Lattice dynamical properties of , , and compounds

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ndPACS:63.20.D-65.40.Ba71.15.NcKeywords:D. Heat capacityD. PhononsD. Thermodynamic properties1. IntroductionUltra-hard materials are used in many applications, from cut-ting and polishing tools to wear-resistant coatings [1]. Unfortu-nately, almost all ultra-hardmaterials (diamond, cubic BN, etc.) areexpensive because they either occur naturally in limited suppliesor have to be made at high pressure synthetically [2]. Therefore,intense research efforts have been carried out to design ultra-hardmaterials [1,2]. The transitionmetal diborides exhibit unique prop-erties such as high melting points, high hardness values, and ex-cellent oxidation resistance [17]. Therefore, they are promisingmaterials for new heat-resistant, corrosion-resistant, and wear-resistant alloys and coatings [8].ScB2, TiB2, and VB2 are members of the transition metaldiborides and they have the hexagonal AlB2 structure with thespace group P6/mmm (space number 191). There are three atomsin the unit cell, all of them on the special positions: X: (0, 0, 0), B:(1/3, 2/3, 1/2) and B: (2/3, 1/3, 1/2).A number of theoretical [923] and experimental [24,25]studies exist in the literature dealing with structural, elastic, andelectronic properties of ScB2, TiB2, and VB2. Specifically, Shein andIvanovskii [9] reported the structural and elastic properties usingthe full-potential linearized augmented plane-wave (FP-LAPW) Corresponding author. Tel.: +90 382 2801232; fax: +90 382 2801246.E-mail address: edeligoz@yahoo.com (E. Deligoz).method with the generalized gradient approximation (GGA) forScB2, TiB2, and VB2. Oguchi [10] studied the systematic trendin lattice constants and heat of formation for these compounds.Vajeeston et al. [11] investigated electronic structure and groundstate properties of these transition metal diborides using theself-consistent tight-binding linear muffin-tin orbital method (TB-LMTO). The elastic properties, electronic structure, electroniccharge distribution and equation of states of titanium diboridewere studied by Milman and Warren [12], Perottoni et al. [13],and Camp et al. [14] by using the first-principles methods.Peng et al. [15] investigated the thermodynamical propertiesof TiB2 using ab initio plane-wave pseudopotential method.Ravindran et al. [16] studied electronic structure using full-potential linearized augmented plane-wave (FP-LAPW)method forTiB2. Wang et al. [17] predicted the elastic and thermodynamicalcharacteristics of TiB2 crystal through the method of densityfunctional theory within the generalized gradient approximation(GGA). Kaur et al. [18] applied the Rigid Ion Model (RIM) to studythe cohesive and thermal properties of TiB2. Munro [19] examinedthe physical, mechanical, and thermal properties of polycrystallineTiB2 and showed that these properties are significantly related tothe density and grain size of the used specimens.Structural, mechanical, elastic behaviour, electronic chargedensity and band structure of VB2 were studied by Mahmudet al. [20] using the ab initio density functional method. Zhouet al. [21] performed first-principles study on the bonding nature,elastic property and hardness of VB2. The band structure and FermiSolid State CommunicatioContents lists availaSolid State Comjournal homepage: wwwLattice dynamical properties of ScB2, TiB2E. Deligoz a,, K. Colakoglu b, Y.O. Ciftci ba Aksaray University, Department of Physics, 68100, Aksaray, Turkeyb Gazi University, Department of Physics, Teknikokullar, 06500, Ankara, Turkeya r t i c l e i n f oArticle history:Received 28 March 2009Received in revised form26 May 2009Accepted 2 July 2009 by S. ScandoloAvailable online 15 July 2009a b s t r a c tWe have investigated the stfirst-principles total energythe bond lengths of XB anstates, and some