Journal of Physics and Chemistry of Solids 73 (2012) 873–880

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Journal of Physics and Chemistry of Solids

0022-36

doi:10.1

n Corr

P. G. Co

fax: þ9

E-m

sps.phy1 Te

journal homepage: www.elsevier.com/locate/jpcs

A comparative study of electronic structure and bonding in transitionmetal monocarbides

Pooja Soni a,b, Gitanjali Pagare c,b,n, Sankar P. Sanyal b,1, M. Rajagopalan d

a Department of Physics, Institute for Excellence in Higher Education, Bhopal 462016, Indiab Condensed Matter Physics Laboratory, Department of Physics, Barkatullah University, Bhopal 462026, Indiac Department of Physics, Government M. L. B. Girls P. G. College, Bhopal 462002, Indiad Crystal Growth Centre, Anna University, Chennai 600025, India

a r t i c l e i n f o

Article history:

Received 3 May 2011

Received in revised form

28 December 2011

Accepted 14 February 2012Available online 1 March 2012

Keywords:

C. Ab initio calculations

D. Elastic properties

D. Electronic structure

D. Thermodynamic properties

97/$ - see front matter & 2012 Elsevier Ltd. A

016/j.jpcs.2012.02.016

esponding author at: Department of Physics

llege, Bhopal 462002, India. Tel.: þ91 755 25

1 755 2661783.

ail addresses: gita_pagare@yahoo.co.in (G. Pag

sicsbu@gmail.com (S.P. Sanyal).

l.: þ91 755 4224989; fax: þ91 755 2491823

a b s t r a c t

The structural, electronic, elastic and bonding properties of four transition metal carbides, ScC, YC

(group III), VC and NbC (group V), have been investigated systematically using the first principles

density functional theory (DFT). The full potential linearized augmented plane wave (FP-LAPW) method

with the generalized gradient approximation (GGA) for the exchange correlation has been used for the

calculation of the total energy. The ground state properties, such as equilibrium lattice constant, bulk

modulus, are computed and compared with theoretical and experimental data. The electronic and

bonding patterns of the two groups of compounds have been analyzed quantitatively and compared

with the available data. It is clear from band structures that all the four transition metal monocarbides

are metallic in nature. Analysis of elastic constants reveals that the carbides of group III are ductile in

nature while those of group V are brittle.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The transition metal carbides (TMCs) and nitrides (TMNs)remained an interesting subject, both in condensed matterphysics, for basic research and materials science from an applica-tion point of view. This is mainly because of their intrinsicproperties they inherited due to the occupancy of d-electrons ofthe transition metal ions. Most of the binary TMX (X¼C, N)compounds crystallize in the NaCl-type structure. The TMNs, infact, have been extensively studied both theoretically and experi-mentally, and are understood fairly well. The TMCs, on the otherhand, have remained relatively less explored. Both classes ofcompounds are predominantly metallic in nature having out-standing physical properties such as high melting point, ultra-hardness, metallic conductivity, chemical stability and highcorrosion resistivity [1–5]. These unusual properties indicate thatthe nature of bonding between TM and X(C, N) atoms is strongand largely covalent. These compounds normally exist in sub-stoichiometric phases with substantial amount of vacancies.Nonmetal deficiency is known to be a major source of the

ll rights reserved.

, Government M. L. B. Girls

60462;

are),

.

nonstoichiometry. The control of the stoichiometry and theunderstanding of its effect on mechanical properties are crucialto the design of hard materials based on these compounds[2,6–8]. Most of the unusual features of the TMCs, in particular,depend on the occupation number of d-electrons in the TM ion. Itis, therefore, interesting to study how the d-electrons (in terms ofoccupancy) influence the electronic properties, nature of bonding,and other macroscopic properties of the TMC compounds of thesame group (or same period).

