Brazilian Journal of Physics ISSN: Sociedade Brasileira de Física Brasil

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1 Brzilin Journl of Physics ISSN: Sociedde Brsileir de Físic Brsil Chihi, T.; Ftmi, M.; Gheouli, B.; Gheouli, M. A.; Bouhemdou, A. Theoreticl Prediction of Structurl, Elstic nd Electronic Properties of M 5 Si 3 (M=Ti, Zr) Compounds Brzilin Journl of Physics, vol. 45, núm. 3, junio, 2015, pp Sociedde Brsileir de Físic Sâo Pulo, Brsil Aville in: How to cite Complete issue More informtion out this rticle Journl's homepge in redlyc.org Scientific Informtion System Network of Scientific Journls from Ltin Americ, the Crien, Spin nd Portugl Non-profit cdemic project, developed under the open ccess inititive

2 Brz J Phys (2015) 45: DOI /s CONDENSED MATTER Theoreticl Prediction of Structurl, Elstic nd Electronic Properties of M 5 Si 3 (M=Ti, Zr) Compounds T. Chihi 1 & M. Ftmi 1 & B. Gheouli 2 & M. A. Gheouli 1 & A. Bouhemdou 3 Received: 6 Decemer 2014 /Pulished online: 14 April 2015 # Sociedde Brsileir de Físic 2015 Astrct Structurl, elstic, electronic nd mechnicl properties of the M 5 Si 3 (M=Ti, Zr) compounds with (Mn 5 Si 3 )16H crystl structure hve een studied with respect to pressure. Our computtionl method is sed on pseudo-potentil plne-wve (PP-PW) method. The exchnge correltion hs een treted using the generlized grdient pproximtion (GGA) in order to work out the densities of sttes. Groundstte quntities, such s lttice prmeter nd ulk modulus, hve een evluted, s well s elstic constnts nd their pressure derivtive. Elstic constnts nd their pressure dependence hve een clculted. Also, ulk nd sher moduli, Young smodulusndpoisson s rtio for idel polycrystlline phses hve een derived. Keywords A-initio clcultions. Bnd structures. Mechnicl properties 1 Introduction An intermetllic compound is phse which crystllizes with structure other thn those of its components. Intermetllic compounds hve een ttrctive cndidtes for hightemperture structurl mterils ecuse of their desirle * M. Ftmi ftmimessoud@yhoo.fr Reserch Unit on Emerging Mterils (RUEM), Setif 1 University, Setif, Algeri Lortory for Studying Surfces nd Interfces of Solids Mterils, Deprtment of Physics, Fculty of Sciences, Setif 1 University, Setif, Algeri Lortory for Developing New Mterils nd their Chrcteriztion, Deprtment of Physics, Fculty of Science, Setif 1 University, Setif, Algeri intrinsic properties. The gret interest in trnsition metl silicide is relted to their ultr-high temperture mterils, good stility, high melting points, low densities, high oxidtion resistnce nd excellent mechnicl strength t elevted temperture [1 3]. The melting temperture of Ti (Zr) is 1841 K (2128 K), tht of silicon is 1687 K, while the temperture of the compound Ti 5 Si 3 (Zr 5 Si 3 ) is higher which is of 2390 K (2520 K). Ti 5 Si 3 which contins wt% (37.5 t.%) Si nd crystllizes in hexgonl D8 8 structure hs the most suitle comintion of high melting temperture (out 2130 C), low density (4.32 g/cm 3 ), high strength nd good oxidtion resistnce t high temperture [4]. Zr 5 Si 3 nd Ti 5 Si 3 hve the sme crystl structure. Hong et l. [5] reported the first principles study for some physicl properties of D8 8 -Ti 5 Si 3 phse, showing tht the ddition of low-rte metlloid toms cn stilize this phse. Willims et l. [6] nlyzed the incorportion of cron, nitrogen or oxygen toms into the lttice of Ti 5 Si 3 nd concluded tht the onding chnges tht occurred on ddition of cron, nitrogen or oxygen cted to decrese the nisotropic therml expnsion mesured. On the other side, Celis et l. [7] studied the potentility of the Mn 5 Si 3 16H crystl structure for high-temperture structurl pplictions choosing the Zr Si system, produced the Zr 5 Si 3, nd concluded tht this compound is good cndidte for high-temperture structurl pplictions. However, nd to the est of our knowledge, some properties such s elstic constnts t high pressures re not well understood y experiments. This prolem cn e solved y -initio theoreticl clcultions. As potentil high-pressure compound, our primry im ws therefore to present the results of theoreticl investigtion of the structurl, elstic, nd electronic properties of Ti 5 Si 3 nd Zr 5 Si 3. However [8], Ti 5 Si 3 nd Zr 5 Si 3 present hexgonl crystl structure Mn 5 Si 3 -type with spce group P6 3 /mcm, N 193;

Brz J Phys (2015) 45:302 307 303 Person symol is hp16 s shown in Fig. 1.