thermodyntemperature-dependent behgood agreement with the ava0038-1098/$ see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2009.07.010s 149 (2009) 18431848ble at ScienceDirectmunications.elsevier.com/locate/ssc, and VB2 compoundsructural and lattice dynamical properties of XB2(X = Sc,V, Ti) by usingcalculations. Specifically, the lattice parameters (a, c) of the stable phase,BB atoms, phonon dispersion curves and the corresponding density ofamical quantities such as internal energy, entropy, heat capacity, and theiraviours, are presented. The obtained results for structural parameters are inilable experimental and other theoretical studies. 2009 Elsevier Ltd. All rights reserved.uiTheorym (GGA) 2.992 3.028 1.01 2.297 1.728(LDA) 2.953 2.965 1.00 2.260 1.705Theoryn (GGA FP-LAPW) 3.007 3.048 1.01Theoryk 2.979 2.995 1.00 2.297 1.772Experimentald 2.998 3.005 1.00Experimentalo 2.997 3.056 1.02a Ref. [9].b Ref. [10].c Ref. [11].d Ref. [24].e Ref. [12].f Ref. [13].g Ref. [14].h Ref. [15].i Ref. [16].j Ref. [17].k Ref. [23].l Ref. [20].m Ref. [21].n Ref. [22].o Ref. [25].surface parameters were studied by Shein and Ivanovskii [22] byusing the self-consistent full-potential linearizedmuffin-tin orbital(FLMTO) method for VB2. Experimentally, the phonon density ofstates of TiB2 and VB2 were measured by Heid et al. [23] by usingthe technique of inelastic neutron scattering.In the present paper, we aim to investigate the phonondispersion relations and thermodynamical properties of ScB2, TiB2,and VB2 compounds in detail and interpret the salient results ofour calculations. The method of calculation is given in Section 2;the results are discussed in Section 3. Finally, the summary andconclusion are given in Section 4.2. Method of calculationOur calculations are based on the density functional formalismand generalized gradient approximation (GGA).Here we use thePerdewBurkeErnzerhof (PBE) functional [26] for the exchange-correlation energy as is implemented in the SIESTA code [27,28].This code calculates the total energies and atomic forces using alinear combination of atomic orbitals as the basis set. The basisset is based on the finite range pseudoatomic orbitals (PAOs) ofthe SankeyNiklewsky type [29], generalized to include multiple-zeta decays. The interactions between electrons and core ionsAtomic pseudopotentials are generated separately for atomsSc, Ti, V, and B by using the 4s2 3d1, 4s2 3d2, 4s2 3d3 and2s2 2p1 atomic configurations, respectively. The cut-off radii for thepresent atomic pseudopotentials are taken as s: 2.84 au, p: 2.84 au,d: 2.54 au f: 2.54 au of Hf, s: 2.71 au p: 2.71 au d: 2.48 au f: 2.48 auof Ti, s: 2.66 au p: 2.66 au d: 2.50 au f: 2.50 au of V and 1.71 au forthe s, p, d and f channels of B.Atoms are allowed to relax until atomic forces were less than0.005 eV 1. The cut-off energy are taken to be 400 Ryd. 792,1014, and 1183 k-points were found to be adequate to obtainingthe total energy to an accuracy of 1 meV/atom for ScB2, TiB2, andVB2, respectively.3. Results and discussion3.1. Structural propertiesFirstly, we optimize the lattice constants to obtain a relaxedstructure. The initial lattice parameters are taken from previoustheoretical and experimental values as a starting point forgeometry optimization. The calculated lattice parameters andbond distances are listed in Table 1, along with the available1844 E. Deligoz et al. / Solid State CommTable 1Calculated equilibrium lattice parameters (a, c in ), the XB and BB bond distance (Reference aScB2 Present (GGA PP) 3.117Theorya (GGA FP-LAPW) 3.114Theoryb (LDA FP-LAPW) 3.114Theoryc (LDA TB-LMTO) 3.154Experimentald 3.148TiB2 Present (GGA PP) 3.024Theorya (GGA FP-LAPW) 3.006Theoryb (LDA FP-LAPW) 3.005Theoryc (LDA TB-LMTO) 3.070Theorye (GGA USP) 3.029(LDA USP) 2.990Theoryf (HartreeFock) 3.027Theoryg (GGA) 2.993Theoryh (GGA PP) 3.015Theoryi (LDA TB-LMTO) 