The electronic structure and bonding properties of TMCcompounds have been studied theoretically by several authorsin the recent past [9–11], using TB-LMTO [12] and the plane wavepseudo-potential method [13,14] within the framework of thedensity functional theory (DFT). Most of these studies reveal acommon feature namely the interactions of TM-d and C-p statesstrongly influence the electronic structure and the nature ofbonding. The aim of the present paper is, therefore, to elucidatethis aspect by considering two groups of TMCs, one having onlyone d-electron (namely ScC and YC) belonging to group III and theother having 3, 4d-electrons (VC and NbC where V has 3d and Nbhas 4d-electrons) belonging to group V. Though such a choice isnot unique, nevertheless, a comparative study of two suchsystems will quantitatively reveal the role of the number ofd-electrons on various features by their electronic properties.We also analyze their elastic properties for a comparative under-standing of the nature of bonding and ductile behavior. Thecomparison between theory and experiment is complicated by

P. Soni et al. / Journal of Physics and Chemistry of Solids 73 (2012) 873–880874

the fact that the stoichiometry i.e. the composition is not perfectin the real crystals, for example, presence of defects, which isignored in the present calculation. The details of the computa-tional method will be presented in Section 2 and the results anddiscussion on ground state properties, band structures and otheranalysis on electronic and bonding properties will be presented inSection 3.

2. Method of calculation

The total energy, ground state properties and electronic bandstructures of the four TMC compounds have been computed withthe density functional theory using the full potential linearizedaugmented plane wave (FP-LAPW) method [15]. The generalizedgradient approximation (GGA) of Perdew, Burke and Ernzerhof (PBE)[16] has been used for the exchange and correlation effects. Theenergy eigenvalue convergence has been achieved by expanding thebasis function up to RMTKmax¼7, where RMT is the smallest atomicsphere radius in the unit cell and Kmax gives the magnitude of thelargest K vector in the plane wave expansion. Muffin-tin (MT)sphere radii (RMT) of 2.1–1.8 and 1.7 Bohr were used for thetransition metal and C atoms, respectively. The semi-core orbitalswere considered as valence states for all TMs. The valence wave

Fig. 1. Variation of the total energy with volu

functions inside the spheres are expanded up to lmax¼10 while thecharge density is Fourier expanded up to Gmax¼12. The self-consistent calculations are considered to be converged when thetotal energy of the system is stable within 10�4 Ry. A mesh of5000 k points is used and the tetrahedral method [17] has beenemployed for the Brillouin zone integration. The total energies arefitted to Birch’s equation of state [18] to obtain the ground stateproperties.

3. Results and discussion

The ground state properties, such as equilibrium lattice con-stant (a0), bulk modulus (B) and its pressure derivative (B0) of allfour TMC (TM¼Sc, Y, V and Nb) compounds have been calculatedin their B1 structure by minimizing the total energy with respectto volume, as shown in Fig. 1. The calculated values of theseproperties are presented in Table 1 and compared with theirexperimental [1–3,19,20] and other theoretical [9,10,12,14,21]results. The calculated values of lattice constants are in goodagreement with the measured values [1–3,19,20] and differ byless than 1% only. Our results are similar to earlier theoreticalresults, cited in the same table, except the one reported inRef. [12] which used LDA for the calculation of the total energy.

me: (a) ScC, (b) YC, (c) VC and (d) NbC.

Table 1Calculated ground state properties of TMCs (TM¼Sc, Y, V and Nb).

Solids a0 (A) B (GPa) B0 N(EF)

(states/Ry/cell)

Total energy

(Ry)

ScC Pre. 4.682 154.81 4.18 19.6 �1604.684

Expt. 4.637a – – – –

4.720b – – – –

O. Theo. 4.684c 148c – 19.9j�1600.896j

4.680d 153d – – –

4.405e 173.6e – – –

YC Pre. 5.086 124.18 4.14 22.72 �6847.634

Expt. 5.11b – – – –

O. Theo. 5.08d 128d – 23.48f –

5.076f – – – –

VC Pre. 4.156 320.67 4.13 15.7 �1974.894

Expt. 4.159b 308–390i – – –

4.163g – – – –

O. Theo. 4.164c 290c – 15.8j�1970.736j

4.154d 304d – – –

3.942e 360.5e – – –

NbC Pre. 4.488 300.69 4.24 10.13 �7717.234

Expt. 4.454g 296–330i – – –

4.471h 331h – – –

O. Theo. 4.492c 293c – – –

4.476d 301d – – –

Pre.¼present, Expt.¼experimental, O. Theo.¼other theoretical.a Ref. [2].b Ref. [3].c Ref. [21].d Ref. [14].e Ref. [12].f Ref. [10].g Ref. [20].h Ref. [19].i Ref. [1].j Ref. [9].

