Physica B 407 (2012) 198–203

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Physica B

0921-45

doi:10.1

n Corr

of Resea

Tel.: þ9

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Theoretical investigation on first-principles electronic and thermalproperties of some CdRE intermetallics

Vipul Srivastava a,b,n, Afroj A. Khan c, M. Rajagopalan d,e, Sankar P. Sanyal b

a Department of Engineering Physics, NRI Institute of Research & Technology, Raisen Road, Bhopal 462 021, Indiab Department of Physics, Barkatullah University, Hoshangabad Road, Bhopal 462 026, Indiac Department of Engineering Physics, Sagar Institute of Science Technology & Engineering, Ratibad, Bhopal 462044, Indiad Crystal Growth Centre, Anna University, Chennai 600 025, Indiae Max-Plank Institute, Fur Festkorperforschung D-70569 Stuttgart, Germany

a r t i c l e i n f o

Article history:

Received 4 August 2011

Received in revised form

11 October 2011

Accepted 12 October 2011Available online 28 October 2011

Keywords:

TB–LMTO

Inter-metallic compounds

Electronic structure

Thermal properties

26/$ - see front matter Crown Copyright & 2

016/j.physb.2011.10.025

esponding author at: Department of Enginee

rch & Technology, Raisen Road, Bhopal, 462

1 755 4224989; fax: þ91 755 2491823.

ail address: vips73@yahoo.com (V. Srivastava

a b s t r a c t

Ab initio calculation on B2-cadmium rare earth (RE), CdRE (RE¼La, Ce and Pr) intermetallics has been

performed at T¼0 K with respect to their structural, electronic and thermal properties. The structural

and electronic properties are derived using self-consistent tight binding linear muffin tin orbital

method at ambient and at high pressure. Other properties like lattice parameter, bulk modulus, density

of states, electronic specific heat coefficient, cohesive energy, heat of formation, Debye temperature and

Gruneisen constant for CdRE are also estimated. The RE–f effect can be seen in CdPr in terms of

variation in the density of states and opens a possibility of structural instability. A pressure induced

variation of Debye temperature is also presented for three cadmium rare earth intermetallics.

Crown Copyright & 2011 Published by Elsevier B.V. All rights reserved.

1. Introduction

Recently, the AB intermetallics and alloys [1–7] have beenconsidered for their significant application in material design due totheir ductile, high tensile strength and thermal properties. The ABintermetallics, where ‘A’ is ‘metal’ and ‘B’ is either ‘rare earth’ (RE) or‘transition metal’ (TM), are formed as unique class of materials.Particularly, Aluminum with RE and TM systems has received a greatattention due to its application in metallic glass technology [6,7]. TheCdRE (RE¼La, Ce and Pr) intermetallics crystallize in cubic cesiumchloride structure (B2–phase, Pm3m, Space Group, 221) [8,9]. In the B2CdRE intermetallics f–electrons present in RE specie, play a crucialrole in exploring structural and electronic properties. Therefore, itbecomes important to understand the electronic properties of thisclass of intermetallics. To the best of our knowledge the structuraland electronic properties of CdRE intemetallics at normal as well as athigh pressure have not been extensively studied and those are yet tobe explored in detail. However, their phase diagram study is available[10]. The experimental thermodynamical investigations of Cd–Lasystem is discussed in Ref. [11]. Our interest in B2 CdRE is motivatedby scarcity of the work and unique thermal and mechanical proper-ties of similar intermetallics [12–14] due to the presence of f–electronof RE element. Under the application of pressure these RE-f states in

011 Published by Elsevier B.V. All

ring Physics, NRI Institute

021, India.

).

the intermetallics may get delocalized and give further possibility ofphase change. The systematic knowledge of the basic thermodynamicproperties of B2-phase CdRE intermetallics is important to under-stand vibrations in lattice. We have, therefore, explored the structural,electronic and thermal properties of CdRE intermetallics undernormal condition and compression.