3 Brz J Phys (2015) 45: Person symol is hp16 s shown in Fig. 1.M 5 Si 3 (M=Ti, Zr) contins two formul units per unit cell nd hs two species of tom tht occupy three types of positions M 1 toms re t x 0 1/4 nd M 2 t 1/3 2/3 0, while Si toms re t x 0 1/4, where x= nd x = [6]. The pper is orgnized s follows. The computtionl method is descried in Sect. 2. In Sect. 3, the results re presented nd compred with ville experimentl nd theoreticl dt. Conclusion is given in Sect Computtionl Method All the electronic structure clcultions re implemented with the Cmridge Seril Totl Energy Pckge (CASTEP) simultion progrmme [9] tht solves the Schrödinger-like Kohn Shm equtions ccording to the formlism of the density functionl theory (DFT) [10, 11]. We used the generlized grdient pproximtion (GGA), nd Perdew Burke Ernzerhof (PBE) scheme [12], for hndling the electronic exchnge correltion potentil energy. Also, the pseudo-potentils were constructed using the -initio normconserving scheme to descrie the vlence electron interction with the tomic core, in which the Ti (3d 2 4s 2 ), Zr (4d 2 5s 2 )ndsi(3p 2 3s 2 ) oritls re treted s vlence electrons. The cutoff energy used for ll structures is 450 ev. Brillouin zone (BZ) smpling is crried out using Monkhorst Pck mesh set [13]. Atomic positions re optimized within density mixing scheme, sed on conjugte grdient (CG) method for eigenvlues minimiztion. Actully, the equilirium lttice prmeter is determined from structurl optimiztion, using the Broyden Fletcher Goldfr Shnno (BFGS) minimiztion technique. This technique provides fst wy of finding the lowest energy structure, with the following thresholds for converged structures: (i) the potentil energy difference etween tomic itertions ws less thn /tom, (ii) the forces on ech tom were typiclly less thn ev/å, (iii) tom displcement during geometry optimiztion less thn Å nd (iv) mximum stress within 0.1 GP. 3 Results nd Discussion 3.1 Structurl Properties TheunitcellisshowninFig.1. The considered M 5 Si 3 (M=Ti, Zr) dopts the hexgonl structure with spce group P6 3 /mcm (N. 193). The results for lttice prmeters nd c re reported in Tle 1 nd compred with experimentl nd previous theoreticl clcultions. Our clculted vlue for lttice prmeters of M 5 Si 3 (M=Ti, Zr) compounds re in excellent greement with the experimentl nd previous theoreticl dt (Tle 2). Fig. 1 Schemtic representtion of M 5 Si 3 silicides with D8 8 structure We re now interested in the pressure effect. We clculted the unit cell volume t vlues from 0 to 40 GP of n pplied hydrosttic pressure in order to construct n eqution of stte (EOS). The EOS ws fitted to third-order Birch Murnghn eqution, s follows: " P ¼ 3 2 B V 7 3 #" ( V þ 3 ) # V B ;ð1þ V 0 V 0 with V 0 corresponding to the vlue determined from the zero pressure dt. We otined the ulk modulus B 0 =B EOS nd its pressure derivtive B t zero pressure (B 0 = GP, B = 3.88) tht is in good greement with the vlues otined vi the elstic constnts (B=138.71). The two rtios / 0 nd c/c 0 of the lttice prmeters hve the sme dependence on the pressure, which cn e explined y the chnges in ond lengths of Ti 5 Si 3 nd Zr 5 Si 3, which hve the sme sensitivity to the pressure Fig Elstic Constnts The clculted independent elstic constnts C ij for the two phses of M 5 Si 3 (M=Ti, Zr) compounds re listed in Tle 1. One condition for mechnicl stility of structure is tht its strin energy must e positive ginst ny homogeneous elstic deformtion. For hexgonl crystl, let us recll tht the generlized elstic stility criteri [17] re s follows: V 0 C 11 > 0; C 33 > 0; C 44 > 0; C 66 > 0; C 11 C 12 > 0; C 11 þ C 33 þ C 12 > 0; ðc 11 þ C 12 ÞC 33 2C 2 13 > 0: ð2þ The fct tht our clculted elstic constnts of the two considered phses stisfy ll the ove criteri indictes tht these two considered phses of M 5 Si 3 (M=Ti, Zr) (Fig. 3)re mechniclly stle up to 40 GP. The C 11 nd C 33 elstic constnts, which correspond to the resistnce to liner compression long the x nd z

