*

Department of Physics,S.S.V.Degree College,Aff i liated to C.C.S.University-Meerut,Hapur,U.P.India

First principle study of structural,electronic and thermodynamic behavior of ternary intermetallic compound:CeMgTl

R.P.Singh*

Department of Physics,S.S.V.Degree College,Aff i liated to C.C.S.University-Meerut,Hapur,U.P.India

To study the structural,electronic and thermodynamic behavior of CeMgTl,full-potential linear augmented plane wave plus local orbital(FPLAPW+lo)methodhasbeenused.Thelatticeparameters(a0,c0),bulkmodulus(B0)anditsf i rstorderpressurederivative(B0′)havebeencalculated for CeMgTl.Band structure and density of states histograms depicts that “5d”orbital electrons of Tl have dominant character in the electronic contribution to CeMgTl.Impact of the temperature and pressure on unit cell volume,bulk modulus,Debye temperature,Gru¨neisen parameter, specif i c heat and thermal expansion coeff i cient(α)have been studied in wide temperature range(0-300 K)and pressure range(0-15 GPa).

Intermetallics;Structural properties;Electronic structure;Thermodynamic properties

1.Introduction

Cerium-magnesium-thallium(CeMgTl)is a 1:1:1 stoichiometry isotypic intermetallic compound belonging to REMgX(RE=elements from rare earth group,X=13th group element)like compounds viz.GdMgIn and GdMgGa) [1].CeMgTl crystallizes in hexagonal ZrNiAl type(P-62m) structure.REMgX intermetallics are formed by replacing late transition element(T)with light main group element viz.Li, Mgetc[1-3]intheequiatomicintermetallicsRETX (RE=rare earth element;T=late transition element; X=main group element)which have potential applications in the sensors,random access memories etc[4].

REMgTl may be the one of superior alternate of RETX from futureaspectsofapplications.SofarhereCeMgTlistakenunder consideration from REMgX compounds for the study.Also CeMgTl and other REMgX intermetalliccompoundshavebeen synthesized successfully and some structural,magnetic studies have been made by R.Kraft et al.[5-7].But these studies provide a very little information on CeMgTl compound and to best our knowledge no more experimental/theoretical study has been made on CeMgTl.Thus,the motivation for the present work is to discuss and give more information on structural and electronic properties along with thermodynamic behavior of unit cell volume(V),Bulk modulus(B),Debye temperature (θD),Gru¨neisen parameter(γ),specif i c heat(CV)and thermal expansion coeff i cient(α)for CeMgTl under pressure and temperature.This work will help in further understanding and controlling the material properties under stress,and also provides reference data for the future experimental/theoretical work on this notable material.

2.Computational approach

Structural,electronic and thermodynamical calculations based on density functional theory(DFT)implemented in WIEN2k and Gibbs2 package have been[8,9]investigated forCeMgTl intermetallic compound. The advanced FPLAPW+lo method[10,11]has been used to linearize the energies which is highly accurate technique based on DFT.For structural and electronic properties,we have used generalized gradient approximation(GGA)[12].ThekandEconvergence are checked by increasing the number ofkpoints and the energy convergence criteria.In the irreducible part of the Brillouin zone,12 × 12 × 18 k points were used to calculate the total and partial density of states and Fermi energy of CeMgTl is found to be 0.3742 eV.

Quasi-harmonic Debye model implemented in Gibbs package [9,13,14]hasbeen used to calculate thermodynamical behavior of CeMgTl.In quasi-harmonic Debye model the non equilibrium Gibbs functionG*(V;P,T)is in the form of

HereE(V)is total energy per unit cell of CeMgTl,PV denotes the constant hydrostatic pressure, θ(V)is the Debye temperature,andAvibis the vibration term which can be expressed using Debye model of the phonon density of states as[14]

Here,nis the number of atoms per unit formula unit,D(θ/T)is the Debye integral.For an isotropic solid, θ can be expressed as[14]

Here,Mis molecular weight per unit cell andBsis the adiabatic bulk modulus,which is nearly equal to static compressibility given by

andf(σ)is given by

The non-equilibrium Gibbs functions as a function of(V;P,T)is minimized with respect to volumeV:

By solving the above equation with respect to volumeV, one can obtain the thermal equation of state(EOS)V(P,T). The specif i c heat at constant volume and pressure(CV,Cp) and thermal expansion coeff i cient α by using the expressions [14]:

Here γ represents the Gru¨neisen parameter,expressed as

3.Results and discussion

3.1.Structural properties

Unit cell structure of CeMgTl generated by “Xcrysden”package[15]has been shown in Fig.1 which shows the hexagonal structure with lattice parameters(a=7.741 ?,c=4.737 ?)[7].The bulk properties of a crystalline material can accordingly be determined by calculating the total energy as a function of unit cell volume.Structural properties viz. lattice parameter(a0,c0),bulk modulus(B0)and its f i rst order pressure derivative(B0′)are calculated by f i tting the total energy according to the Birch-Murnaghan's equation of state [16,17]given by:

where,and are the energy and volume at equilibrium.and are the equilibrium bulk modulus and its f i rst order pressure derivative.

Fig.1.Unit cell structure of CeMgTl generated by Xcrysden.

Fig.2.Total energy as a function of unit cell volume for CeMgTl with GGA approximation.

The energy vs.volume(obtained using optimization method)curve for CeMgTl has been shown in Fig.2 with displaying equilibrium volume(V0),bulk modulus(B),pressure derivative of bulk modulus(BP)and minimum energy (E0).The shape of energy vs volume curve shows good relaxed optimization of CeMgTl.The calculated lattice parameters (a0,c0),bulk modulus and its f i rst order pressure derivative are also shown in Table 1 in whicha0,c0show good agreement with experimental values[7].

Table 1Lattice constant,a0,c0(?),Bulk modulus,B0(GPa),Pressure derivative of bulk modulus,B0′(GPa)equilibrium condition(at 0 K)for CeMgTl using GGA.

Fig.3.Contour plots of the total valence charge density in 3-dimentional(100) plane for CeMgTl.

3.2.Electronic properties

Fig.4.(a)Electron dispersion curves along high symmetry direction in the Brillouin zone for CeMgTl.

The charge density calculations are necessary to determine the character of chemical bonds between the constituentelements.In order to investigate the charge distribution and also the nature of chemical bonding,we have calculated the electronic charge density for CeMgTl.The contour plots of charge density have been shown in Fig.3.From Fig.3,it appears that the La-Tl bond is typically ionic,this is because there is no much bonding charge to link the La and Tl atoms and almost spherically symmetric around each atom.Again, no bonding charge is found to link between La-Mg andTl-Mg.Thus,La-Mg and Tl-Mg bonds also have complete ionic character.The ionic character is a consequence of metallic character,indicating that CeMgTl is a metallic material.

Fig.5.(i)Calculated total density of states for(a)CeMgTl(b)Ce,Mg and Tl(ii)partial density of states for(c)Ce-“s”,“p”orbital(d)Ce-“d”orbital(e)Ce-“f”orbital(f)Mg-“s”, “p” orbital(g)Tl-“s”, “p” orbital(h)Tl-“d”orbital.

The results on the electronic behavior of CeMgTl have been shown in terms of energy bands and total,partial density of states.The calculated band structure of CeMgTl along the high symmetry directions Γ,M,KandAin the Brillouin zone have been shown in Fig.4.It can be seen from Fig.4 that there is no band gap is found at the Fermi level as valence and conduction bands overlap signif i cantly near the Fermi level,as a result,CeMgTl exhibit the metallic character.

Fig.6.(i)Temperature induced variation in(a)volume,V(c)Bulk modulus,B(e)Debye temperature,θD(ii)pressure induced variation in(b)volume,V(d)Bulk modulus,B(f)Debye temperature.