P. Soni et al. / Journal of Physics and Chemistry of Solids 73 (2012) 873–880 875

The calculated values of bulk modulus for ScC and YC could not becompared due to the lack of experimental data. Nevertheless,these values are comparable to those reported by Isaev et al.[14] and Vojvodic and Ruberto [21] using the pseudo-potentialmethod and GGA. In the case of VC and NbC, our calculated valuesof the bulk modulus are in good agreement with the range ofmeasured data [1] and other theoretical results [12,14,21]. As inthe case of other binary compounds, the calculated values of B0

are close to 4.0, and the measured or theoretical values are notavailable for comparison. Also the computed values of latticeenergies for ScC and VC (listed in Table 1) agree excellently withthose reported by Haglund et al. [9].

The self-consistent energy band structures along the highsymmetry directions are shown in Fig. 2. The overall band profilesof ScC–YC and VC–NbC are very similar. As we go along the sameperiod (i.e. from Sc to V and from Y to Nb), the number of valenceelectrons increase and the position of TM-d bands shift towardsthe lower energy side with respect to the Fermi level (EF). In thevalence region, for both groups of carbides, the C-s states aresituated at the lower energy side and do not contribute much tothe bonding. An examination of the band structure of ScC (Fig. 2a)reveals that at the G-point the Sc-d orbitals split into two parts:one with a two fold degenerate state (G12) and the other with athree fold degenerate state (G250). The three fold degenerate state(G15) which lie below (G250) state contains the substantial con-tribution from C-2p orbitals. Similar features are also observed inthe other three TMCs considered in the present study, exceptthat in the case of ScC and YC, the ‘d’ band maximum for valenceelectrons occurs above the Fermi level (Fig. 2a and b) while in thecase of VC and NbC it is below the Fermi level (Fig. 2c and d).

The total and partial density of states (DOS) of the four TMCsare plotted and presented in Fig. 3. For ScC, the C-s states aresituated at lower energies with a very small amount of TM-d

states. The states just below the Fermi level (EF), are dominatedby strongly hybridized TM-d and C-p states. The states aboveFermi level (EF) are contributed by partially filled or antibondingstates, dominated by TM-d states with a clear contribution fromC-p states. For YC, the DOS is very similar to that for ScC. One ofthe interesting features that emerges from Fig. 3 is that for groupIII TMCs (namely ScC and YC), the Fermi level (EF) resides belowthe pseudo-gap while in the case of group V TMCs (namely VCand NbC), the EF lies above the pseudo-gap. It is due to the furtherfilling up of ‘d’ orbitals. The density of states at the Fermi level(N(EF)) is presented in Table 1 and compared with other theore-tical results [9,10]. It is clear from Table 1 that the value of N(EF) isgreater for group III TMCs than for group V TMCs, which isconsistent with the findings reported in other theoretical results[9] (see Table 1). We can also correlate this trend of variation inN(EF) to the facts that for group III TMCs, the Fermi level (EF)crosses the strong bonding states whereas for group V TMCs, EF

crosses the weak antibonding or unoccupied states (i.e. below EF

they are occupied and above partially filled or unoccupied) whichis originated mainly from the hybridization of TM-d and C-p

states. It is easily understood that this difference is deduced fromthe different numbers of valence electrons. It is interesting topoint out and can clearly be seen in Fig. 3 that for group III TMCs(ScC and YC), the major contribution to N(EF) comes from the C-p

states, but for group V TMCs (VC and NbC), TM-d states contributemostly.