4 304 Brz J Phys (2015) 45: Tle 1 Clculted lttice prmeters ( nd c,inå,ndv 0, in Å 3 ) of one unit formul; ulk modulus * B 0 for single crystl nd its pressure derivtive B ; elstic constnts; sher modulus; Young s modulus;b/g rtio; longitudinl, trnsverse nd verge sound velocities in (m/s) nd Deye temperture (Θ in Kelvin) clculted from the men sound velocities for Ti 5 Si 3 nd Zr 5 Si 3 in comprison with reported experimentl nd theoreticl dt Property Ti 5 Si 3 Zr 5 Si 3 Present Expt. Others Present Expt. Others [6] [14] c [6] [14] V [7] * B * B C C C C C C ** B [14] ** B G v G R G [14] E [14] B/G V l 10, V T V M Θ D crystllogrphic xes, re significntly lrger thn other elstic constnts, resulting in pronounced elstic nisotropy in our compounds. However, C 11 elstic constnt lwys remins much lrger thn C 33, indicting tht c-xis is more compressile thn - xis. In ddition, the ulk nd sher moduli (B nd G) re two importnt mechnicl quntities for technologicl nd engineering pplictions. The ltter (G), which is relted to ond ending, depends on the nture of the ond nd decreses ccording to the iconicity. To evlute the elstic nisotropy of these compounds, we used the sher nisotropic fctor which provides mesure of the degree of nisotropy in the connection etween the cron toms in different plnes. For M 5 Si 3 (M=Ti, Zr) with the hexgonl structure, sher nisotropy fctor A(C ij ), defined s A=4C 44 /(C 11 +C 33 2C 13 )for the {100} plnes of sher etween the <011> nd <010>directions [18]. For crystls with isotropic elstic properties A= 1, while vlues smller or greter thn unity mesure the degree of elstic nisotropy, s well s the rtio etween liner compressiility coefficients for hexgonl crystls, i.e. k c / k =(C 11 +C 12 2C 13 )/(C 33 C 13 )[19]. The computed sher nisotropic fctors re listed in Tle 3. Tle 2 Clculted tomic coordintes of M 5 Si 3 (M=Ti, Zr) in comprison with reported dt Atom Site Ti 5 Si 3 Zr 5 Si 3 x/ y/ z/c x/ y/ z/c M1 4d 1/3 2/3 0 1/3 2/3 0 1/3 [15] 2/3 [15] 0 [15] M2 6g / / [16] 0 [16] 1/4 [16] [15] 0 [15] 1/4 [15] [5] 0 [5] 1/4 [5] Si 6g / / [16] 0 [16] 1/4 [16] [15] 0 [15] 1/4 [15] [5] 0 [5] 1/4 [5]

