Ground state parameters, electronic properties and elastic constants of CaMg3: DFT study

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Available online at www.sciencedirect.com

JournalofMagnesiumandAlloysxxx(xxxx)xxx

www.elsevier.com/locate/jma

Full

Length

Article

Ground

state

parameters,

electronic

properties

and

elastic

constants

of

CaMg

3

:

DFT

study

H. Rekab-Djabri

a,b

,

Manal

M. Abdus

Salam

c,∗

,

S. Daoud

d

,

M. Drief

e

,

Y. Guermit

e

,

S. Louhibi-Fasla

a

a_Laboratory_of_Micro_and_Nanophysics_(LaMiN),_National_Polytechnic_School_Oran,_ENPO-MA,_BP_1523,_El_M’Naouer,₃₁₀₀₀_Oran,_Algeria b_Faculty_of_Nature_and_Life_Sciences_and_Earth_Sciences,_Akli_{Mohand-Oulhadj}_University,₁₀₀₀₀_Bouira,_Algeria

c_Department_of_Applied_Physics,_Tafila_Technical_University,_Tafila_66110,_Jordan

d_Laboratory_of_Materials_and_Electronic_Systems,_Mohamed_El_Bachir_El_Ibrahimi_University_of_Bordj_Bou_Arreridj,_Bordj_BouArreridj_34000,_Algeria e_Laboratoire_des_Matériaux_{Magnétiques,}_Département_de_Physique,_{Faculté des}_Sciences,_{Université Djilali}_LIABES_de_{Sidi-Bel-Abbès,}_Sidi_Bel_Abbes

22000,Algeria

Received 24 February 2020; received in revised form 4 June 2020; accepted 17 June 2020 Available online xxx

Abstract

Thepresentstudyaimstoinvestigate theequationofstate(EOS)parametersofCaMg3 inαReO3 (D09),AlFe3 (D03),Cu3Au(L12) and

CuTi3 (L60) structures,using fullpotential linear muffin-tin orbitals (FP-LMTO) approach based on the densityfunctional theory (DFT).

The localdensityapproximation (LDA) and thegeneralized gradientapproximation (GGA) wereboth appliedforthe exchange-correlation

potential term.The calculatedequation ofstate parametersatequilibrium, in general, agreedwell withthe available data ofthe literature.

ThecalculationsshowedthatundercompressionCaMg3 transformsfromD03toD09 atabout29.96GPa,and25.1GPausingLDAandGGA,

respectively.

The elastic constants Cij, aggregate moduli, Vickers hardness, sound velocity,and Debye temperature of CaMg3 in D03 structure were

alsoreported, discussedand analyzed. Using LDA(GGA), the calculatedvalues of HV and θD were found ataround 5.80GPa (5.93GPa)

and 393.44K(389.91K),respectively.

Electronic band structure, total density of states (TDOS) as well as the partial density of states (PDOS) have been also obtained.

The electronic band structure confirms the metallic behavior of CaMg3 in D03 phase, the valence bands are dominated by the maximum

contribution of‘d’ likestatesofCain theenergy rangingfrom2to 3eVforGGA,and from4.5to5eVforLDA, respectively.

Thisisanopenaccess articleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Peerreviewunderresponsibilityof ChongqingUniversity

Keywords: CaMg 3 compound; Electronic properties; Phase transition; Elastic constants; FP-LMTO.

1. Introduction

Magnesium(Mg)isanabundantelementintheworld com-paredwithothercommonlyusedmetals;itisoneofthe light-est among several commonly used structural metals. One of itsmajoradvantagesisthelowdensity,itisaboutonequarter that of steels and two thirds that of aluminum [1]. Further-more, Mg-based alloys, like the binary systems of Mg–Cu

∗_{Corresponding author.}

E-mailaddress:manalab70@yahoo.com (M.M.A. Salam).

and Mg–Ca, have gained more attention in the last decade. Thesematerials have severalapplications in engineering, es-pecially inaerospace manufacturing fieldand automotive in-dustry[2–6].ThebinarysystemofMg-Ca ispotentiallyused as biomaterial and has been a subject of research by many investigators inrecent years[7],[8].

ZhouandGong[8] havestudied the electronicproperties, mechanicalmoduli,chemicalbondingandmanyother param-eters of Mg-Ca systemindifferent configurations. Their cal-culationsshowedthat bothBCC(AlFe3 -typestructure(D03 )) and FCC (Cu3 Au-type structure (L12 ) phases of CaMg3 https://doi.org/10.1016/j.jma.2020.06.007

2213-9567/© 2020 Published by Elsevier B.V. on behalf of Chongqing University. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer review under responsibility of Chongqing University

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2 H.Rekab-Djabri,M.M.A.SalamandS.Daoudetal./JournalofMagnesiumandAlloysxxx(xxxx)xxx

Table 1

Plane wave number NPLW, muffin-tin radius (RMT) (in a.u.) and the energy cut-off (in Ry) used in our calculation.