We have plotted the Fermi surfaces (FS) (3D plots) from thebands which cross EF and shown them in Figs. 4 and 5 for groupsIII and V TMCs, respectively. We can see the similar featuresfor both the compounds of group III (ScC, YC) and group V (VC,NbC). As we have seen in Fig. 2a and b, three bands (namely bands6, 7 and 8) are crossing the Fermi level (EF) for group IIITMCs. These three bands give rise to three FS sheets as depictedin Fig. 4. For ScC, filled bonding states cross EF. So for band 6the filled electrons states lie inside the Brillouin zone aroundthe center point G. The energy range for band 6 is 0.23–0.65 Ry.The energy ranges for bands 7 and 8 of ScC are 0.38–0.65 Ry and0.42–0.65 Ry, respectively. Similar behavior is observed for YC.The energy ranges are 0.31–0.65 Ry for band 6, 0.46–0.65 Ry forband 7 and 0.47–0.65 Ry for band 8 of YC. Similarly, for VC andNbC (Fig. 2c and d), only two bands (namely bands 9 and 10) arecrossing the Fermi level EF. The FS from these two bands arepresented in Fig. 5 where the outer sheet of FS is formed by holepockets and the inner sheet of FS is from electron pockets. Theenergy ranges are 0.84–1.05 Ry and 0.89–1.20 Ry for band 9 ofVC and NbC, respectively. For band 10, the energy ranges are0.91–1.09 Ry and 0.99–1.27 Ry of VC and NbC, respectively. Wecan easily observe the nesting feature of FS for both groups ofTMCs (Fig. 6 for ScC and VC), where the inner electron sheet of FSis parallel to the outer hole sheet of FS. A common feature that wehave observed from the band structure and FS is a metallicbehavior for all the studied TMCs.

The electronic charge distribution from the metal and non-metal atomic spheres and from the interstitial region are pre-sented in Fig. 7 for all TMCs. We have observed that as we goalong a group (from 3d ScC or VC to 4d YC or NbC), the chargecontribution by metallic sphere increases due to the large numberof electrons. In order to visualize the nature of the bondingcharacter and to explain the charge transfer, we have plottedthe valence charge density for all TMC’s along the (1 1 0) plane inFig. 8. The charge density plot for ScC is largely different from allother TMCs. The covalent bonding between TM and C in ScC isweak. In the same group, for YC, the covalent character is slightly

Fig. 2. Band structures of TMCs: (a) ScC, (b) YC, (c) VC and (d) NbC.

P. Soni et al. / Journal of Physics and Chemistry of Solids 73 (2012) 873–880876

increased as compared to ScC because the small fraction of thebonding states becomes occupied. The large value of bulk mod-ulus of these carbides can be explained by the strong covalentbonding in them. The covalent bonding between TM and Csystematically becomes prominent in VC and NbC. It is clear fromthe figure that the covalent bonding is between the TM and the Catoms. It can be seen that for each material, there is an increase inelectron charge distribution at C site and a net decrease in the TMsite. This indicates a charge transfer from TM to C atom, which isdue to the high electronegativity of C atom as compared to TMatom. This direction of charge transfer is confirmed experimen-tally by the near edge X-ray absorption fine structure (NEXAFS)measurements [22].

The elastic constants provide a link between the mechanical anddynamical behavior of crystals and provide important information

concerning the nature of the interatomic forces. In particular,they provide information on the stability and stiffness of materials.Recently, Mankad et al. [23] have reported the structural,elastic, electronic and phonon properties of platinum carbide withinthe DFT scheme. We have calculated the second order elasticconstants (SOECs) of TMCs at ambient pressure using the methodof tetrahedral and rhombohedral distortions on the cubic structures[15]. In the present work, the calculated values of SOECs aregiven in Table 2. It is noticed from Table 2 that our calculatedelastic constants satisfy the stability criteria: C11–C1240, C4440,C11þ2C1240 and C12oBoC11 [24,25] and indicate the stability ofthese compounds in B1 phase. To the best of our knowledge noexperimental or theoretical information on the elastic propertiesof ScC and YC are available in the literature. However, for VC andNbC very limited data is available [26–30]. It can be seen from Table 2

Fig. 3. Total and partial density of states: (a) ScC, (b) YC, (c) VC and (d) NbC.

P. Soni et al. / Journal of Physics and Chemistry of Solids 73 (2012) 873–880 877

that the calculated values of elastic constants are in good agreementwith the experimental and theoretical data [26–30]. There is adecreasing trend of variation of all the three elastic constants of ScCand YC but in the case of VC and NbC, C12 and C44 decrease while C11

remains nearly the same. The ductile and brittle behaviors ofmaterials can be explained from the knowledge of bulk and shearmoduli, defined as [31,32]

B¼ 13ðC11þ2C12Þ ð1Þ

G¼GVþGR

2ð2Þ

where the Voigt shear modulus

GV ¼C11�C12þ3C44

5ð3Þ

and the Reuss shear modulus

GR ¼5C44ðC11�C12Þ

4C44þ3ðC11�C12Þð4Þ

It was shown empirically by Pugh [33] that, the materialwill be ductile if the ratio of B/G is greater than 1.75 otherwisebrittle. From Table 2, we notice that for ScC and YC, the ratio ofB/G41.75 and Cauchy’s pressure (C12–C44) is positive, hence theyare expected to be ductile in nature, whereas the brittle nature ofVC and NbC can be correlated to their B/Go1.75 and negativeCauchy’s pressure.