The total density of states(TDOS)plots for CeMgTl,Ce, Mg,and Tl have been shown in Fig.5(a)and(b).The partial density of states(PDOS)have been displayed in(i)Fig.5(c) for Ce-s,p(ii)Fig.5(d)for Ce-d(iii)Fig.5(e)for Ce-f(iv)Fig.5(f)for Mg-s,p(v)Fig.5(g)for Tl-s,p and(vi)Fig.5(h) for Tl-d orbitals respectively.The dotted lines in all DOS Fig.5(a)-(h)at 0 eV show the Fermi level.It is clear from Fig.5(a)that most of the DOS lie in the energy range approximately between-11.0 eV and 0 eV.The states which lie at around-11.0 eV are due to Tl atom(see blue line contribution in Fig.5(b)).An interesting feature of the DOS of CeMgTl is the strong hybridization between Ce-p and Mg-s states at about-7.0 eV below Fermi level.Hybridization also exists between Ce-p and Mg-p states at around-1.0 eV. Furthermore,since large difference is existed between the energies of Ce-p and Tl-s,d states.Thus,no hybridization isfound between Ce-p and Tl-s,d states.Some Ce-d,f states are empty above the Fermi level for the conduction(see Fig.5(d), (e)).In overall DOS Figures,it is clear that Tl-d states have dominant character in contribution to electronic conduction in CeMgTl.

Fig.7.(i)Temperature induced variation in(a)Debye temperature, θD(c)Gru¨neisen parameter, γ and(e)specif i c heat,CV(ii)Pressure induced variation in(b) Debye temperature, θD(d)Gru¨neisen parameter, γ and(f)specif i c heat,CV.

3.3.Thermodynamic properties

The effect of temperature and pressure on unit cell volume (V),Bulk modulus(B),Debye temperature(θD),Gru¨neisen parameter(γ),specif i c heat(CV)and thermal expansion coeff i cient(α)for CeMgTl have been studied using QuasiharmonicDebyemodelin awidetemperaturerange 0-300 K and pressure range 0-15 GPa at different temperatures(0 K,75 K,150 K,225 K and 300 K).The results on effect of temperature and pressures have been shown in Figs.6 and 7.

Fig.6(a)and(b)depict that unit cell volume(V)increases with temperature but decreases with pressure.Unit cell volume increases with temperature due to the expansion of its dimensions with temperature.Unit cell volume,Vdecreases with pressure as dimensions of unit cell are compressed with increasing the pressure.The bulk modulus(B) is a material property indicating the degree of resistance of a material to compression.Larger the bulk modulus,greater is the degree of resistance.Fig.6(c)and(d)show that bulk modulus,Bdecreases with increasing the temperature but increases with increasing the pressure.This change in bulk modulus is caused by change in unit cell volume with temperature and pressure.This also shows that degree of resistance of CeMgTl decrease with increasing the temperature but increase with increasing the pressure.Debye temperature is related to the maximum thermal vibration frequency of a solid.The variation of Debye temperature with temperature and pressure ref l ects the fact that the thermal vibration frequency of the particles changes with pressure and temperature.It can be seen from Fig.6(e)and(f)that Debye temperature, θDdecreases slowly with increasing the temperature and increase with increasing the pressure rapidly. The slow variation in θDwith temperature ref l ects small effect of temperature on θDwhile rapidly variation in θDwith pressure shows good impact of pressure on θD.In case of CeMgTl,the variation of Debye temperature, θDwith temperature and pressure ref l ects the fact that the thermal vibration frequency of the particles changes slowly with temperature but changes rapidly with pressure.

The Gru¨neisen parameter could describe the alteration in vibration of a crystal lattice based on the increase or decrease in volume as a result of temperature or pressure.Fig.7(a)and (b)shows the variation in Gru¨neisen parameter with temperature and pressure.It can be seen from these Figures that Gru¨neisen parameter increase with temperature but decrease with pressure rapidly.Gru¨neisen parameter increases almost monotonously with temperature,indicating that temperature has no signif i cant effect on Gru¨neisen parameter.The temperature and pressure dependent behavior of the calculated heat capacity at constant volume(CV)has been shown in Fig.7(c)and(d).One can see from Fig.7(c)thatCVincreases with the temperature up toT≈ 150 K(follows DebyeT3law)and at temperature,T＞ 150 approaches a constant value(Dulong-Petit limit),indicating that temperature has more impact on the heat capacity.Fig.7(d)indicates that pressure has opposite inf l uences on the heat capacity,CVand the effect of temperature on the heat capacity is more signif i cant than that of pressure.Fig.7(e) shows the variation of thermal expansion coeff i cient, α as a function of temperature.The thermal expansion coeff i cient (α)increases rapidly especially at temperature below 150 K, whereas it gradually tends to a slow increase at higher temperatures.Fig.7(f)shows that thermal expansion coeff i cient, α decrease slowly with pressure,indicating small impact of pressure on thermal expansion coeff i cient,α.Furthermore,It can be observed from temperature dependent thermodynamic characteristics(Fig.6(a),(c),(e))and((b),(d),(f))that temperature has negligible effect onV,B, θD,γ and α in the temperature range 0-50 K.

