Static and dynamical properties of SiC polytypes

PHYSICAL REVIEW B, VOLUME 65, 125202

Static and dynamical properties of SiC polytypes

E. Halac,1,* E. Burgos,1,2,† and H. Bonadeo1,‡

1Departamento de Fı´sica, Comisioń Nacional de Energıá Atomica, Avda. Gral. Paz 1499,(1650) San Martıń,Pcia. de Buenos Aires, Argentina

2CONICET, Rivadavia 1917, (1033)Buenos Aires, Argentina~Received 12 June 2001; published 11 March 2002!

Static and dynamical properties of 3C, 2H, 4H, 6H, and 15R SiC polytypes have been calculated usinga modification of Tersoff ’s potential for covalent multicomponent systems. Structures and vibrational frequen-cies are in good agreement with experimental results. For uniaxial polytypes, the dependence of the opticalfrequencies on propagation angle has been studied. The relative intensity of Raman bands has been calculatedusing a bond polarizability model. Isotope shifts, phonon eigenvectors and relative phase for longitudinal andtransversal modes of 3C SiC for the dispersion branches along the^111& direction have been determined andcompared with experimental results. The success of this calculation opens up the possibility of calculatingstatic and dynamical properties of disordered Si-C systems.

DOI: 10.1103/PhysRevB.65.125202 PACS number~s!: 63.20.2e, 34.20.Cf

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I. INTRODUCTION

In the past decade Tersoff has developed semiempiinteratomic potentials modeling the energetics of covalsystems.1–3 Though progress in more accurateab initiomethods or tight binding calculations, together with the dvelopment of faster computers would seem to make unnessary the use of semiempirical potentials in a great numof crystalline systems, these potentials are still today invaable for the treatment of complex systems~diamondlikeamorphous carbon films, polymerized fullerenes, etc.! or theaccomplishment of extended simulations.

In a previous work4 we have shown the unaccuracythese potentials, parametrized by Tersoff or Brenner,5 to ad-equately describe dynamical properties of carbon copounds. As this problem is intrinsic of the potential form, whave modified it by introducing a torsionlike term whictogether with an adequate reparametrization, allowed astantial improvement in the agreement between calculaand observed dynamical properties. Such a potential, wrefined parameters to fit static and dynamic propertiesgraphite, diamond, londsdalite, and some theoretical carstructures~single cubic, bcc, fcc! has shown to be suitable tdescribe with similar precision other carbon compoundsperimentally known: structures and vibrational propertiesC36, C60, and C70 molecules,4,6 as well as crystal structureof the three observed phases of one-dimensional~1D! and2D polymerized C60-based fullerites. More recently, thespotentials have allowed one to explore a great numbepossible structures of three-dimensional C60-based fullerites;three superhard phases were predicted and then confirmefirst-principle calculations.7

Among the systems whose complexity makes it difficto performab initio calculations, there is a growing interein the simulation of the deposition of amorphous carbfilms, or more recently amorphous carbon-silicon films,silicon substrates.8–11 The properties of Si-C polytypes havbeen widely studied using a variety of methods: on one haab initio calculations,12–15 and on the other force constanmethods such as the bond charge model16 or the linear-chain

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model.17 However, these calculations cannot be easilytended to amorphous or disordered systems. Semiempicalculations using Tersoff’s parameters for C andsystems18 do not reproduce well the dynamical propertiesthese systems. In this paper we propose a potential for clent C-Si systems, showing that it is suitable for the calcution of structures and dynamical properties of experimentaknown C-Si compounds, as is the case of the different crtalline SiC polytypes.

Silicon carbide is a wide band gap semiconductor andcurrently being explored as an attractive material for higtemperature high-power electronics and optical sensors inultraviolet region. There is a large number of crystallipolytypes for the SiC crystal. Their identification durincrystal growth and further characterization is important bcause the band gap energy and electrical properties areferent for different polytypes.