Parameters αReO3 (B0 9) AlFe 3 (D0 3) Cu 3Au (L1 2) CuTi 3 (L6 0)

Total NPLW LDA 6120 5002 5421 7454 GGA 11,042 11,212 12,546 14,838 RMTS (Ca) LDA 2.62 3.21 3.17 3.224 GGA 2.63 3.15 3.284 3.224 RMTS (Mg) LDA 2.71 2.75 3.07 3.103 GGA 2.82 2.70 3.098 3.103 Ecut LDA 12.02 11.5 14.5 71 GGA 14.1 12.1 14.8 75 K-Point LDA (44, 44, 44) (46, 46, 46) (38, 38, 38) (44, 44, 49) GGA (40, 40, 40) (40, 40, 40) (42, 42, 42) (44, 44, 49)

are mechanically stable at equilibrium. Also, they investi-gated phase transition and found that both AlFe3 -type struc-ture (D03 ) and Cu3 Au-type structure (L12 ) transform to the hexagonalclosepackedHCP-typestructure(A3) atpressures around 29.47GPa and26.44GPa, respectively. Actually, it is known that under the effect of hydrostatic compression, the crystaloftentransformsfromthe mostenergetic stablephase toanothercrystallographic configuration [9],[10].

Groh [11] has investigated several physical, mechanical, and thermal properties of pure Calcium (Ca) and Mg–Ca binary system, in the framework of the second nearest-neighborsmodifiedembedded-atommethod(MEAM).His re-sults showed also that both AlFe3 -type structure (D03 ) and Cu3 Au-type structure (L12 ) of CaMg3 are mechanically sta-bleat equilibrium.

Furthermore, to the best of our knowledge, the CaMg3 phasesof Mg–Cabinarysystemisnotsynthesizeduntilnow, anditis verydifficult toobtain thephysical propertiesof all phases by using the experimental measurement. Under this situation, the first-principles calculations can be applied to computevariousphysicalpropertiesofdifferent phasesbased on the crystal structural information. In order to further im-provethepropertiesof Mg-Ca alloys, asystematic investiga-tionandaccurateinformationonthemechanicalpropertiesof someotherstructuresinbinaryMg-Casystemisthe prerequi-site.In the presentstudy,we investigatethe equationof state (EOS) parameters, pressure-induced phase transition, elastic constantsand electronic properties of CaMg3 compound us-ingfirst-principlestotalenergy calculations intheframework ofdensityfunctionaltheory(DFT),withinboththelocal den-sity approximation (LDA) [12] and the generalized gradient approximation(GGA) [13].

InSection2weintroduceabriefdescriptionofthemethod used in this work. Then, we present, in Section 3 our ob-tained results of the structural parameters, high-pressure in-ducedphasetransitions, elasticconstants aswell as the elec-tronic properties of CaMg3 compound. Finally, a brief con-clusionis giveninSection 4.

2. Computational details

The chemical and physical properties of system are de-termined by the inter-atomic interactions, which can be de-scribed by the inter-atomic interaction potential. The

calcu-lations inthe present workwere made using the all-electron fullpotentiallinear muffin-tinorbital(FP-LMTO)augmented by a plane-wave basis (PLW) [14] within the framework of density functionaltheory (DFT).Unlike the previous LMTO methods,thepresentversiontreatsboththeinterstitialregions andthecoreregionson thesamefooting[14].The exchange correlation energy of electrons is described using both the local density approximation (LDA) [12] andthe generalized gradient approximation (GGA) as parameterized by Perdew etal. [13].

In FP-LMTO approach, the non-overlapping muffin tin spheres MTS potential is expanded in terms of spherical harmonics inside the spheres of radius RMTS, while in the interstitialregion,thes,panddbasisfunctionsareexpanded in a number (NPLW) of plane waves determined automati-cally by the cut-off energies. The details of calculations are as follows: the charge density and the potential are repre-sented inside the muffin-tin (MT) spheres by spherical har-monics up to lmax =6. The self-consistent calculations are considered tobe converged when totalenergy of the system in stable within 10−4 Ry, while self consistent convergence of forces was achieved to within 2× 10−3 Ry/bohr in ionic minimization.Atotalenergy convergencetestsare performed by varying both: plane waves’ number PW and cut-off en-ergy Ecut. The number of plane waves (NPLW), total

cut-off energies, andthe muffin-tin radius(RMT) valuesused in ourcalculationforourmaterialof interestaresummarizedin

Table1.