The Debye temperature (yD) may be estimated from the averagesound velocity vm by the method outlined in Refs. [32,34] from thefollowing equation:

yD ¼h

kB

3n

4PVa

� �1=3

vm ð5Þ

Fig. 4. Fermi surface plots for ScC and YC.

Fig. 5. Fermi surface plots for VC and NbC.

Fig. 6. Illustration of the nesting feature of the Fermi surface sheets for ScC

and VC.

Fig. 7. Charge contribution (in %) in the metallic, non-metallic and in the

interstitial region: (a) ScC and YC and (b) VC and NbC.

P. Soni et al. / Journal of Physics and Chemistry of Solids 73 (2012) 873–880878

where h is Planck’s constant, kB is Boltzmann’s constant, Va is theatomic volume, and n is the number of atoms per formula unit. Wehave derived the Debye temperatures (yD) for the four TMCs in theirB1 phase using the calculated elastic constants and listed them in

Table 2. It is noticed that the increase in the Debye temperature (yD)is directly related to the increase in the elastic constants. Thesevalues agree fairly well with the predicted and measured data[26–30].

4. Conclusion

The FP-LAPW method has been used to systematically inves-tigate the structural, electronic, elastic and bonding properties offour TMC (TM¼Sc, Y, V and Nb) compounds. The calculatedground state properties in the B1 phase are in good agreementwith experimental values and other theoretical results. Thecomputed band structures and density of states show that thestudied TMCs are metallic in nature. For group III TMCs, the C-p

states play the main role in the density of states at the Fermi level(EF) while TM-d states contribute mainly for group V. From theFermi surface plots, we can conclude that the different behaviorsof both groups of carbides are directly related to hybridizationbetween the p–d states near Fermi level (EF). With the increasednumber of ‘d’ electrons from groups III to V, the decrease instability indicates the filling of antibonding states. We haveobserved a charge transfer from TM to C atom from the calculatedcharge densities. The covalent bonding between TM and C in ScCand YC is weak. The covalent bonding systematically becomesprominent in VC and NbC. Our calculated elastic constants obeythe traditional mechanical stability conditions for cubic crystals.Group III TMCs (ScC and YC) are ductile in nature while group VTMCs (VC and NbC) are brittle. The elastic and thermal properties

ScC YC

VC NbC

Fig. 8. Charge density plots of TMCs (TM¼Sc, Y, V and Nb).

Table 2Calculated elastic and thermal properties of TMCs (TM¼Sc, Y, V and Nb).

Solids C11 (GPa) C12 (GPa) C44 (GPa) B/G C12–C44

(GPa)yD (K)

ScC Pre. 310.41 77.01 62.31 1.92 14.7 665.0

O. Theo. – – – – – –

YC Pre. 238.05 67.25 56.09 1.86 11.2 467.6

O. Theo. – – – – – –

VC Pre. 648.28 156.88 209.99 1.43 �53.1 967.3

O. Theo. – – 155–192a – – –

578.2b 147.2b 176.3b – – 901.0b

NbC Pre. 651.13 125.47 161.32 1.53 �35.9 739.6

O. Theo. 620c 200c 150c – – 689.7c

– – 153–205d – – –

667e 163e 161e – – 731.1e

Pre.¼present, O. Theo.¼other theoretical.a Ref. [27].b Ref. [26].c Ref. [28].d Refs. [27,29].e Ref. [30].

P. Soni et al. / Journal of Physics and Chemistry of Solids 73 (2012) 873–880 879

for ScC and YC are reported for the first time in this work and willbe tested in the future experimentally.

Acknowledgments

The authors are thankful to S.S. Chouhan for computationalassistance and useful discussions. The authors are thankful toMPCST, Bhopal, and University Grants Commission (UGC), NewDelhi, for the financial support. SPS is thankful to CSIR and UGC(SAP), New Delhi, for financial assistances. MR acknowledges CSIRfor Emeritus Fellowship.

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