3.070Theoryj (GGA PP) 3.030Theoryk 2.998Experimentald 3.038VB2 Present (GGA PP) 3.002Theorya (GGA FP-LAPW) 2.970Theoryb (LDA FP-LAPW) 2.969Theoryc (LDA TB-LMTO) 2.983Theoryl (LCAO) 3.008are simulated with the separable TroullierMartins [30] norm-conserving pseudopotentials.nications 149 (2009) 18431848n ) along with the available experimental and theoretical values for XB2 .c c/a XB BB3.407 1.09 2.478 1.7993.512 1.133.465 1.113.523 1.11 2.533 1.8213.517 1.123.154 1.04 2.352 1.7453.212 1.073.186 1.063.262 1.06 2.409 1.4433.219 1.063.152 1.053.240 1.073.147 1.053.222 1.073.262 1.063.232 1.063.188 1.07 2.409 1.7733.239 1.072.999 0.99 2.291 1.7333.029 1.022.989 1.013.047 1.01 2.297 1.7223.068 1.02 2.317 1.736experimental and theoretical data for comparison. It is seen thatthe present lattice parameter (a) is in good agreement with theuTotalBVVB2 K A H L AM K A H L AM50FREQUENCY [THz]253020151050FREQUENCY [THz]6 8 10 12 14 16 18 20 224 2624 2810 15 20FREQUENCY [THz]5 25.04.020.20.22.18.16.14.12.10.08.06.04.02AMPLITUDE [1/Hz].240Fig. 1. Calculated phonon dispersions, total and partial density of states for ScB2 , TiB2 , and VB2 .theoretical experimental values, but lattice parameter (c) is lowerthan the experimental and other theoretical values [911] by% 23 for these compounds. This small difference may stemfrom the different density functional based electronic structuremethods. Also, it is clear from this table that, as we go along thesame row of the periodic table from Sc to V, the lattice constantsslightly decrease in accordance with the mass of the transitionmetal atoms.The bond lengths of ScB2, TiB2, and VB2 are consistent with thetheoretical data. The bond lengths decrease on going from ScB2 toVB2. In fact, it is known that the short bond length strongly affectssome physical properties, such as melting point, elastic constants,and hardness [31].3.2. Phonon dispersion curvesMany physical properties of solids depend on their phononproperties such as specific heat, thermal expansion, heat conduc-tion, and electronphonon coupling. Though several works werereported, as stated above, on the structural and electronic prop-erties of these compounds, neither experimental nor theoreticalstudies are exist in literature on the lattice dynamics of these com-report the results of the lattice dynamic7al calculations of thesecompounds. The present phonon dispersion curves without LO/TOsplitting and other related quantities are calculated by using thePHONON software [32]. This code is compatible with SIESTA anduses the Direct Method [33] and the HellmannFeynman forceson atoms for generating the phonon dispersion and the density ofstates (DOS). Its theoretical and applicational details can be foundin the PHONON manual and references therein.The optimized rhombohedral supercell with 36 atoms andbasis vectors are given as as = 2a + c , bs = 2b + c , cs =2a 2b + c , where a, b, c are the primitive translation vectorsof the hexagonal unit cell. [34]. The symmetry of the selectedrhombohedral supercell breaks the two fold symmetry axis, andlowers the P6/mmm space group to R3m. Here, the displacementamplitudes are taken as 0.03 , and the positive and negativeatomic displacements along x, y, and z axes are taken into account.TheHellmannFeynman forces acting on the atoms in the supercellare evaluated to construct the force constants and dynamicalmatrix. Parlinski [34] used the samemethod and PHONON code forobtaining the correct phonon dispersion relations of the hexagonalMgB2.The calculated phonondispersion curves and the correspondingE. Deligoz et al. / Solid State Comm2520ScB2TiB215105 K A H LM0FREQUENCY [THz]FREQUENCY [THz]25201510pounds, except for density of states (DOS) of TiB2 and VB2 [23],to compare with the present results. Therefore, it is important tonications 149 (2009) 18431848 1845TotalBScTotalBTiA 6 8 10 12 14 16 18 20 22FREQUENCY [THz]4 24.20.18.16.14.12.10.08.06.04.02AMPLITUDE [1/Hz]0AMPLITUDE [1/Hz].20.22.18.16.14.12.10.08.06one-phonon DOS for XB2 along the high-symmetry directions areillustrated in Fig. 1. The unit cell of XB2 contains three atoms,u1846 E. Deligoz et al. / Solid State Commv=13.47 THz v=13.56 THzv=18.64 THzv=24.11 THzFig. 