The structures with cubic symmetry (αReO3 -type (D09 ), AlFe3 -type (D03 ), which is described as cubic close packed (CCP) cell and Cu3 Au-type (L12 ) which is described as an ordered CCP cell), have only one structural parameter (the latticeconstanta)that isused todescribethe unitcell, while the CuTi3 -type (L60 ) having tetragonal symmetry,two struc-turalparameters(aandc/aratio)areusedtodescribetheunit cell.

The locations of atoms for each crystallographic config-uration are also presented in Fig. 1, while the positions of different atoms, as well as the space group for each of the considered structures of CaMg3 are summarized in Table 2. From Fig.1,we can observe that the unit cell con-tains four molecules of CaMg3 in D03 structure, while in all other phases (D09 , L12 and L60 ) onlyone molecule was observed.

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H.Rekab-Djabri,M.M.A.SalamandS.Daoudetal./JournalofMagnesiumandAlloysxxx(xxxx)xxx 3 Table 2

Location of different atoms and space group of each type of structure of CaMg 3 compound. Pm −3m : Cubic Primitive ( cP), Fm −3m : Face-centered

cubic (FCC), P4/mmm: Tetragonal Primitive ( tP).

Ca Mg Space Group

1st atom 1st atom 2nd atom 3rd atom

αReO3 (D0 9) 0.0; 0.0; 0.0 0.0; 0.0; 1/2 1/2 ; 0.0; 0.0 0.0; 1/2; 0.0 Pm-3m

AlFe 3(D0 3) 0.0; 0.0; 0.0 1/2; 0.0; 0.0 −1/4;−1/4;−1/4 1/4; 1/4; 1/4 Fm-3m

Cu 3Au (L1 2) 0.0; 0.0; 0.0 0.0; 1/2; 1/2 1/2; 0.0; 1/2 1/2; 1/2; 0.0 Pm-3m

CuTi 3 (L6 0) 0.0; 0.0; 0.0 1/2; 1/2; 0.0 0.0; 1/2; 1/2 1/2; 0.0; 1/2 P4/mmm

Fig. 1. Cubic and tetragonal crystal structures of CaMg 3: Ca atoms in red, and Mg atoms in green.

From Fig.1,we canobserve thatinD03 structure,the Ca atomsoccupy thepositions of theCCP structure andtheMg atoms fill all of the octahedral voids; while inL12 structure the Ca atoms occupy the cell vertices, while the Mg atoms occupythefacecenters.TheL12 structureisjustthatofcubic Perovskite (CaTiO3 (E21 )) without the Titanium atoms, and replacing the atoms of Oxygen O per those of Magnesium Mg.

The L60 structure (a=b = c) is a tetragonal distortion of L12 structure (a=b=c), so when c=a, the atoms are at the positions of a face centered cubic lattice, and with consequence L60 structure becomes that of L12 .

3. Results and discussion

3.1. Equationofstate parameters

In order to investigate the ground state parameters, the total energy at different volumes (E-V) around the equilib-rium one is usually determined [15]–[17], and this is how we obtained the structural parameters of different phases of CaMg3 compoundinthe presentwork.Theseparameterscan be also predicted from ab-initio calculation of the pressure versusunit cellvolume(P-V)data [18].The equilibrium lat-ticevolumeV0 , bulk modulus B0 andthe pressurederivative

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Fig. 2. Total energy versus volume for different structures of CaMg 3 com-

pound using LDA.

Fig. 3. Total energy versus volume for different phases of CaMg 3compound

using GGA.

of the bulk modulus B0 havebeen computed by minimizing

the total energy by means of Murnaghan’s equation of state (EOS),whichcan beexpressed as [15]:

E(V)=E(V0 )+B0 V B 0 (V0 /V)B0 B 0 − 1 + 1 − B0 V0 B 0 − 1 (1) InEq.(1),E0 istheenergyofthegroundstate, correspond-ing to the equilibrium volume V0, and B0 (B0 = ∂B/∂P, at P=0) isthe first pressurederivative of the bulk modulus B. The bulk modulus B determines the compressibility and is calculatedusing [16]: B= V∂ 2 _E ∂V2 (2) In fact, the bulk modulus B is quantity that defines the strengthof bondsinsolids;itis ameasureofthe solid resis-tancetoexternal deformation[15].