Their crystal structures are classified into three groucubic (C), hexagonal (H), and rhombohedral (R). The cubic3C polytype is the simplest one, having a zinc blende strture, space groupTd

2 . Hexagonal and rhombohedral polytypes are uniaxial, with different stacking sequencesdouble atomic planes of Si-C along thec direction. They arerepresented by the number of Si-C double layers in thecell and the letterH or R specifies the lattice type. The hexagonal polytypes (2H, 4H, 6H, 8H, etc.! have space groupP63mc ~factor group C6v); the rhombohedral polytype~15R, 21R, etc.! are described by space groupR3m ~factorgroupC3v). The primitive unit cell ofnH or 3nR polytypescontainsn formula units~Si-C!. Thec axis of these polytypescorresponds to the111& direction of the cubic structurethey can be regarded as superlattices.

The vibrational properties of 3C SiC have been extensively studied. Recently, high quality 4H and 6H SiC wafershave been grown, allowing Raman investigation of thepolytypes. The phonon studies of these and higher polytyhave demonstrated that polytype identification by Ramscattering is possible.17,19–22

The dispersion curves of the phonon modes propagaalong thec direction in higher polytypes can be approx

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E. HALAC, E. BURGOS, AND H. BONADEO PHYSICAL REVIEW B65 125202

mated by folded dispersion curves in the basic Brillouin zoof 3C-SiC in the ^111& direction. The unit cell of 3C-SiCcontains one formula unit and there should be one tridegenerate optical mode, corresponding to the oppositeplacement of the two atoms in the primitive cell, as in othcrystals having the diamond structure~diamond, silicon!.However, in contrast to usual homopolar covalent semicductors, cubic SiC shows ‘‘ionicity’’ due to the heteropolbond asymmetry, in such a way that the phonon propagaintroduces a preferential direction that splits the optimodes in one longitudinal~LO! and one transversal doubldegenerate~TO! mode.

In uniaxial polytypes, the crystal anisotropy should girise to axial~A! nondegenerate and planar~P! doubly degen-erate modes. In addition, the phonon propagation direcintroduces an additional asymmetry, due to the bond ionicwhich depends on the propagation angleu between the wavevectork and the unique crystal axis. We can distinguish thcases:~i! u50, the axial modes are longitudinal (AL) andthe planar modes are doubly degenerate transversal (PT);~ii ! u5p/2, the axial modes are transversal (AT), and thedegeneration of planar modes can be broken becausesplit into longitudinal (PL) and transversal components; a~iii ! arbitraryu, planar modes with polarization perpendiclar to the (k,c) plane are transversal, and the remainimodes could mix axial and planar displacements. In the lacase, longitudinal and transversal frequencies depend ouaccording to Loudon’s formula23

nL25nAL

2 cos2u1nPL2 sin2u, ~1a!

nT25nAT

2 sin2u1nPT2 cos2u. ~1b!

For 3C-SiC, the wave vector at the Brillouin zone bounary along the 111& direction (L point! is ukLu5p/d, wheredis the distance between Si~or C! planes of successive bilayers. In the primitive unit cell ofnH and 3nR polytypes~con-taining n Si-C formula units!, thec axis isnd. Accordingly,the Brillouin zone in theG-L direction is reduced to 1/n ofthe basic~3C! Brillouin zone, and the dispersion curvesthe phonon modes propagating along thec direction inhigher polytypes are approximated by folded disperscurves in the basic Brillouin zone. This folding results innumber of new phonon modes at theG point (k50) forhigher polytypes, which correspond to phonon modes insor at the edge of the basic Brillouin zone.

For nH and 3nR polytypes, a folded mode correspondsa phonon mode having wave vectork5(2m/n)kL (m is aninteger less or equal ton/2) along the basic Brillouin zoneTherefore the optical modes of uniaxial polytypes canclassified with a reduced wave vectorx5k/kL52m/n. In allpolytypes, including 3C-SiC, there are only six modes atx50, three of them are acoustic translational modes andother three are optical vibrational modes with all Si-C unoscillating in phase.

II. THE POTENTIALS

The structures and phonon frequencies of SiC polytyhave been calculated using Tersoff ’s potential for coval

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multicomponent systems3 with the modifications introducedin Ref. 4. These consist essentially in the addition of a tsional term and reparametrization, in order to obtain accuresults for the dynamical properties of carbon systems.