2. E1u , A2u , B1g , and E2g phonon modes for ScB2 at the 0 point.which give rise to a total of 9 phonon branches. It is seen fromFig. 1 that ScB2, TiB2, and VB2 have very similar dispersion curvesand the corresponding density of states. It is well known that themass difference between anions and cations strongly affects themaximumandminimumvalues of the acoustic and optic branches,and a clear gap is formed between them as in Fig. 1. Here, whilethemaximumvalue of the phonon frequencies for optical branchesfollows the order from larger to smaller as VB2TiB2ScB2, themagnitude of gap between the acoustic and optic branches followsthe order of ScB2 TiB2 VB2. Interesting features of opticalphonon modes are observed at 0 point. Optical phonon branchesare nearly flat at the 0 point. E2g mode (The highest frequencyoptical branch) which evolve from the doubly degenerate at the0 point for these compounds. Along the AL and 0M directionsthese branches are not degenerate.Besidesmany similarities, one can observe some distinctive dif-ferences on the phonon dispersion curves of these compounds.Forexample, for TiB2 and VB2, a clear gap is formed between opticalbranches at 0 point, but this situation is not seen for ScB2.The calculated 0 point frequencies of that ScB2, TiB2, and VB2are given Table 2 along with the available experimental data forcomparison. The 0 point frequencies of the E1u, A2u, and B1g are, ingeneral, favorable agreement with the experimental [23] ones. E2gis higher than experimental [23] values by 5% for VB2. The atomicmotions of gamma-point normalmodes are displayed in Fig. 2 onlyfor ScB2 to save space in journal. Graphical package Molekel [35,36] is used to analyze the individual normal modes. E1u and A2uphonon modes include atomic vibrations from both X (X = Sc, V)and B atoms. The situation is slightly different for TiB2. Here E1uphonon modes contain the vibrations from B atom; the A2u modescontain the vibrations from both Ti and B atom. B1g modes includethe atomic vibrations from B atoms for X (X=Sc, V) and includethe atomic vibrations from both Ti and B atoms for TiB2. Due tothe mass difference between X=(Sc, Ti, V) and B atoms, the E2gmodes contain only vibrations of B atom. B1g and E2g gamma-pointmodes involve atomic vibrations along the directions of z and y,respectively for these compounds. The other gamma-pointsmodesinvolve atomic vibrations along the directions as E1u:z, A2u:y forScB2, E1u:z, A2u:x for TiB2 and E1u:x, A2u:z for VB2.The figures on the right side of the phonon dispersion curvesshow the corresponding total and partial density of phonon statesnications 149 (2009) 18431848TotalBScTotalBTiTotalBVINTERNAL ENERGY [meV/unit cell]INTERNAL ENERGY [meV/unit cell]0 200 400 600 800 1000TEMPERATURE [K]1200 1400 1600 1800 20001600140012001000800600400200 400 600 800 1000 1200 1400 1600 1800TEMPERATURE [K]0 2000200 400 600 800 1000 1200 1400 1600 1800TEMPERATURE [K]0 20001600140012001000800600400140012001000800600400200INTERNAL ENERGY [meV/unit cell]16000Fig. 3. Temperature dependence of internal energy for XB2 .Table 2Calculated 0 point frequencies in meV for ScB2 , TiB2 , and VB2 .E1u A2u B1g E2gScB2 Present 55.72 56.11 77.08 99.72TiB2 Present 65.11 66.21 67.52 111.85Experimentala 65.5 66.4 70.0 112.8VB2 Present 60.69 62.69 65.24 120.22Experimentala 60.6 62.1 69.6 114.9a Ref. [23].for each compound. While the main contribution to acousticphonons results from the transition metals sublattice, the high-frequency phonons stem from the boron ions. This is expectedbecause the boron atom