The variation of the totalenergy as a functionof the unit cellvolumewasplottedinFigs. 2 and3 for different phases of CaMg3 using LDA and the GGA, respectively. One can notice that CuTi3 -type (L60 ), AlFe3 -type (D03 ) and Cu3 Au-type (L12 ) structures have almost the same minimum en-ergy, inboth LDA andGGA, while the minimum energy of

αReO3 -type(D09 )structureisslightlyhigherinboth approx-imations.Ourresultsoftheequilibriumstructuralparameters,

bulkmodulusandthepressurederivativeofthebulkmodulus of CaMg3 in D03 , L12 , L60 and B09 structures are summa-rizedinTable3 together withthoseof the literature[8],[11]. From Table 3, we can see that the lattice constant a0 of both AlFe3 -type (D03 ) and Cu3 Au-type (L12 ) configurations are invery goodagreement comparedtoothertheoretical re-sults [8],[11]. Our value (7.482A)˚ obtained with GGA for cubic AlFe3 -type (D03 ) structure overestimates the theoret-ical value (7.48A)˚ reported by Zhou and Gong, using PP-PAW (GGA) [8] by less than 0.03%, andunderestimates the theoretical result (7.494A)˚ reportedby Grohusing (MEAM)

[11] by about 0.16%; while our obtained value (4.775A)˚ of cubic Cu3 Au-type(L12 ) phaseunderestimates the theoretical result(4.78A)˚ reportedbyZhouandGong[8]byabout0.1%, and overestimates the theoretical value (4.76A)˚ reported by Groh[11] byabout 0.32%.

The calculated values of the bulk modulus B0 of both

D03 andL12 structures, as listed inTable3,are slightly dif-ferent from those obtained by other theoretical approaches

[8],[11]; where for example, our value (33.72GPa) obtained with GGA for D03 structure overestimates the theoretical value(29.57GPa) reportedbyZhouandGongusingPP-PAW (GGA) [8] by about 14%. To best of our knowledge, there are no other data existing in the literature on the structural parameters, bulk modulus and the pressure derivative of the bulk modulus for CaMg3 compound in both L60 and D09 structures.Ourfindingsregardingthestructuralparametersof CaMg3 inbothL60 andD09 structuresphasesperhapscanbe used to predict most of the physicalproperties of this mate-rial.Thisisduetothefactthatmostofthephysicalquantities ofcompoundsandalloysarerelatedtothebondingof atoms, whichis directlyrelatedto the structural parameters.

3.2. Structural phasetransition

Itiswellknownthat highpressuresinfluencecrystal pack-ing and electronic structure and as a result it plays an im-portant role in materialsproperties, such as superconducting phenomenon, elastic properties, and structural phase transi-tion[8].Inordertogetmoreinformationaboutthe pressure-induced phasetransition of crystals, we havetocalculate the Gibbs free energies G of different considered phases,which canexpressed as follows [19]–[22]:

G=E+PV− TS (3)

Here E, P, V, T and S symbolize the total internal en-ergy,pressure,volume,temperature,andentropy,respectively. Since thepresent calculationswere performedat T=0K,the term TSbecomes null,andwithconsequence, the Gibbsfree energybecomesequaltotheenthalpyH[19–22]:H=E+PV. For CaMg3 compound, the transition pressure (Pt) between

AlFe3 -type(D03 ) configurationandαReO3 -type(D09 ) phase werecalculatedusing theenthalpydifferenceasafunctionof the pressure with respect to D03 structure. Using both LDA andGGA,thevariationoftheenthalpydifferencesas a func-tion of pressureare plottedin Fig.4.

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H.Rekab-Djabri,M.M.A.SalamandS.Daoudetal./JournalofMagnesiumandAlloysxxx(xxxx)xxx 5 Ta b le 3 Structural parameters (equilibrium lattice constants a and c/a ratio), bu lk modulus B 0 and the pressure deri v ati v es of the bu lk modulus B 0 ’ for D0 9 , D0 3 , L1 2 and L6 0 phases for CaMg3 compound. a-Ref. [8] using PP-P A W (GGA), b-Ref. [11] using modified embedded-atom method (MEAM). Phase CaMg 3 Structure T ype α ReO 3 -type AlFe 3 -type Cu 3 Au-type CuT i 3 -type Symmetry Cubic Cubic Cubic T etragonal (D0 9 ) (D0 3 ) (L1 2 ) (L6 0 ) P arameter LD A GGA LD A GGA LD A GGA LD A GGA a0 ( ˚ A) This wo rk 6.395 6.694 7.315 7.482 4.680 4.775 4.908 5.014 Other wo rk s – – 7.48 a, 7.494 b 4.78 a, 4.76 b –– B0 (GP a) This wo rk 20.077 14.27 35.31 33.72 24.6 18.70 120.36 107.39 Other wo rk s – – 29.57 a, 29.87 b 27.45 a, 29.29 b –– B0 ’ This wo rk 2.876 2.837 3.301 2.832 3.617 3.726 3.294 3.434 c/a This Wo rk ––––––0 .8 7 6 0.865 Emin (eV) This Wo rk − 34,733.2 − 34,844.0 − 34,734.8 − 34,848.3 − 34,737.7 − 34,848.2 − 34,737.6 − 34,848.3

Fig. 4. Variation of the enthalpy differences H as a function of pressure for CaMg 3 compound in αReO3 type (D0 9) phase using both LDA and GGA.