Potential parameters for C and Si have already been inpendently determined by fitting a single parametrized pottial to the respective elemental data. A preliminary calcution showed that the potential proposed by Tersoff for siliccompounds3 gives a reasonable dynamical description of tSi crystal@see Fig. 1~bottom!#. Therefore, Si parameters havbeen taken without modification, while for the C parametwe have used the modified potential of Ref. 4@see Fig.1~top!#. The mixture parameters for C-Si bonds have beobtained by averaging both sets, following the procedproposed by Tersoff for heteropolar bonds;3 arithmetic aver-ages were taken for the torsional parameter. The additioparameter, which strengthens or weakens the heteropbond, was initially taken as a fitting parameter; the optimvalue happened to be very close to that proposed by Ter

FIG. 1. Calculated dispersion curves for diamond~top! and sili-con ~bottom! along G-X and G-L directions. Experimental data~solid circles! from Refs. 31 and 32, respectively.

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STATIC AND DYNAMICAL PROPERTIES OF SiC POLYTYPES PHYSICAL REVIEW B65 125202

Therefore, this parameter was taken from Tersoff’s potenwithout modification.

For SiC, a fairly covalent crystal, the static charge transis small, and its effects are implicitly taken into accountthe parametrization of the semiempirical potential. Howevthis is a short-range potential and cannot account fordynamic ionicity, exhibited in the splitting of the longitudinaand transverse optical modes at the zone center. In orddescribe this effect, charges6q are associated to C and Snuclei following a Born effective charge model. This modleaves invariant the trace of the dynamic matrix in suchway that the triply degenerate frequency calculated forcubic SiC (3C phase! with the semiempirical potential muslie between the observed longitudinal and transversal moOn the other hand, it is well known that for binary systemwith cubic symmetry13

nL22nT

25q2/pVm, ~2!

TABLE I. Lattice parameters~in Å) and cohesion energies.

Calculated Experimental

3C SiC a(cub) 4.3709 4.360a

a(hex) 3.0907 3.083dC-Si 1.8927 1.8879

E (eV/at) 26.6135 26.341b

2H SiC a 3.0906 3.061a

c 5.0475 5.034a

dC-Si 1.8928 1.8877E (eV/at) 26.6135

4H SiC a 3.0906 3.073a

c 10.0938 10.052a

dC-Si 1.8926 1.8847E (eV/at) 26.6135

6H SiC a 3.0906 3.081a

c 15.1405 15.120a

dC-Si 1.8926 1.8900E (eV/at) 26.6136

15R SiC a 3.0906 3.073c

c 37.8512 37.700dC-Si 1.8926 1.885

E (eV/at) 26.6117b-Si a(cub) 5.4320 5.429d

a(hex) 3.8410 3.8389dSi-Si 2.3521 2.3508

E (eV/at) 24.6296 24.63e

C ~diamond! a(cub) 3.5218 3.5657a(hex) 2.4903 2.5213

dC-C 1.5250 1.544f

E (eV/at) 28.555 27.37g

aReference 24.bReference 25.cReference 26.dReference 27.eReference 28.fReference 29.gReferences 28,30.

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where nL and nT are longitudinal and transversal frequecies,m is the reduced mass, andV the primitive cell volume.Taking experimental values of the volume and the longitunal and transversal frequencies, we found that the magniof the splitting is well approximated byq5e.

With this potential we have calculated dynamical propties for 3C, 2H, 4H, 6H, and 15R SiC polytypes. Foruniaxial polytypes, we have studied the dependence ofoptical frequencies on propagation angleu. We have alsostudied the relative intensity of Raman bands using a bpolarizability model. Assuming all Si-C bonds to be equivlent, the relative intensities depend only on the eigenvecof the dynamical matrix; therefore this calculation could prvide an additional test for the interaction potential.

III. RESULTS

Calculated and experimental lattice parameters and cosion energies per atom for diamond, silicon and five Spolytypes are given in Table I. For all SiC polytypes, calclated cell parameters are slightly greater than observed obut the disagreement is less than 1%. As expected, thehesion energy is practically the same for all polytypes athe c parameter is proportional to the number of bilayersthe primitive cell.