is lighter than transition metal atoms,which leads to but not very strong electronphonon interactions.The covalent character of the BB bonding is also decisive for thehigh frequency of phonons involving the boron atoms.uE. Deligoz et al. / Solid State CommTotalBScTotalBTiTotalBVTEMPERATURE [K]2000 200 400 600 800 1000 1200 1400 1600 1800 2000TEMPERATURE [K]0 200 400 600 800 1000 1200 1400 1600 1800 2000TEMPERATURE [K]0 200 400 600 800 1000 1200 1400 1600 1800 20000-200-400-600-800-1000-1200-1400FREE ENERGY [meV/unit cell]-16002000-200-400-600-800-1000-1200-1400FREE ENERGY [meV/unit cell]-16000200-200-400-600-800-1000-1200-1400FREE ENERGY [meV/unit cell]400Fig. 4. Temperature dependence of free energy for XB2 .3.3. Temperature dependence of the thermodynamical quantitiesAs is known, the first-principles phonon calculations are limitedto T = 0 K, and the thermodynamic properties of the crystalscan be determined in detail by phonons. The quasi-harmonicapproximation can describe such properties quite satisfactorily. Inthe Gibbs free energy expression G(T , P) = F(T , V )+ PV , the freeenergy F(T , V ) of a crystal is the sum of the ground state energyE(V ) and the phonon free energy Fphon(T , V ). The temperature Tappears in F(T , V ) via the phonon term. Themore explicit forms ofthe internal energy, free energy, entropy and heat capacity in theharmonic approximation per primitive unit cell are given [32] asfollows:E = 12r 0g()(h) coth(h2kBT)d, (1) [ ( )]F = rkBT0g() ln 2 sinhh2kBTd, (2)nications 149 (2009) 18431848 1847Total18161412108642BScTotalBTiTotalBVTEMPERATURE [K]0 200 400 600 800 1000 1200 1400 1600 1800 2000TEMPERATURE [K]0 200 400 600 800 1000 1200 1400 1600 1800 2000TEMPERATURE [K]0ENTROPY [k_b/unit cell]ENTROPY [k_b/unit cell]181614121086420ENTROPY [k_b/unit cell]1816141210864200 200 400 600 800 1000 1200 1400 1600 1800 2000Fig. 5. Temperature dependence of entropy for XB2 .S = rkB 0g(){(h2kBT)[coth(h2kBT) 1] ln[1 exp(h2kBT)]}d, (3)C = rkB 0g()(hkBT)2 exp ( hkBT )[exp(hkBT) 1]2 d, (4)where r is the number of degree of freedom in the unit cell, h isthe Planck constant, kB is the Boltzmann constant, and T is thetemperature.The above formulae are used to plot the temperature depen-dence of the internal energy (E), free energy (F ), entropy (S), andheat capacity at constant volume (Cv) for each compound and theirconstituent atoms, by using the data obtained from SIESTA andPHONON software.The calculated internal energies for XB2 as a function oftemperature are depicted in Fig. 3. It is seen that, above 300 K, thetotal internal energies increase almost linearly with temperature.u1848 E. Deligoz et al. / Solid State CommTotalBScTotalBTiTotalBVTEMPERATURE [K]0 200 400 600 800 1000 1200 1400 1600 1800 2000TEMPERATURE [K]987654321HEAT CAPACITY [k_B/unit cell]0HEAT CAPACITY [k_B/unit cell]0 200 400 600 800 1000 1200 1400 1600 1800 20009876543210TEMPERATURE [K]HEAT CAPACITY [k_B/unit cell]98765432100 200 400 600 800 1000 1200 1400 1600 1800 2000Fig. 6. Temperature dependence of heat capacity for XB2 .The obviously, the contributions to internal energy from B isdominant than transition metal atoms. At high temperatures theinternal energy tends to kBT behaviour.Fig. 4 shows the variations of phonon free energy withtemperature for the same compounds. The overall profiles ofall plots are similar and the phonon free energy decreases withincreasing temperature.The variations of entropy with temperature for XB2 are given inFig. 