The reference enthalpy in set for D0 3 phase.

ThetransitionfromAlFe3 -typephase(D03 )toαReO3 -type (D09 ) may occur at pressures of 29.96GPa (from LDA cal-culations), and 25.1GPa (from GGA calculations) as shown onFig.4.Atthesepressurestheenthalpiesofbothstructures becomeequal;andthe enthalpydifferencesbecomenull.Our resultsof thetransition pressures(Pt)areinconsistencewith

the results of Zhou and Gong [8], which found that both AlFe3 -type structure (D03 ) and Cu3 Au-type structure (L12 ) transform to the hexagonal close packed HCP-type structure (A3)at pressureof around 29.47GPa and26.44GPa, respec-tively.

Tobest ofourknowledge, thereareno otherdata existing in the literature on the pressure-induced phase transition for CaMg3 compound.Our findingsregardingthe pressurephase transition of CaMg3 compound may be used as a reference for futureworks.

3.3. Electronicproperties

The electronic band structures of CaMg3 compound in D03 structure at the calculated equilibrium lattice constants alongthe highsymmetry directionsinthe Brillouin zoneare presented in Fig. 5, using both LDA and GGA. One of the mostimportant toolstoinvestigatethe electronicstructure of ametallicmaterialisthe Fermisurface;whichrepresentsthe surface of constant energy in k-space [23]. The Fermi level (EF), the dashed line in Fig.5, was setto zero energy. It is noticed that the CaMg3 in D03 structure has a metallic be-havior since a number of valance and conduction bands are overlappingat the Fermilevel, andno bandgaps exist.

Elasticconstants, engineeringmoduli andseveralother re-lated physical properties are directly related with the nature of atomic bonding in material, which can be analyzed and explained using both the total density of states (DOS) and the local density of states (LDOS) [24],[25]. The electronic densityof states(EDOS) elucidatestheelectronic featuresof materials (elements, compounds, alloys, etc.) [26], the total densityofstates(TDOS)andpartialdensityof states(PDOS) of CaMg3 in D03 structure are calculated and presented in Fig.6.Thisfigure showsthat thelowestlyingbands aredue

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Fig. 5. Band structure of CaMg 3 compound in D0 3 structure using both LDA and GGA approaches.

Fig. 6. Density of states (TDOS and PDOS) of CaMg 3 in D0 3structure using both LDA and GGA.

to mainly ‘s’ like states of Ca and do not contribute much tobonding. The valence bands in the energy range between 2eV, and 3eV are dominated by the maximum contribution of‘d’likestatesof Cafor GGAapproximation,andbetween 4.5and5eV are dominatedby the maximumcontributionof ‘d’ like statesof Causing the LDAapproximation.

Moreover, the DOS at Ef of CaMg3 in D03 phase were calculated, they are to be around 0.469 and 0.454 using the LDA and GGA, respectively. Our obtained values agree well with the results of Zhou and Gong [8]. The differ-ences are around - 0.061and - 0.076using LDA and GGA, respectively.

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Calculated CijB,GV,GR,GH,E are all expressed in GPa, ν,A and G/ B are without unity. Values with § are calculated using Cijof Ref. [8], while

those with ∗are calculated using Cijof Ref. [11].

Parameter C 11 C 12 C 44 B G V G R G H E ν A G/ B Our work LDA 42.34 27.25 46.65 32.28 31.01 15.18 23.09 55.94 0.21 6.18 0.72 GGA 37.82 25.75 48.84 29.77 31.72 12.73 22.22 53.39 0.20 8.09 0.75 Ref. [8] 37.77 25.47 47.81 29.57 31.15 12.89 22.02 52.92 0.20 7.77 § _0.74 Ref. [11] 38.77 25.84 24.77 30.15 ∗ 17.45 ∗ 11.62 ∗ 14.53 ∗ 37.56 ∗ 0.29 ∗ 3.83 ∗ 0.48 ∗ 3.4. Elasticproperties

3.4.1.Elasticconstants, some aggregatemoduli and Vickers

hardness

Theformationof solidsisgovernedbytheforcesbetween the atoms, ions, and/or molecules, which are related to both the structural parameters of its crystal structure and to its chemicalcomposition[27].Theelasticpropertiesplayan im-portant role in the structural stability and stiffness of mate-rials. In cubic structures, as in the case of D03 structure of CaMg3 compound, there are three independent elastic stiff-nessconstants, namely: C11 ,C12 andC44 ,thatwere obtained inthepresent workbycalculatingthe totalenergyas a func-tion of strain[16],[28].The determinationof the elastic con-stants Cij needs the knowledge of the nature of the strain,

whichis expressedas follow [16]: ¯ε= ⎡ ⎢ ⎣ δ 0 0 0 −δ 0 0 0 ₍₁_−δδ22) ⎤ ⎥ ⎦ (4)

Applying this tensorstrain modifies the totalenergy from its unstrained value tothe followingexpression[16]:

E(δ)=E(0)+3(C11 − C12 )V0 δ2 +0δ4 (5)

where E(0) isthe energy ofthe unstrained latticeof unit cell volumeV0 .

