Journal of Physics and Chemistry of Solids 72 (2011) 810–816

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Journal of Physics and Chemistry of Solids

0022-36

doi:10.1

n Corr

E-m

sps.phy1 Te

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

Structural, high pressure and elastic properties of transition metalmonocarbides: A FP-LAPW study

Pooja Soni a,b,1, Gitanjali Pagare a,n, Sankar P. Sanyal b,1

a Department of Physics, Government M.L.B. Girls P.G. College, Bhopal 462002, Indiab Condensed Matter Physics Laboratory, Department of Physics, Barkatullah University, Bhopal 462026, India

a r t i c l e i n f o

Article history:

Received 15 January 2011

Received in revised form

12 March 2011

Accepted 11 April 2011Available online 16 April 2011

Keywords:

C. Ab initio calculations

C. High pressure

D. Elastic properties

D. Phase transition

D. Thermodynamic properties

97/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jpcs.2011.04.006

esponding author. Tel.: þ91 755 2560462; fa

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

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

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

a b s t r a c t

The structural, elastic and thermal properties of four transition metal monocarbides ScC, YC (group III),

VC and NbC (group V) have been investigated using full potential linearized augmented plane wave

(FP-LAPW) method within generalized gradient approximation (GGA) both at ambient and high

pressure. We predict a B1 to B2 structural phase transition at 127.8 and 80.4 GPa for ScC and YC along

with the volume collapse percentage of 7.6 and 8.4%, respectively. No phase transition is observed in

case of VC and NbC up to pressure 400 and 360 GPa, respectively. The ground state properties such as

equilibrium lattice constant (a0), bulk modulus (B) and its pressure derivative (B0) are determined and

compared with available data. We have computed the elastic moduli and Debye temperature and

report their variation as a function of pressure.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The transition metal carbides (TMCs) and nitrides (TMNs) haveremained an interesting subject, both in condensed matterphysics for basic research and materials science from applicationpoint of view. This is mainly because of their intrinsic propertiesthey inherited due to the occupancy of ‘d’ electrons of thetransition metal ions. Both the class of compounds are predomi-nantly metallic in nature having outstanding physical propertiessuch as high melting point, ultra hardness, metallic conductivity,chemical stability and high corrosion resistivity [1–5]. All theseproperties make them good candidates for hard materials andlead them to use in coatings for cutting tools, high power energyindustry and optoelectronics. Nowadays the advancements in theab initio quantum mechanical calculations of the atomic andelectronic structures allow the theoretical modeling of newmaterials which permits the prediction of their properties andthe suggestion of new syntheses. So far as the structural proper-ties are concerned, several TMCs and TMNs have been investi-gated experimentally [1–3,6,7]. Besides the experimental studies,several theoretical calculations based on the density functionaltheory (DFT) have been used to obtain interesting electronic andbonding properties [8–13]. The lattice dynamical properties oftransition metals and their compounds have been reviewed by

ll rights reserved.

x: þ91 755 2661783.

gare),

Weber [14]. The mechanical properties of TMCs and TMNs havebeen examined theoretically by Wu et al. [15]. Amongst theseproperties, elastic constants represent a good test for estimatingthe quality of a theoretical approach [16,17]. Recently, Ham andLee [18] reviewed the applications of TMCs and TMNs as electrodematerials for low temperature fuel cells. In the present work westudy the TMCs namely, ScC, YC, VC and NbC, at high pressure tolook into the structural and elastic properties.

The mechanical properties under pressure for these carbideshave received less or no attention. To the best of our knowledge,neither theoretical nor experimental work exists to explore thehigh pressure properties for these materials. Moreover, theore-tical or experimental data on elastic and thermal properties arenot available for ScC and YC. A step towards a better under-standing the physics of these TMCs will be to investigate theirelastic and thermal properties. In the present paper, we aim toinvestigate the high pressure behavior, elastic and thermalproperties of four TMCs (TM¼Sc, Y, V and Nb) which crystallizein NaCl-type structure. The rest of the paper is organized asfollows: In Section 2, we briefly describe the method of calcula-tion. The results obtained for the structural, elastic and thermalproperties are presented and discussed in Section 3. Finally, inSection 4, we summarize the main conclusion of our work.