5 Brz J Phys (2015) 45: Tle 3 The rtio etween liner compressiility coefficients, k c /k,of hexgonl M 5 Si 3 (M=Ti, Zr). The sher nisotropy fctor A, C 11 C 12, B/ G nd Poisson s rtio ν were otined. All constnts re dimensionless k c /k A C 11 C 12 B/G ν Ti 5 Si Zr 5 Si [15] Fig. 2 The two rtios / 0 nd c/c 0 of the lttice prmeters versus pressures for M 5 Si 3 The polycrystlline Poisson s rtio (ν) Young s modulus (E) nd Lmè constnts (λ) re often used for polycrystlline mterils when investigting their hrdness. These quntities re clculted from the following formul: Fig. 3 Elstic constnts under pressures of M 5 Si 3 (M=Ti, Zr) E ¼ 9BG 3B þ G ; ν ¼ 3B E 6B ; λ ¼ νe ð1 þ νþð1 2νÞ : ð3þ Pugh [20]proposed the B/G rtio to represent mesure of Bmchinle ehviour^. A high B/G vlue is then ssocited with ductility nd low vlue with rittleness. The criticl vlue which seprtes ductile nd rittle ehviours is t out For instnce, dimond hs B/G of 0.80 [21], while luminium, colt, rhodium nd iridium present B/G rtios of 2.74, 2.43, 1.77, nd 1.74, respectively [20]. In Tle 3, we hve clculted the B/G Pugh rtio for M 5 Si 3 (M=Ti, Zr) compounds under study. It cn e seen tht our clcultions show tht M 5 Si 3 (M=Ti, Zr) compounds re of rittle chrcter. Besides B/G, itiswellknownthtc 11 C 12 is lso very significnt chrcteristic for mechnicl properties of mterils [22]. Pointing out tht Zr 5 Si 3 hs lower vlues of oth C 11 C 12 (172 GP) nd Young s modulus(e=216 GP) thn Ti 5 Si 3 (C 11 C 12 =190 GP nd Young s moduluse=251 GP), we my conclude for etter plsticity s fr s Zr 5 Si 3 is concerned. On the contrry, Ti 5 Si 3 most proly presents the much poorest plsticity. To e complete, the five singlecrystl elstic constnts of Ti 5 Si 3 nd Zr 5 Si 3 re listed in Tle 3. The Deye temperture my e estimted from the verge sound velocity V m [23]. Θ ¼ h k 1 3n N A ρ 3V m ; ð4þ 4π M where h is Plnck sconstnts,k is Boltzmn sconstnt,n A is Avogdro s numer, n is the numer of toms per formul unit, M is the moleculr mss per formul unit, ρ is the density nd V m is otined from V m ¼ V 3 þ 1 1 3; ð5þ s V 3 l where V s nd V l re the sher nd longitudinl sound velocities, respectively. The rithmetic verge of the Voigt nd the Reuss ounds is clled the Voigt Reuss Hill (VRH) verge ndiscommonlyusedtoestimteelsticmoduliof

6 306 Brz J Phys (2015) 45: polycrystls. The polycrystlline moduli re the rithmetic of the vlues of the Voigt nd Reuss moduli [24]: G H ¼ 1 ð 2 G R þ G V Þ; B H ¼ 1 ð 2 B R þ B V Þ: ð6þ Therefore, the prole vlues of the verge sher nd longitudinl sound velocities cn e clculted from Nvier s eqution [14]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffi B H þ 4 u G H 3 G H t V S ¼ ; V l ¼ : ð7þ ρ ρ The longitudinl, trnsverse nd verge sound velocities nd Deye temperture of M 5 Si 3 (M=Ti, Zr) hve een clculted nd listed in Tle 1. The otined vlues of the polycrystlline Deye temperture clculted with Eq. (4) is 909 K nd 804 K for Ti 5 Si 3 nd Zr 5 Si 3, respectively, t mient pressure. The pressure dependence of the Θ is shown in Fig. 4. Θ represents the temperture t which nerly ll modes of virtions in solid re excited; this increse of Θ implies n increse in the rigidity of these compounds with the pressure. 3.3 Electronic Structures The nd structures of M 5 Si 3 (M=Ti, Zr) long the vrious symmetry lines within the GGA scheme re given in Fig. 5.It cn e seen tht ll these mterils re metllic. The electronic DOS of M 5 Si 3 (M=Ti, Zr) compounds re shown in Fig. 6. The upper curve presents the totl density of sttes per formul unit, nd the lower curves give the contriution from Ti, Zr nd Si toms to the totl DOS. We oserve tht there re four distinct structures. The sttes which re pproximtely locted etween 10 nd 6 ev elow the Fermi level originte from the onding of Si-s nd Ti(Zr)-d sttes with smll Ti(Zr)-p contriution. Between 6 evnd the top of the vlence nds reflects the onding of Si-p nd Ti(Zr)-d sttes. The two totl density of sttes (TDOS) hve some similrities; however in Fig. 6, nd Fig. 5 M 5 Si 3 nd structure long high-symmetry directions in reciprocl spce in Zr 5 Si 3 compound, the TDOS t the Fermi level hs the smllest n(e f ) with 8.6 sttes/ev/tom; the Ti 5 Si 3 phse hs Fig. 4 Deye temperture under pressures of M 5 Si 3 (M=Ti, Zr) Fig. 6 Totl nd prtil stte densities of s-si, p-si, s-m, p-m nd d-m in M 5 Si 3 (M=Ti, Zr)