In the present paper, the ground state electronic and thermalproperties of B2-type CdRE are investigated by employing the non-spin polarized first-principles tight binding linear muffin tin orbital(TBLMTO) method (within ASA and LDA) and Debye–Gruneisen (DG)model. The calculated lattice parameters, bulk modulus, electronicspecific heat coefficient, cohesive energy, heat of formation, Debyetemperature and Gruneisen constants are presented and open a scopefor experimentalists and other theoreticians for further investigations.The organization of paper is as follows. Section 2 deals with the first-principles computational details for describing electronic propertiesand description of DG model to calculate Debye temperature.The results with discussion on structural, electronic, thermal proper-ties for all CdRE intermetallics are presented in Section 3. Section 4deals with the concluding remarks of the present work.

2. Method of calculations

2.1. Computational details

The total energy, band structure and density of states for CdRE arecalculated in a manner similar to our previous work [12,13] using the

rights reserved.

-0.94

-0.9

-0.86

-0.82

-0.78

0.65 0.75 0.85 0.95 1.05

CdLa

V/V0

0.65 0.75 0.85 0.95 1.05

V/V0

Tot

al e

nerg

y in

Ry

(-28

156.

0+)/

f.u.

-1.04

-1

-0.96

-0.92

-0.88

0.65 0.75 0.85 0.95 1.05

CdCe

V/V0

Tot

al e

nerg

y in

Ry

(-28

890.

0 +)

/f.u

.

-0.44

-0.4

-0.36

-0.32

-0.28

CdPr

Tot

al e

nerg

y in

Ry

(-29

644.

0+)/

f.u.

Fig. 1. Variation of total energy as a function of relative volume for CdRE (RE¼La, Ce and Pr) intermetallics.

V. Srivastava et al. / Physica B 407 (2012) 198–203 199

TB–LMTO method [15,16] within the local-density approximation(LDA) [17]. The von–Barth and Hedin [18] parameterization schemewas used for exchange correlation potential. The CdRE intermetallicscrystallize in the B2-type structure, and are positioned at Cd: (0, 0, 0 )and RE: (0.5, 0.5, 0.5). The Wigner–Seitz sphere was chosen in such away that the sphere boundary potential is minimum and the chargeflow between the atoms is in accordance with the electro-negativitycriteria. The E and k convergence was consequently checked. Thetetrahedron method [19] of Brillouin zone integration was used tocalculate the density of states. The total energy was computed byreducing the volume from 1.05 V0 to 0.65 V0, where V0 is theequilibrium cell volume. The calculated total energy was fitted tothe Birch equation of state [20] to obtain the pressure volumerelation. The pressure was obtained by taking volume derivative ofthe total energy. The bulk modulus, B¼�V0 dP/dV was also calcu-lated from the P–V relation.

2.2. Debye–Gruneisen model

To calculate thermal properties of a vibrating Debye latticewe used the Debye–Gruneisen (DG) model [21]. The Debye

temperature YD is calculated using the following expression

YD ¼ 67:48

ffiffiffiffiffirB

M

rð1Þ

where r is the Wigner radius (in atomic unit), B is bulk modulus(in kbar) and M is the average atomic weight, which is theweighted arithmetical average of the masses of the species forcompounds. The calculated value of (YD)0 using the aboveexpression, at r¼r0 (at equilibrium) and substituting the calcu-lated (or experimental) value of bulk modulus deviates from theexperimental value to a larger from the experimental value forbinary compounds. To overcome this situation a scaling factor hasbeen introduced. In this modified expression, the theoreticalvalue of r at r¼r0 and bulk modulus derived from the abovefirst-principle calculation at r0 are substituted and the Debyetemperature (YD)0 can be calculated using the expression

YD ¼ 41:63

ffiffiffiffiffiffiffiffir0B

M

rð2Þ

and the Gruneisen constant g0 is calculated using

g0 ¼@lnYD

@lnVð3Þ

V. Srivastava et al. / Physica B 407 (2012) 198–203200

where V is the volume of the solid. More details about theformulation can be obtained from Refs. [6,7].