4.Conclusions

Full potential linearize potential augmented plane wave method using generalized gradient approximation(GGA)and quasiharmonic Debye model has been used to study the structural,electronic and thermodynamic properties of CeMgTl isotypic intermetallic compound.The calculated lattice parameters are consistent with experimental/theoretical values.Ionic character is dominant on CeMgTl which is consequence of metallic character.The bands are found to be overlap at Fermi level,indicating metallic character of CeMgTl.In DOS,Tl-d orbital electrons are found to be dominant character.The bulk modulus,Bfound to be decreases with increasing the temperature but increase with increasing the pressure,indicating degree of resistance of CeMgTl decrease with increasing the temperature but increase with increasing the pressure.Temperature has found to be small impact onB, θD, γ but good impact onCVand α in temperature rangeT＜ 150 K.Pressure has found to be high impact onB, θD,and small impact on γ,CVand α.

[1]F.Canepa,M.L.Fornasini,F.Merlo,M.Napoletano,M.Pani,J.Alloys Compd.12(2000)312.

[2]A.Czybulka,G.Steinberg,H.U.Schuster,Z.Natur-forsch.34b(1979) 1057.

[3]G.Steinberg,H.U.Schuster,Z.Natur-forsch.34b(1979)1165.

[4]C.Falser,F.G.Fecher,B.Balke,Chem.Int.Ed.46(2007)668.

[5]R.Kraft,M.Valldor,R.Pottgen,Z.Natur-forsch.58b(2003)827.

[6]R.Kraft,M.Valldor,D.Kurowski,R.D.Hoffmann,R.Pottgen,Z.Naturforsch.59b(2004)513.

[7]R.Kraft,R.Pottgen,Z.Natur-forsch.60b(2005)265.

[8]P.Blaha,K.Schwarg,G.K.H.Madsen,D.Kvasnicka,J.Luitz,WIEN2k, an Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties,Technical University,Wien,Austria,2001,ISBN 3-9501031-1-2.

[9]M.A.Blanco,E.Francisco,V.Lua~na,Comput.Phys.Commun.158 (2004)57.

[10]K.Schwarz,P.Blaha,Comput.Mater.Sci.28(2003)259.

[11]G.K.H.Madsen,P.Blaha,K.Schwarz,E.Sj¨ostedt,L.Nordstr¨om,Phys. Rev.B 64(2001)195134.

[12]J.P.Perdew,Y.Wang,Phys.Rev.B 45(1992)3244.

[13]A.Otero-de-la-Roza,D.Abbasi-Prez,V.Luaa,Comput.Phys.Commun.182(10)(2011)2232.

[14]M.Flrez,J.M.Recio,E.Francisco,M.A.Blanco,A.M.Pends,Phys.Rev. B 66(2002)144112.

[15]A.Kokalj,Comput.Mater.Sci.28(2003)155.

[16]F.Birch,Phys.Rev.71(1947)809.

[17]F.D.Murnaghan,Proc.Natl.Acad.Sci.U.S.A.30(1944)244.

Received 23 August 2014;revised 20 October 2014;accepted 21 October 2014 Available online 4 December 2014

*Tel.:+91 9411901677;fax:+91 122 2306645.

E-mail address:rishisingh79@gmail.com.

Peer review under responsibility of National Engineering Research Center for Magnesium Alloys of China,Chongqing University.

http://dx.doi.org/10.1016/j.jma.2014.10.004.

2213-9567/Copyright 2014,National Engineering Research Center for Magnesium Alloys of China,Chongqing University.Production and hosting by Elsevier B.V.All rights reserved.

Copyright2014,NationalEngineeringResearchCenterforMagnesiumAlloysofChina,ChongqingUniversity.ProductionandhostingbyElsevier B.V.All rights reserved.