7 Brz J Phys (2015) 45: the lrgest n(e f ) with 12.5 sttes/ev/tom. The electronic sttes etween 0 nd 6 ev re dominted minly y Ti(Zr)-d sttes. In the rest of the conduction nd, the electronic sttes re dominted y Ti(Zr)-p with the mixed chrcter of Ti(Zr)- d, Ti(Zr)-s, Si-s nd smll contriution of Si-p chrcter. 4 Conclusion Structurl properties, elstic nd electronic structure of M 5 Si 3 (M=Ti, Zr) compounds re studied y DFT clcultions. The clculted lttice constnts re found to e in very good greement with experimentl results. As result, the two rtios / 0 nd c/c 0 of the lttice prmeters hve the sme dependence on the pressure, which cn e explined y the chnges in ond lengths of Ti 5 Si 3 nd Zr 5 Si 3 which hve the sme sensitivity to the pressure. Elstic nd mechnicl properties re otined, which show tht our compounds re mechniclly stle. The ggregte elstic modulus B, G, E, Deye temperture Θ nd elstic nisotropies were evluted. Unfortuntely, there is no experimentl dt for comprison. The results will stimulte further experimentl nd theoreticl work in the future. References 1. M.E. Schlesinger, Chem. Rev. 90, 607 (1990) 2. P.J. Meschter, D.S. Schwrtz, JOM 41, 52(1989) 3. J.J. Petrovic, A.K. Vsudevn, Mter. Sci. Eng. A 261, 1 (1999) 4. G. Frommeyer, R. Rosenkrnz, C. Luecke, Z. Met. 81 H(5), 307 (1990) 5. L. Hong, Y. Ye, H. Gu, J. Nt. Sci. 3(4), 433 (1998) 6. J.J. Willims, M.J. Krmer, M. Akinc, S.K. Mlik, J. Mter. Res. 8(8), 1773 (2000) 7. P.B. Celis, K. Ishizki, J. Mter. Sci. 26, (1991) 8. P. Villrs, L.D. Clvert, Persons Hndook of Crystllogrphic Dt for Intermetllic Phses, 2nd edn. (ASM Interntionl, Mterils Prk, 1991) 9. M.D. Segll, P.J.D. Lindn, M.J. Proert, C.J. Pickrd, P.J. Hspin, S.J. Clrk, M.C. Pyne, J. Phys. Condens. Mtter 14, 2717 (2002) 10. P.Hohenerg,W.Kohn,Phys.Rev.B13(6), 864 (1964) 11. W. Kohn, L.J. Shm, Phys. Rev. A 140, 113 (1965) 12. J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 78, 1396(E) (1997) 13. H.J. Monkhorst, J.D. Pck, Phys. Rev. B 13, 5188 (1976) 14. K.B. Pnd, K.S. Rvi Chndrn, Comput. Mter. Sci. 35, 134 (2006) 15. Y.Z. Nie, Y.Q. Xie, H.J. Peng, X.B. Li, Chin. J. Nonferrous Metls 17, 1495 (2007) 16. W. Person, P. Villrs, L.D. Clvert, Person s Hndook of Crystllogrphic Dt for Intermetllic Phses (Americn Society for Metls, Metls Prk, 1985) 17. Q.K. Hu, Q.H. Wu, Y.M. M, L.J. Zhng, Z.Y. Liu, J.L. He, H. Sun, H.T. Wng, Y.J. Tin, Phys. Rev. B 73, (2006) 18. P. Rvindrn, L. Fst, P.A. Korzhvyi, B. Johnsson, J. Wills, O. Eriksson, J. Appl. Phys. 84, 4891 (1998) 19. J.Y. Wng, Y.C. Zhou, Z.J. Lin, T. Lio, L.F. He, Phys. Rev. B 73, (2006) 20. S.F. Pugh, Philos. Mg. 45, 823 (1954) 21. J. Hines, J.M. Leger, G. Bocquillon, Synthesis nd design of superhrd mterils. Annu. Rev. Mter. Res. 31, 1 23 (2001) 22. A.Sumer,J.F.Smith,J.Appl.Phys.33, 2283 (1962) 23. O.L. Anderson, J. Phys. Chem. Solids 24, 909 (1963) 24. R. Hill, Proc. Soc. London A 65, 350 (1952)

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