Figure 1 shows calculated dispersion curves for diamoand silicon along theG-L andG-X directions together withexperimental results.31,32 Diamond curves were calculate

FIG. 2. Calculated dispersion curves withq5e for natural 3CSiC along theG-L direction. Calculated optical frequencies for Sipolytypes~open circles! and Raman experimental data from Ref. 2~solid circles!.

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using the modified potential of Ref. 4 and silicon curvwere calculated using the potential proposed by Ters3

without modification. It can be seen that, for both crystacalculated curves are a good fit to experimental pointsoptical and longitudinal acoustic branches. Only the callated transversal acoustic branch exhibits a greater dispethan the observed one, especially in the case of silicwhere experimental frequencies present a plateau that issent in calculated curves.

Figure 2 shows calculated dispersion curves for cubicalong the^111& direction and optical frequencies for highpolytypes. These results are compared with observed Rafrequencies.20 It can be seen that the reparametrization ofpotential and the inclusion of the charges improvesagreement with experiment with respect to earlcalculations.18 As in Fig. 1, the largest difference is observbetween the calculated transversal acoustic branch andcorresponding observed values for optical frequencieshigher polytypes.33 For 4H, 6H, and 15R polytypes, withx50 or 1, bands are grouped into doublets. For modes

FIG. 3. Calculated dispersion curves for natural 3C SiC alongG-L direction withq50 andq5e.

TABLE II. Calculated and observed~Ref. 19! dependence of theoptical frequencies on propagation angle (u) for 6H SiC.

u590° u545° u50°

L E1L 988.97~970! (A11E1)L 988.42~967! A1L 987.86~964!T1 E1T 835.13~797! E1T 835.13~797! E1T 835.13~797!T2 A1T 833.79~788! (A11E1)T 834.46~792! E1T 835.13~797!

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longing to 3C acoustic branches, all Raman active doublhave been observed with splittings ranging from 110 cm21, while for those belonging to optical brancheonly one doublet has been observed. Calculations are

FIG. 4. Calculated and experimental~Ref. 20! relative Ramanintensities for SiC polytypes.

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agreement with these results, showing greater splittingsacoustic than for optical modes, even though calculatedues are smaller than observed ones in all cases.

Dispersion curves for cubic SiC along the^111& directionhave been also calculated assuming the dynamical charq50. This would be the simplest calculation to deal wamorphous or complex SixC12x systems, where C~Si! atomsmay have a variety of coordinations. Figure 3 compathese results with those previously shown forq5e and ex-perimental values. Of course,q50 calculations cannot account for the observed LO-TO splitting at theG point. How-ever, the results show that the effect associated to dynamcharges is only relevant for optical branches and decrewith k. The LA branch is nearlyq independent and in particular the TA branch withq50 fits better experimental values than the one calculated withq5e.

FIG. 5. ~a! Calculated isotope shifts~solid lines! and experimen-tal data~Ref. 34! ~circles! for 3C longitudinal modes.~b! Calcu-lated eigenvectors~solid lines! and relative phase~dotted line!. Ex-perimental data for eigenvectors~solid circles! and relative phase~open circles!.

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As was stated before, we have studied the dependencthe optical frequencies on propagation angle (u) for uniaxialsystems. This effect has been observed for the 4H, 6H, and15R polytypes.19 In all uniaxial polytypes studied we havfound that only the longitudinal mode and one transvermode corresponding tox50 shift appreciably with the direction of k. We find that the shifts increase with the hexagoncharacter of the polytype~cubic →6H→15R→4H→2H),in agreement with experiment. However, our calculashifts are about 5 times smaller than experimental onesthough they also are small with respect to the unshiftedquencies. Results for 6H SiC are summarized in Table II. Au changes from 0° to 90°, the frequency of the longitudinx50 mode increases by 6 cm21, while for one transversax50 mode the frequency decreases 9 cm21.

Calculated and experimental Raman intensities for fi

FIG. 6. ~a! Calculated isotope shifts~solid lines! and experimen-tal data~Ref. 34! ~circles! for 3C transversal modes.~b! Calculatedeigenvectors~solid lines! and relative phase~dotted line!. Experi-mental data for eigenvectors~solid circles! and relative phase~opencircles!.