3. Results and discussion

3.1. Lattice parameter and bulk modulus

The total energy of CdRE intermetallics is estimated using thefirst-principle TB-LMTO method. The total energy is plotted fordifferent compressions and shown in Fig. 1. The minimum of all thecurves defines the equilibrium volume V0, which is found to be56.11 A3 for CdLa, corresponding to the lattice parameter 3.83 A,which is underestimated by 2% with the Wycoff value [8,9]. Similarly,for CdCe and CdPr intermetallics the estimated equilibrium volumesare 51.53 and 50.81 A3 with the corresponding lattice parameters as3.72 and 3.70 A, respectively. These results are tabulated in Table 1.The present calculated values of lattice parameters are in goodagreement with the Wycoff values [9], although the reported valuesof lattice parameters of CdRE intermetallics are slightly higher ascompared to our results. This is due to usage of LDA [17] in thepresent calculation. Regarding the LDA contraction, it is often foundthat the LDA leads to some overbinding that yields lattice parametersthat are somewhat smaller when compared with experiment.

The bulk modulus for CdLa, CdCe and CdPr are calculated to be64.77, 65.25 and 57.46 GPa, respectively, and are given in Table 1.These values could not be compared with the experimental valuesfor the reason that no study is reported so far.

Fig. 2. Band structure along the high symmetry directions at ambient pressure for

CdRE intermetallics.

3.2. Band structure and density of states

The band structures along the high symmetry directions for allthe CdRE intermetallics are presented in Fig. 2. For CdLa, thelowest energy bands are due to ‘Cd-d’ states and at Fermi levelsome of the ‘La-d’ states can be seen, which hybridize with ‘Cd-p’states near the Fermi level. However, small amount of ‘La-f’ states(fractional values are presented in Table 4) of RE can be seenabove Fermi level. Similar BS is obtained for CdCe and CdPr butwith clear picture of RE–f (RE¼Ce and Pr) states. The f–states ofCe and Pr can be seen at Femi level and hybridize with ‘Cd–p’states, which shows increased metallic character. To confirm thiswe have calculated partial DOS along with the total DOS at Fermilevel (Ef) for all the CdRE intermetallics under ambient conditionsand depicted in Fig. 3. A sharp localized peak of f–states of Ce andPr can be seen at the Femi level, which hybridize with ‘Cd–p’states. From the present calculations we have found that numberof states increases as we go from –La to –Pr. In Fig. 4 we haveplotted total density of states under compression for all the CdRE

Table 1Calculated ground state properties: Lattice constant a0, Bulk modulus B0, Density of states N(Ef), Electronic specific heat cofficient g Cohesive energy Ec, Heat of formation

-DH Wigner radius r0, Debye Temperature YD (K) and Gruneisen constants g0, of CdRE in B2 phase.

Solid a0 B0 N(Ef) g Ec �DH r0 YD g0

(a.u.) (GPa) (states/Ry/f.u.) (mJ/mol K2) (Ry) (kcal/mol) (A) (K)

CdLa 7.232 64.77 42.64 7.38 1.17 83.07 1.797 197.25 1.013

7.370a

CdCe 7.032 65.25 69.69 12.07 1.20 85.20 1.753 208.59 1.025

7.294a

CdPr 6.999 57.46 625.43 107.94 1.21 85.91 1.764 194.21 1.009

7.219a

a Ref. [9].

Fig. 3. Partial density of states at Fermi level for CdRE intermetallics.

25

30

35

40

45

CdLa

V/V0

Den

sity

of

Stat

es (

Stat

es/R

y.ce

ll)

10

30

50

70

90

CdCe

V/V0

Den

sity

of

Stat

es (

Stat

es/R

y.ce

ll)

100

200

300

400

500

600

700

CdPr

V/V0

Den

sity

of

Stat

es (

Stat

es/R

y. c

ell)

1.050.950.850.750.651.050.950.850.750.651.050.950.850.750.65

Fig. 4. Variation of density of states at Fermi level with relative volume for CdRE (RE¼La, Ce and Pr) intermetallics.

V. Srivastava et al. / Physica B 407 (2012) 198–203 201

intermetallics. From this figure one can notice a linear decrease inDOS at Ef as we compress these three intermetallics.