5 for the same temperature range. The total and partial entropygraphs exhibit a similar trend and the contributions to internalenergy from transitionmetal atoms are dominant thanboron atom.The contributions to the total heat capacity from the latticevibrations of XB2 are illustrated in Fig. 6. It is realized from figuresthat when T < 700 K, Cv increases very quickly with temperature;nications 149 (2009) 18431848when T > 700 K, Cv increases slowly with temperature andit almost approaches a constant called the DulongPetit limit athigher temperatures. It is seen that while the contribution tothe transition metal atoms (Sc, Ti, V) to the heat capacity curvesdecreases on going fromSc to V, the contribution to the boron atomincreases. The temperature is limited to 2000 K to minimize thepotential influence of anharmonicity in all graphs.4. Summary and conclusionThe structural, vibrational, and thermodynamical propertiesfor ScB2, TiB2, and VB2 are presented by using the first-principles calculations implemented in SIESTA within the GGAapproximation. The results, generally, are in agreement with theavailable experimental and theoretical data. The original aspectsof the present calculations concern the phonon dispersion curvesand thermodynamical results at various temperatures,which is notconsidered previously. ScB2, TiB2, and VB2 exhibit similar, but notvery strong electronphonon interactions.AcknowledgmentThiswork is supported byGazi University Research-Project Unitunder Project No: 05/2008-42.References[1] R.B. Kaner, et al., Mater. Sci. 308 (2005) 1268.[2] W. Zhou, et al., Phys. Rev. B 76 (2007) 184113.[3] H.Y. Chung, et al., Science 316 (2007) 436.[4] R.W. Cumberland, et al., J. Am. Chem. Soc. 127 (2005) 7264.[5] M.I. Matkovich, et al., in: M.I. Matkovich (Ed.), Boron and Refractory Borides,Springer-Verlag, Berlin, 1977.[6] G.V. Samsonov, I.M. Vinitskii, Handbook of Refractory Compounds, Metal-lurgiya, Moscow, 1976.[7] T.I. Serebryakova, et al., High-Temperature Borides, Metallurgiya, Moscow,1991.[8] I.R. Shein, A.L. Ivanovskii, Phys. Rev. B 73 (2006) 144108.[9] I.R Shein, A.L. Ivanovskii, J. Phys.: Condens. Matter 20 (2008) 415218.[10] T. Oguchi, J. Phys. Soc. Japan 71 (2002) 1495.[11] P. Vajeeston, et al., Phys. Rev. B 63 (2001) 045115.[12] V. Milman, M.C. Warren, J. Phys.: Condens. Matter 13 (2001) 5585.[13] C.A. Perottoni, et al., J. Phys.: Condens. Matter 12 (2000) 7205.[14] P.E. van Camp, D.E. van Doren, High Pressure Res. 13 (2005) 335.[15] F. Peng, et al., Physica B 400 (2007) 83.[16] P. Ravindran, et al., Phys. Rev. B 64 (2001) 224509.[17] W. Chun-Lei, et al., Chin. Phys. B 18 (2009) 1674.[18] N. Kaur, et al., Mod. Phys. Lett. B 21 (2007) 885.[19] R.G. Munro, J. Res. Natl. Inst. Stand. Technol. 105 (2000) 709.[20] S.T. Mahmud, et al., J. Phys.: Condens. Matter 16 (2004) 2335.[21] X. Zhou, et al., Physica B 404 (2009) 1527.[22] I.R. Shein, A.L. Ivanovskii, Phys. Solid State 44 (2002) 1833.[23] R. Heid, et al., Phys. Rev. B 67 (2003) 180510.[24] P. Villars, Pearsons Handbook: Crystallographic Data for Intermetallic Phases,ASM International, Materials Park, OH, 1997.[25] I.R. Shein, A.L. Ivanovskii, Preprint cond-mat/0109445(2001).[26] J.P. Perdew, et al., Phys. Rev. Lett. 77 (1996) 3865.[27] P. Ordejn, et al., Phys. Rev. B (Rapid Comm.) 53 (1996) R10441.[28] J.M. Soler, et al., J. Phys.: Condens. Matt. 14 (2002) 2745.[29] O.F. Sankey, D.J. Niklewski, Phys. Rev. B 40 (1989) 3979.[30] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993.[31] X. Zhang, et al., Comput. Mater. Sci. 44 (2008) 411.[32] K. Parlinski, Software PHONON (2003), and references therein.[33] K. Parlinski, et al., Phys. Rev. Lett. 78 (1997) 4063.[34] K. Parlinski, Acta Phys. Polon. A 100 (2001) 767.[35] Molekel 4.3 P. Flkiger, H.P. Lthi, S. Portmann, J. Weber, Swiss Center forScientific Computing, Manno (Switzerland), 2000.[36] S. Portmann, H.P. Lthi, Molekel: An interactive molecular graphics tool,Chimla 54 (2000) 766.Lattice dynamical properties of ScB2 , TiB2 , and V B2 compoundsIntroductionMethod of calculationResults and discussionStructural propertiesPhonon dispersion curvesTemperature dependence of the thermodynamical quantitiesSummary and conclusionAcknowledgmentReferences

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