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SiC polytypes are shown in Fig. 4. The overall agreemengood. The major disagreement is observed for 4H and 6Hpolytypes in the low frequency region: calculations predrelatively intense bands near 200 cm21 while experimentalones are very weak. As stated before, there are no additiadjustable parameters in this calculation, and the relativetensities depend only on the eigenvectors given by the mopotential.

The phonon eigenvectors~ion displacements of Si and Cand their relative phase! of 3C SiC for the dispersionbranches along the111& direction have been determinemeasuring Raman spectra of several SiC polytypes34 (3C,6H, and 15R) made from natural silicon and30Si. Thesemeasurements provide an additional test for our potentsince different lattice-dynamical models, although reproduing successfully experimental phonon frequencies, can digreatly in calculated eigenvectors.

Figures 5 and 6 show isotope shifts, phonon eigenvec(eSi ,eC) and relative phase (f) for longitudinal and trans-versal modes, respectively, with the convention

~eSi ,eC!5~mSi1/2uSi ,mC

1/2uC!,

eC5ue~C!uexp@ i ~f1l!#,

eSi5ue~Si!uexp@ il#,

whereuSi , anduC are atomic displacements. The eigenvetors of the optic modes can be obtained by the replacemf→f1p and ue(C)u↔ue(Si)u. For longitudinal modes, itcan be seen that the isotope shift of the acoustic modecreases withk and that of the optical mode decreases.similar behavior is observed for the Si eigenvector, aspected@Fig. 5~b!#. For transversal modes the change of tisotope shift withk @Fig. 6~a!# is appreciably smaller than folongitudinal modes, associated with the fact that C anddisplacement amplitudes are almost constant@see Fig. 6~b!#.

Calculated eigenvectors are in all cases in good agreemwith experimental ones.34 Also calculated isotope shifts folongitudinal and transversal modes are in good qualitatagreement with observed values; only for the transver

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acoustic mode there are appreciable differences~about3 cm21). The calculated relative phase for longitudinmodes shows some differences with experimental resultsprevious calculations,12,34 but it is qualitatively right. Fortransversal modes calculated relative phases reproducewell experimental values and previous calculations.

IV. CONCLUSIONS

We have studied several SiC crystalline polytypes, findithat a potential as that proposed by Tersoff, with a suitareparametrization and modification, accounts well for a wivariety of physical properties: structures, cohesion energphonon frequencies, and their dependence on propagaangle, Raman intensity profiles, isotopic frequency shifts acomposition of vibrational modes~amplitude and phase!.

Our calculations show an overall qualitatively gooagreement with experimental values. Although in some cathere are quantitative disagreements, as is the case offrequency dependence on propagation angle, calculaticorrectly predict the sign and the ordering of the shifts. Aother calculated properties are also quantitatively well repduced.

As was pointed out before, the static charge transferSiC polytypes is small, and is taken into account in the prametrization of the semiempirical potential, in such a wthat structures have been calculated assuming no additiocharges on the atoms (q50). On the other hand, calculatefrequencies withq50 deviate less than 10% from expermental values. It may be reasonably hoped that, given thfeatures, this potential could be successfully applied tostudy of more complex systems, as is the case of smallclusters or amorphous materials where it would be difficto estimate a value for the dynamical charges.

ACKNOWLEDGMENTS

E.B. has been partially supported by CONICET of Argetina, Grant No. PICT-PMT0051; E.H., E.B. and H.B. werpartially supported by SEPCYT, Grant No. PICT 12-0697

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.

,

v. B

ev.

ev.

ick,

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]. Tersoff, Phys. Rev. Lett.61, 2879~1988!.2J. Tersoff, Phys. Rev. B37, 6991~1988!.3J. Tersoff, Phys. Rev. B39, 5566~1989!.4E. Burgos, E. Halac, and H. Bonadeo, Chem. Phys. Lett.298, 273

~1998!.5D. W. Brenner, inAtomic Scale Calculations in Materials Sci

ence, edited by J. Tersoffet al., MRS Symp. Proc. No. 141~Materials Research Society, Pittsburgh, 1989!, p. 59.