The partial number of electrons and total density of states arealso calculated as functions of cell volume and presented in Tables2–4. The value of DOS for CdLa decreases from 44 to 30.4 states/Rycell at compression value of V/V0¼1.05–0.75. It suddenlyincreases to 36.2 states/Rycell at V/V0¼0.70. In the case of CdCeonly, decrease in DOS is noticed from 97.2 to 20.2 states/Rycellbetween compression value of V/V0¼1.05 and 0.65. As far as suchvariation for CdPr is concerned, an increase in DOS is noticed from451.4 to 646.5 states/Rycell at V/V0¼1.05–0.90 and then adecrease to 200 states/Rycell. Such an interesting feature couldbe the indication of structural instability under compression. Thevariation of DOS under compression is strongly related to the f-

states of rare earth atom. To understand such variation of DOSunder compression in CdLa and CdPr, we have estimated the

electronic DOS of f states. In Fig. 5 we have plotted them for thecase of CdPr. One can see the fluctuations in Pr-f states undercompression. Due to compression the f-like electrons may getdelocalized. The delocalization of the Pr-f-like states could be thereason of variation in total DOS at Ef under compression.

3.3. Electronic specific heat coefficient, cohesive energy, heat of

formation and Debye temperature

From the density of states the electronic specific heat coeffi-cient is calculated using the expression:

g¼p2K2

BNðEf Þ

3The values are tabulated in Table 1. An inspection of Table 1

reveals that electronic specific heat coefficient increases as

Table 2Partial number of electrons and total density of states for CdLa in CsCl structure.

Partial number of electrons Partial number of electrons DOS

Cd La (States/Ry-cell)

V/V0 s p d f s p d f

1.05 1.136 0.932 9.853 0.011 0.537 0.542 1.722 0.263 43.83

1.00 1.125 0.935 9.841 0.012 0.527 0.537 1.752 0.268 43.44

0.95 1.115 0.940 9.829 0.015 0.514 0.530 1.781 0.273 41.71

0.90 1.102 0.938 9.813 0.017 0.502 0.526 1.816 0.282 39.10

0.85 1.090 0.939 9.797 0.021 0.487 0.523 1.849 0.291 36.16

0.80 1.074 0.935 9.777 0.025 0.472 0.523 1.887 0.303 33.19

0.75 1.056 0.928 9.754 0.030 0.455 0.527 1.928 0.318 30.36

0.70 1.036 0.898 9.727 0.038 0.438 0.537 1.991 0.332 36.21

0.65 1.002 0.868 9.694 0.049 0.410 0.551 2.068 0.353 28.21

Table 3Partial number of electrons and total density of states for CdCe in CsCl structure.

Partial number of electrons Partial number of electrons DOS

Cd Ce (States/Ry-cell)

V/V0 s p d f s p d f

1.05 1.120 0.947 9.829 0.015 0.528 0.526 1.784 1.248 97.26

1.00 1.110 0.952 9.817 0.017 0.516 0.522 1.835 1.227 78.96

0.95 1.098 0.953 9.803 0.020 0.504 0.519 1.891 1.208 67.54

0.90 1.085 0.953 9.787 0.024 0.491 0.517 1.950 1.189 60.88

0.85 1.071 0.943 9.766 0.032 0.477 0.512 2.033 1.162 55.51

0.80 1.057 0.940 9.747 0.037 0.464 0.510 2.090 1.151 29.61

0.75 1.038 0.933 9.721 0.045 0.448 0.515 2.153 1.143 25.59

0.70 1.016 0.920 9.689 0.056 0.431 0.523 2.221 1.140 22.94

0.65 0.989 0.914 9.654 0.067 0.411 0.538 2.272 1.151 20.56

Table 4Partial number of electrons and total density of states for CdPr in CsCl structure.