Theidentificationof C44 isthroughthevolume-conserving tetragonal straintensor [16]:

¯ε= ⎡ ⎢ ⎣ δ 2 0 0 0 −δ₂ 0 0 0 ₍_4−δδ22) ⎤ ⎥ ⎦ (6)

The totalenergy is givenas follow [16]

E(δ)=E(−δ)=E(0)+1

2C44 V0 δ

2 ₊₀_δ4 ₍₇₎

Our resultsof the elasticconstants(C11 ,C12 andC44 ) ob-tainedfrombothLDAandGGAapproximationsarepresented in Table 4. Except of two theoretical works based on DFT and MEAM, by Zhouand Gong [8] and Groh [11], respec-tively, there are no other theoretical or experimental values available, to the best of our knowledge, for the elastic con-stants of CaMg3 in D03 phase. Using the calculated values of the elastic constants, other elastic parameters can be cal-culated suchas:bulkmodulus B,Voigt,ReussandHill shear moduli (GV ,GR ,GH ),Young’smodulus E,Poisson’s ratio ν, anisotropyfactorAandPugh’sratio(G/B)usingthefollowing equations [16],[28],[29]: B= C11 +2C12 3 (8) GV = C11 − C12 +3C44 5 , (9) GR= 5C44 (C11 − C12 ) 4C44 +3(C11 − C12 ) (10) GH= 1 2(GV +GR) (11) E = 9BG G+3B (12) ν = 3B− 2G 2(3B+G) (13) A= 2C44 C11 − C12 (14)

Our results of the elastic stiffness constants Cij and the

other elastic parameters above obtained for D03 phase of CaMg3 phaseusing both LDAand GGA approximationsare presented in Table 4. Except of two theoretical works based on DFT and MEAM, realized by Zhou and Gong [8] and Groh [11], respectively, to the best of our knowledge, there are no other theoretical or experimental values available for the elasticconstants of CaMg3 inD03 phase.

Alook on Table 4, shows that the calculated elastic con-stants Cij satisfied elastic stability criteria: (C11 - C12) > 0,

C11 >0,C44 >0,(C11+2C12)>0,C12 <B<C11 [30],[31]. At normal conditions, the mechanical stability of CaMg3 in D03 phase was also found in previous calculations reported by Zhouand Gong [8] and Groh [11].One can also notice, from Table 4, that the elastic constant C11 (related to the axial compressionalong the principal crystallographic direc-tions [32]) isslightly lower thanC44 ,indicating that CaMg3 compound inD03 phase presents a higher resistance topure shear deformation compared to the axial compression resis-tance.ThesamebehaviorwasalsonoticedbyZhouandGong

[8]usingthesameapproach(DFT)likeours,butnotobserved in the work of Groh [11], which used a different formal-ism (the second nearest-neighbors modified embedded-atom method(MEAM)).

Another note, from Table 4, that our values of the elas-tic stiffness constant Cij and other elastic moduli of CaMg3 compound in D03 structure are in excellent agreement with the results of Zhouand Gong [8] and Groh [11], exceptfor C44 , where for example, our value (29.77GPa) of the bulk

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modulus B obtained with GGA overestimates the theoretical value (29.57A)˚ reported by Zhou and Gong [8] by around 0.68%.

The Poisson’s ratio ν is small (usually v< 0.1)for cova-lent materials, while for ionic materials, v is 0.25 [33],[34]. Therefore,inCaMg3 compoundinD03structure(where v∼ 0.20),a higher ioniccontribution in an intra-atomicbonding isexpected.

Young’s modulus E is an important indicator on elastic-ity;materialshaving highervaluesofE,aremorestiffer.Our obtainedvalue of E for CaMg3 compound inD03 phasewas found at around 55.94GPa using LDA, and 53.39GPa us-ing GGA, respectively. These two values are slightly higher thantheYoungmodulus(41.43GPa)reportedbyDaoudetal.

[35]for MgCa inB2 phase.