6E. Halac, E. Burgos, and H. Bonadeo, Chem. Phys. Lett.299, 64~1999!.

7E. Burgos, E. Halac, R. Weht, H. Bonadeo, E. Artacho, andOrdejon, Phys. Rev. Lett.85, 2328~2000!.

8G. Galli, Comput. Mater. Sci.12, 242 ~1998!.9Chu-Chun Fu and M. Weissman, Phys. Rev. B60, 2762~1999!.

.

10S.M. Bennington, N. Kitamura, M.G. Cain, M.H. Lewis, and MArai, Physica B263-264, 632 ~1999!.

11H.U. Jager and K. Albe, J. Appl. Phys.88, 1129~2000!.12K. Karch, P. Pavone, W. Windl, O. Schu¨tt, and D. Strauch, Phys

Rev. B50, 17 054~1994!.13G. Wellenhofer, K. Karch, P. Pavone, U. Ro¨ssler, and D. Strauch

Phys. Rev. B53, 6071~1996!.14K. Karch, F. Bechstedt, P. Pavone, and D. Strauch, Phys. Re

53, 13 400~1996!.15S. Limpijumnong and W. Lambrecht, Phys. Rev. B57, 12 017

~1998!.16M. Hofmann, A. Zywietz, K. Karch, and F. Bechstedt, Phys. R

B 50, 13 401~1994!.17S. Nakashima, H. Harima, T. Tomita, and T. Suemoto, Phys. R

B 62, 16 605~2000!.18L.J. Porter, Ju Li, and S. Yip, J. Nucl. Mater.246, 53 ~1997!.19D.W. Feldman, J.H. Parker, Jr., W.J. Choyke, and L. Patr

2-6

on

d

l.

ly,mhe

s.

ds

m

a,


Phys. Rev.170, 698 ~1968!; 173, 787 ~1968!.20S. Nakashima and H. Harima, Phys. Status Solidi A162, 39

~1997!.21J.C. Burton, L. Sun, F.H. Long, Z.C. Feng, and I.T. Fergus

Phys. Rev. B59, 7282~1999!.22T. Tomita, S. Saito, M. Baba, M. Hundhausen, T. Suemoto, an

Nakashima, Phys. Rev. B62, 12 896~2000!.23R. Loudon, Adv. Phys.13, 423 ~1964!.24Physics of Group IV Elements and III-V Compounds, edited by O.

Madelunget al., Landolt-Bornstein, New Series, Goup II, Vo17a ~Springer, Berlin, 1982!.

25D. D. Wagman, W. H. Evans, V. B. Parker, I. Halow, S. M. Baiand R. H. Schumm,Selected Values of Chemical Thermodynaics Properties, Tables, for the First Thirty-Four Elements in tStandard Order of Arrangement, Natl. Bur. of Stand. Tech. NoteNo. 270-3~U.S. GPO, Washington, D.C., 1968!.

12520

,

S.

-

26N.W. Tibault, Am. Mineral.29, 327 ~1944!.27Natl. Bur. Stand.~U.S.! Monogr. 25, 13, 35~1976!; Swanson,

Fuyat, Natl. Bur. Stand.~U.S.! Circ. 539, 11, 6~1953!.28M.T. Yin and M.L. Cohen, Phys. Rev. B26, 5668~1982!.29L. Bragg, The Crystalline State~Bell and Sons, London, 1966!,

Vol. I.30M.T. Yin, Phys. Rev. B30, 1773~1984!.31J.L. Warren, J.L. Yarnell, G. Dolling, and R.A. Cowley, Phy

Rev.158, 805 ~1967!.32G. Dolling, Inelastic Scattering of Neutrons in Solids and Liqui

~IAEA, Vienna, 1963!, Vol. 1, p. 37.33Numerical results are available upon request fro

[email protected]. Widulle, T. Ruf, O. Buresch, A. Debernardi, and M. Cardon

Phys. Rev. Lett.82, 3089~1999!.

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Static and dynamical properties of SiC polytypes