Partial number of electrons Partial number of electrons DOS

Cd Pr (States Ry-cell)

V/V0 s p d f s p d f

1.05 1.152 0.998 9.841 0.013 0.544 0.513 1.506 2.429 451.43

1.00 1.141 1.002 9.830 0.015 0.531 0.507 1.563 2.406 471.06

0.95 1.131 1.002 9.817 0.020 0.517 0.498 1.632 2.378 606.96

0.90 1.111 0.981 9.795 0.025 0.508 0.497 1.724 2.356 646.53

0.85 1.109 0.996 9.785 0.033 0.484 0.484 1.783 2.322 471.61

0.80 1.094 0.989 9.766 0.040 0.468 0.482 1.861 2.296 332.06

0.75 1.078 0.980 9.744 0.050 0.449 0.478 1.944 2.273 226.95

0.70 1.057 0.975 9.717 0.061 0.430 0.483 2.015 2.258 200.10

0.65 1.031 0.968 9.686 0.072 0.410 0.494 2.086 2.250 192.69

Fig. 5. Variation of Pr–f states under compression in CdPr.

V. Srivastava et al. / Physica B 407 (2012) 198–203202

one goes from La to Pr in this series and it is calculated forCdPr to be 107.94 mJ K�2 mol�1. One can understand thereason for the high value of electronic specific heat coefficientfor CdPr as DOS is calculated as 625.43 states/Ry cell for CdPr.Since there are no experimental observations available to thebest of our knowledge, a comparison is not possible in thetables now.

Further, cohesive energy is the measure of the strength of theforcesthat bind the atoms together in the solid state. In thisconnection, the cohesive energy of all CdRE intermetallics in B2

phase is estimated. The cohesive energy (Ecoh) in B2 phase isdefined as the difference in the total energy of the constituentatoms at infinite separation and total energy of that particular

phase, such that

EABcoh ¼ ½E

AatomþEB

atom�EABtotal�where A¼ Cd; B¼ RE

where EABatom is the total energy of compound CdRE at equilibrium

lattice constant and EAatom and EB

atom are atomic energies of thepure constituents. In order to study the stability of phase theformation energy is also calculated. To determine the heat offormation we first calculated the total energy of elemental Cd,and La, Ce, Pr corresponding to their respective equilibrium latticeparameters. At zero temperature, there is no entropy contribution

1500 100 200 300 400 500

175

200

225

250

CdLa

CdCe

CdPr

Deb

ye T

empe

ratu

re (

K)

Pressure (Kbar)

Fig. 6. Variation of Debye temperature as a function of pressure for CdRE

intermetallics.

V. Srivastava et al. / Physica B 407 (2012) 198–203 203

to the free energy; therefore, the heat of formation can beobtained from the following relation:

EABf orm ¼ ½E

ABtotal�ðE

AsolidþEB

solid�

where EABtotal refers to the total energy of CdRE (RE¼La, Ce and Pr)

intermetallic at equilibrium lattice constant and EAsolid, EB

solid aretotal energies of the pure elemental constituents. The calculatedvalues of cohesive energies Ecoh and heat of formation (DH) aregiven in Table 1 for all CdRE intermetallics. However, a compar-ison is not possible for the want of experimental data.

Many physical properties like specific heat, melting temperature,thermal conductivity and vibrational properties of solids can beunderstood by considering the Debye temperature of solids. Wehave, therefore, calculated the Debye temperature and Gruneisenconstant for three intermetallics using the DG model [21]. In thismethod, the Debye temperature YD can be obtained from thecalculated values of bulk modulus and are presented in Table 1 forall the three CdRE intermetallics. The variation of YD with respect topressure for all the CdRE intermetallics is presented in Fig. 6, whichshows that as pressure increases YD also increases for CdLa, CdCeand CdPr. The Gruneisen constant g0 can be obtained from Eq.(3) discussed in Section 2.2. The bulk modulus (B0), Wigner–Seitzradii (r0), Debye temperature (YD)0 and Gruneisen constants (g0) atabsolute temperature are calculated for all CdRE intermetallics. In thepresent paper most data for CdRE intermetallics are presented for thefirst time and open a scope for future experimental verification.