On the other hand, the shear modulus G, which can be obtainedfrom the measureof resistance tothe reversible de-formation under the applied shearing stress, plays a domi-nantroleinpredictingthe hardnessof the material[36].The Pugh’s ratio G/B has been extensively used as an empirical parametertoexpressthe brittleness/ductility of materials[8]. ThecriticalvalueofG/Bratiothatseparatesthebrittle/ductile behavioris0.57(B/G=1.75);alargerG/Bvaluemeansmore brittleness,andviceversa[8].ItcanbeseenfromTable4that our valuesof G/B ratio obtained using bothLDA and GGA for CaMg3 compound in D03 structure are in good agree-ment with the results of Zhou and Gong [8]. The values of

G/Bratio are greater than0.57, indicatingthe brittleness na-tureof CaMg3 compoundinD03 phase.Thisconclusionwas also confirmed from the values of the Poisson’s ratio (v ∼ 0.20)whichissmallerthanthecritical valuev=0.26 [16].It should be noted that, if we use the elastic stiffness constant Cij obtained by Groh [11], a value of ∼ 0.48 for the G/B ratio willbe found.

Liu etal.[37]reportedthat the ductilityis ashear-related mechanicalpropertyofmaterial,itisassociatedwithboththe elasticconstantC44 andthe densityof states (DOS)atFermi energy, while Daoud et al. [35] showed an anti-correlation between the elastic constant C44 and the DOS at Fermi en-ergy in MgCa intermetallic compound under compression. Thisanti-correlationperhaps explainedbythefactthat asthe total DOS at Fermi level increases covalent/ionic behavior, graduallytransformedinto metallicbehavior,thusturningthe brittlephase intoa ductileone[35].

The Vickers hardness HV measurementis oneof themost

techniquesusedinthemechanicalcharacterizationofthe ma-terials [35]. Like refractive index and density, hardness is a intrinsicpropertyof thegivencrystal[36].TheVickers hard-ness HV, the bulk modulus B and the shear modulus G are

relatedbythe empirical formula [35]:

HV =0.92(G/B)1 .137 xG0.708 (15)

Using our values of the bulk modulus B and the shear modulusGobtainedfromthe LDAandGGA,the present re-sultsoftheVickershardnessHVforCaMg3 compoundinD03 phaseare: 5.80 and5.93GPa, respectively. These two values are slightly higher than the Vickers hardness HV (4.82GPa)

of MgCaintermetalliccompoundinB2phase[35].Asfar as we know,thereare nodataavailable relatedtoVickers hard-ness HV in the literature for CaMg3 in D03 phase, therefore ourcalculatedvaluescanbeconsideredas predictionforthis propertyfor this material.

3.4.2.Elasticwave speedsand Debye temperature

The Debye temperature θD parameter is related to many important physical properties of solids, such as specific heat andmelting temperature [32].It is either measured from the elasticconstants, orfromthe specificheatmeasurement[38]. However, at low temperatures bothmethods give almost the same value of θD, since at low temperature the vibrational excitations arise from acoustic modes only. The Debye tem-peratureθDmaybeestimatedfromtheaveragesoundvelocity

vm bythe followingequation [16]:

θD= h kB 3 4πVa 1 3 vm (16)

where hisPlank’sconstant,kB Boltzmann’sconstant,andVa

is the atomicvolume.

Usually, the average sound velocity vm of the

aggre-gate material can be calculated from the longitudinal (com-pressed) vl andtransverse (shear) vt sound velocities as

fol-lows[39],[40] vm= 1 3 2 v3 t + 1 v3 _l ₋1 3 (17) The crystaldensity ρ isusually expresseding/cm3 (or in kg/m3 ); itis givenas follow [41–43]

ρ =MZ/NAV (18)

where M is the molecular weight, usually expressed in 10−3 kg, Z is the number of molecules per unit cell, NA (= 6.022× 1023 mol−1 )istheAvogadro’snumber,whileVisthe unit cell volume usually expressed in m3 . For CaMg3 com-pound in D03 phase, the number of molecules per unit cell

Z was taken equal to four, and the unit cell volume V was takenequal toa3 ,while aislatticeconstant.Adachi[44]has mentioned that the lattice parametersare related tothe pres-surebyMurnaghanequationof state,andtheyare influenced bythe crystalline perfection, suchas:impurities, stoichiome-try, dislocations andsurface damage. Table 5 summarize the results of the crystaldensity ρ, the average elastic wave ve-locityvm,thelongitudinalwavevelocityvl andthe transverse

acoustic wave velocity vt as well as the Debye temperature θD of CaMg3 compound in D03 phase, which could not be compared duetounavailability of the measured data.

A look on Table 5, the Debye temperature θD of CaMg3 inD03 phasewas foundat around 393.44Kusing LDA,and 389.91K using GGA, respectively. Since the Debye temper-ature θD correlates with the Young’s modulus E in cubic perovskite-type RBRh3 (R are Sc, Y, La and Lu) materials [45],thesetwo valuesof θD arealso slightlyhigher thanthe Debye temperature θD (328.65K) reported by Daoud et al. [35]forMgCainB2phase.Althoughthisrationalizationmay

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Crystal density ρ, sound velocities vl,vt,vm Debye temperature θD and the limiting angular vibrational frequency ωD for CaMg 3 compound in D0 3

phase. Values with § are calculated using the data of Ref. [8], while those with ∗are calculated using the data of Ref. [11].

