underlying networkappears as a slightly distorted tetrahedral lattice. A detaanalysis of the spatial distribution of the two constitueatomic species has proved that no segregation occursthe entire composition range here explored. The absencSi-rich or C-rich regions observed in our computer-generasamples is consistent with experimental evidence.6

Valuable information on the chemical ordering of theloys can be obtained from Fig. 6 and from the calculationthe partial coordination numbersnC-C, nC-Si, nSi-C, andnSi-Si, already introduced in Sec. II A. In Table II we repo

FIG. 3. Total pair correlation functions fora-Si0.625C0.375 ~top!,a-Si0.375C0.625 ~middle!, a-Si0.25C0.75 ~bottom! computed by meansof the modified Tersoff potential at density predicted by Eq.~8!~dashed line! and by Eq.~11! ~full line!.

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our computed coordination numbers. ThenC-Si value equalsthe nSi-C one only for the stoichiometric alloy, as expecteThere is, however, a disparity in the evolution ofnC-Si andnSi-C, which is reflected in the C and Si average coordintion. For any investigated content of carbon, the positionthe first C-C peak~Fig. 6, right! occurs at;1.5 Å, while

FIG. 4. Total pair correlation function fora-Si12xCx .

FIG. 5. Total bond-angle distribution functiong(u) fora-Si0.5C0.5.

.

PRB 58 10 361STRUCTURE AND CHEMICAL ORDER OF BULK . . .

FIG. 6. Partial pair correlation functions fora-Si12xCx . Left panel: C-C pairs; central panel: Si-C pairs; right panel: Si-Si pairs

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the average C coordination numbernCave5nC-C1nC-Si is al-

ways slightly smaller than four. In particular, forx50.5 wehave nC

ave53.7, to be compared with the valuenCave53.85

reported by Finocchiet al. Both features are in good agrement withab initio results10 for the stoichiometric alloy andstand for a prevalent diamondlike behavior of the carbsublattice. The~small! amount of threefold-coordinated Catoms diplay a nice graphiticlike angular coordination~herenot shown!.

The behavior of the network formed by Si atoms appemore complex. First of all, the first coordination peak of tSi-Si pair correlation function is remarkably broad~exceptfor very high content of Si!, indicating that Si atoms canaccommodate at rather different relative positions. Furthmore, as shown in Fig. 7, the partialg(u) formed by Si

TABLE II. Partial coordination numbers fora-Si12xCx .

Stoichiometry nC-C nC-Si nSi-C nSi-Si

x50.125 0.25 3.30 0.46 3.20x50.250 0.51 3.07 0.86 2.70x50.375 0.91 2.70 1.60 2.14x50.500 1.40 2.30 2.36 2.03x50.625 1.90 1.90 3.00 0.95x50.750 2.38 1.22 3.70 0.55x50.875 2.97 0.62 3.98 0.17

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triplets is characterized by three main peaks and no preence for tetrahedral coordination is shown. Finally, the avage coordination numbernSi

ave5nSi-C1nSi-Si can vary over awide range of values~see Table II!. These results indicatethat the Si network is very distorted and are again in exclent agreement withab inito data.10

We have further characterized the alloy ordering asfunction of x by means of the parametert ~see Sec. II A!.From Fig. 8~bottom!, it is clear how the degree of chemica

FIG. 7. Partial bond-angle distribution function for Si tripletsa-Si0.5C0.5.

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10 362 PRB 58MURA, COLOMBO, BERTONCINI, AND MULA

disorder increases as a function of the C content in the cpound. However, up to very high content of carbon, the alshows a substantial degree of chemical ordering. For contrationsx@0.7, the parametert is no longer useful since thvery small value ofnC-Si drives t to unphysically high val-ues. Interestingly enough, the maximum degree of chemdisorder (t51) is, however, reached for the slightly C-rica-Si0.325C0.625 compound.

Finally, we have also introduced a new parameterdSi

512nSi-Si4 , wherenSi-Si

4 is the partial coordination numbewhich counts the relative occurrence of Si atoms bondedother four Si atoms. Our aim was to monitor the distortionthe Si network at differentx values. WhendSi→0 we havemainly fourfold-coordinated Si sites, i.e., there is small dtortion similar to that observed ina-Si ~which is by far more

FIG. 8. Variation of the structural parameterdSi ~top! andchemical parametert ~bottom! as a function of Si content.

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tetrahedrally coordinated than the present Si sublattices.18,19!On the contrary, whendSi→1 we can observe a much largedistortion. There is, of course, an obvious limitation to thdefinition: it holds only for sufficiently high Si content, sthat we can think of a connected network of Si atoms~inother words, the parameter is meaningless forx@0.5). TheparameterdSi is plotted in Fig. 8~top!. Its evolution parallelsthe variation oft: the higher Si-network distortion is observed atx50.625 in correspondence to the higher chemidisorder. This result seems to indicate that the two quantiare closely related: the chemical randomness of the alloybe obtained by affecting the short-range tetrahedral coonation of the Si sublattice.

IV. CONCLUSIONS

We have presented a systematic computational investtion of the structural properties of amorphous silicon carballoys. Our results, based on a modified release of the Terpotential, provide a rationale to understand the variationsthe structure and chemistry of the alloy as a function ofcomposition. In particular, we have found that, for very hicontent of carbon, the chemical disorder of the alloy is linkto a sizable distortion of the Si sublattice.

ACKNOWLEDGMENTS

One of us~L.C.! acknowledges partial financial suppoby project PRA-CLASS of INFM. The Italian National Research Council~CNR! and Regione Autonoma Sardegna ain turn acknowledged for partial financial support by D.MR.B., and G.M.

v.

S.

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