4. Conclusion

The CdRE intermetallics have been investigated theoreticallywith respect to their electronic aspect such as band structure,

density of states, partial number of electrons and other ground stateproperties like equilibrium lattice parameter, bulk modulus, heat offormation, electronic specific heat coefficient, cohesive energy,Debye temperature and Gruneisen constants in B2 phase. It is foundfrom the present study that CdRE intermetallics are metallic innature and metallicity increases from La to Pr. A variation in DOSnumbers is noticed in CdLa and CdPr under compression, which maybe due to the delocalization of ‘f’ electrons under pressure. On theother hand, we have used the Debye–Gruneisen model to calculatethe Debye temperature and Gruneisen constant using the bulkmodulus and the sphere radius obtained from the TB–LMTOmethod. All CdRE (RE¼La, Ce and Pr) show linear increase in YD

under high pressure. The aim of this study was to derive technolo-gically important material parameters, which could not be obtainedfrom experimental measurements so far.

Acknowledgments

VS is thankful to Madhya Pradesh Council of Science andTechnology (MPCST), Bhopal, India for the award of researchproject (MPCST Project no. 1904/CST/R&D/2009) and for financialsupport. Also thankful to Shri D. Subodh Singh and Dr. B. B. Saxena,NRI Group of Institutions, Bhopal, India, for their encouragement.Also thankful to Mr. Chandra Shekhar, I.I.Sc., Bangalore, Indiafor providing valuable help in exploring research articles. SPSgratefully acknowledges Council of Science and Industrial Research(CSIR), New Delhi, India. MR is thankful to CSIR, New Delhi, India,for the award of emeritus Professor.

References

[1] X. Tao, Y. Ouyang, H. Liu, Y. Feng, Y. Du, Z. Lin, Solid State Commun. 148 (2008)314.

[2] W. Hu, H. Xu, X. Shu, X. Yuan, B. Gao, B. Zhang, J. Phys. D 33 (2000) 711.[3] K.A. Gschneidner Jr., Min Ji, C.Z. Wang, K.M. Ho, A.M. Russell, Ya. Mudryk,

A.T. Becker, J.L. Larson, Acta Mater. 57 (2009) 5876.[4] K. Gschneidner Jr., A. Russell, A. Pecharsky, J. Morris, Z. Zhang, T. Lograsso,

D. Hsu, C.H. Chester Lo, Y. Ye, A. Slager, D. Kesse, Nat. Mater 2 (2003) 587.[5] M. Divi, J. Kuriplach, Physica B 205 (1995) 353.[6] X. Tao, Y. Ouyang, H. Liu, F. Zeng, Y. Feng, Z. Jin, Physica B 399 (2007) 27.[7] X. Tao, Y. Ouyang, H. Liu, F. Zeng, Y. Feng, Z. Jin, Comp. Mater. Sci 40 (2007)

226 (and references therein).[8] P. Villars, L.D. Calvert, Pearson’s Handbook of Crystallographic Data for

Intermetallic Phases, vols. 1-4, ASM International, Materials Park, OH, 1991.[9] R.W.G. Wyckoff, 2nd edn.,Cryst. Struct., Vols. 1-2, Inter-science, 1963 (1965).

[10] K.A. Gschneidner, F.W. Calderwood, J. of Phase Equilib. 9 (1988) 32 9(1988)21; 9 (1988) 130.

[11] Klaus W. Richter, Sonia Besana, Gabriella Borzone, Herbert Ipser, J. AlloysCompd. 365 (2004) 181.

[12] G. Pagare, V. Srivastava, S.P. Sanyal, M. Rajagopalan, Physica B 406 (2011) 449.[13] V. Srivastava, G. Pagare, S.P. Sanyal, M. Rajagopalan, Phys. Status Solidi B 6

(2009) 1206.[14] G. Pagare, S. Chohan, S.P. Sanyal, M. Rajagopalan, Physica B 406 (2011) 339.[15] O.K. Andersen, Phys. Rev. B 12 (1975) 3060.[16] O.K. Andersen, O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571.[17] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133.[18] U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629.[19] O. Jepsen, O.K. Andersen, Solid State Commun. 9 (1971) 1763.[20] F. Birch, J. Geophys. Rev. 83 (1978) 1257.[21] P. Debye, Ann. Phys. 39 (1912) 786.