Parameter ρ (kg/m3₎ _v

t (m/s) vl (m/s) vm (m/s) θD (K) ωD(10 13rad/s)

This work (LDA) 1917.42 3470.48 5735.34 3835.99 393.44 5.93

This work (GGA) 1791.88 3521.67 5757.77 3888.34 389.91 5.88

Ref. [8] 1793.31 § _3503.9§ _5732.26§ _3868.97§ _388.07§ _5.85§

Ref. [11] 1783.28 ∗ 2854.61 ∗ 5269.92 ∗ 3185.47 ∗ 318.91 ∗ 4.81 ∗

be usefulfor chemicallyrelatedcompounds,compounds that are significantly different in chemical nature perhaps should not be necessaryexpected tofollow thesame correlation.

We have also calculated the Debye temperature θD for CaMg3 inD03 phaseusing asemi-empiricalformulabetween

θD and elastic constants Cij first proposed by Blackmann [46], and latter used by Siethoff and Ahlborn [47] after im-provements forseveralcrystalswithdifferentstructures. This semiempirical formula canbe written as [48]

θD=CB(aGB/M)1 /2 ,

GB =[C44 (C11 − C12 )(C11 − C12 +2C44 )]1 /3 , (19)

where a is the lattice constant, M is the atomic weight (for compounds, M is the weighted arithmetical average of the masses of the species), and CB=3.89× 1011 × n–1/6 h/kB

is a model parameter. In this model parameter, h (= 6.62617× 10−34J.s),kB (=1.38062× 10−23J.K− 1)andn are

Planck’s constant, Boltzmann’s constant and the number of atomsintheunitcell,respectively.Moredetailscanbefound in Refs.[48].

Using Eq. (19), the Debye temperature θD of CaMg3 in D03 phase was found at around 417.16K using LDA, and 408.91KusingGGA,respectively.ThesetwovaluesofθDare

slightlyhigher thanthe values393.44K(LDA)and389.91K (GGA) obtained from Eq. (16).

From Debyetemperature onecanestimate theDebye cut-off frequency(the limiting angularvibrationalfrequency) ωD

by the followingexpression[49]:

ωD=kB θD/¯h (20)

wherekB isBoltzmann’sconstant,andћ=h/2π,hisPlanck’s constant.

Substitutingthevalues(393.44K,and389.91K)ofthe De-bye temperature θD, obtained from the LDA and GGA, in

Eq. (20), the present results of the limiting angular vibra-tional frequency ωDfor CaMg3 compound inD03 phase are: 5.93× 1013 and 5.88× 1013 rad/s, respectively. These results as wellas thosecalculatedfromthedataofRefs.[8],[11]are also summarized inTable5.

The same as in the case of the elastic constants and the structural parameters, the longitudinal, transverse and aver-agesoundvelocitiesas wellasthe Debyetemperature ofour materialof interestareinexcellentagreement withthose cal-culated using the data of Ref. [8] obtained from the same approach (DFT). To the best of our knowledge, thereare no othertheoreticalorexperimentaldataexistingintheliterature onthe soundvelocity,Debyetemperature, andDebyecut-off

frequency for CaMg3 in D03 structure. So,we thinkthat our findingsregardingthesequantitiescanbeused topredict and explainmost of the physicalproperties of thismaterial.

4. Conclusion

In this work, we have investigated the equilibrium struc-turalparametersof CaMg3 compoundinαReO3 -type(D09 ), AlFe3 -type (D03 ), Cu3 Au -type (L12 ) and CuTi3 -type (L60 ) configurations using an ab-initio FP-LMTO method, within bothlocaldensityapproximation(LDA),andgeneralized gra-dientapproximation(GGA).Atequilibriumourresultsforthe EOS parameters, in general, agreed well with other data of the literature.

Theresults of the present workconcerned withthe possi-bility of phase transition at high pressure show that CaMg3 transforms from AlFe3 -type structure (D03 ) to αReO3 -type (D09 ) at pressure of around 29.96GPa using LDA, and at around 25.1GPa using the GGA.

Both LDA and GGA approaches for the electronic band structures, the totaldensity of states (TDOS) as well as the partial density of states (PDOS) showed that CaMg3 inD03 phasehas ametallic behavior.

The elastic constants, Young’s modulus, shear modulus, Poisson’s ratio, index of ductility, Vickers hardness, sound velocities,Debyetemperature,andthelimitingangular vibra-tionalfrequency of CaMg3 inD03 phase werealso reported. Ourfindingsontheelasticconstantswerealsoinagreedwell withothertheoretical data of the literature.

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