2. Method of calculation

The total energy, ground state properties and electronic bandstructures of four TMC compounds have been computed using full

P. Soni et al. / Journal of Physics and Chemistry of Solids 72 (2011) 810–816 811

potential linearized augmented plane wave (FP-LAPW) methodwithin the density functional theory (DFT) [19]. The generalizedgradient approximation (GGA) in the scheme of Perdew et al. [20]has been used for the exchange and correlation effects. Theenergy eigen value convergence has been achieved by expandingthe basis function up to RMT

n Kmax¼7, where RMT is the smallestatomic sphere radius in the unit cell and Kmax gives the magni-tude of the largest K vector in the plane wave expansion. Thevalence wave functions inside the spheres are expanded up tolmax¼10 while the charge density is Fourier expanded up toGmax¼12. The self-consistent calculations are considered to con-verge when the total energy of the system is stable within 10�4

Ry. A mash of 5000 k points is used and the tetrahedral method[21] has been employed for the Brillouin zone integration. Toobtain the ground state properties, the total energies are fitted toBirch [22] equation of state

P¼3B0

2

V0

V

� �7=3

�V0

V

� �5=3" #

1þ3

4ðB00�4Þ

V0

V

� �2=3

�1

( )" #ð1Þ

where P is the pressure, V is the volume at pressure P, V0 is thevolume at ambient pressure, B0 is the bulk modulus at ambientpressure and B00 is the pressure derivative of bulk modulus B0.

The knowledge of elastic constants is essential for manypractical applications related to the mechanical properties suchas internal strain, sound velocities, fracture toughness andthermo-elastic stress [23]. We have, therefore, calculated thesecond order elastic constants (SOECs) of TMCs at ambientpressure by using the method of tetrahedral and rhombohedraldistortions on the cubic (fcc) structure. [19]. In this method, thesystem is fully relaxed after each distortion in order to reach to

Fig. 1. Variation of total energy with volum

the equilibrium state [24]. The elastic stability criteria for cubiccrystal at ambient conditions are: C11�C1240, C4440,C11þ2C1240 and C12oBoC11 [25,26]. Apart from this, theelastic moduli like Young0s (E) and Shear modulus (G), Poisson0s(u) and anisotropic ratio (A) [25–28], which are the most inter-esting mechanical properties for application point of view, aregiven by

E¼9BG

3BþGð2Þ

G¼GVþGR

2ð3Þ

where Voigt shear modulus

GV ¼C11�C12þ3C44

5ð4Þ

and Reuss shear modulus

GR ¼5C44ðC11�C12Þ

4C44þ3ðC11�C12Þð5Þ

u¼3B�2G

2ð3BþGÞð6Þ

A¼2C44

C11�C12ð7Þ

The Debye temperature (yD) is a fundamental thermal prop-erty that is closely related to many physical properties such aselastic constants, specific heat and melting temperature etc. Oneof the standard methods to calculate the Debye temperature is

e for (a) ScC (b) YC (c) VC and (d) NbC.

P. Soni et al. / Journal of Physics and Chemistry of Solids 72 (2011) 810–816812

from elastic constants data. The sound velocities are obtained byusing the elastic constants as follows [27,29]:

Longitudinal sound velocity

nl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC11þð2=5Þð2C44þC12�C11Þ� �

r

sð8Þ

Transverse sound velocity

nt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC44�

15 ð2C44þC12�C11Þ

� �r

sð9Þ

Here C11, C12 and C44 are SOECs and r is mass density per unitvolume. By using these velocities, the average sound velocity vm

is approximately calculated from [28,29]

nm ¼1

3

2

n3t

þ1

n3l

!" #�1=3

ð10Þ

the Debye temperature (yD) may be estimated from the averagesound velocity vm by the following equation [28,29]

yD ¼h

kB

3n

4PVa

� �1=3

nm ð11Þ

where h is a Plank0s constant, kB is Boltzmann0s constant, Va is theatomic volume, n is the number of atoms per formula.

3. Results and discussion

For calculating the ground state properties of four TMCs(TM¼Sc, Y, V and Nb), the total energies are calculated as a

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

Solids a0(A) B(GPa) B’

ScC B1 phase Pre. 4.682 154.81 4.18

Expt. 4.637a – –

4.720b – –

O. theo. 4.684c 148c –

4.680d 153d –

4.405e 173.6e –

B2 phase Pre. 2.876 149.0 4.01

YC B1 phase Pre. 5.086 124.18 4.14

Expt. 5.11b – –

O. theo. 5.08d 128d –

5.076f – –

B2 phase Pre. 3.100 120.2 4.12

VC B1 phase Pre. 4.156 320.67 4.13

Expt. 4.159b 308–390i –

4.163g – –

O. theo. 4.164c 290c –

4.154d 304d –

3.942e 360.5e –

NbC B1 phase Pre. 4.488 300.69 4.24

Expt. 4.454g 296–330i –

4.471h 331h –

O. theo. 4.492c 293c –

4.476d 301d –

Pre.¼present, Expt.¼experimental and O. theo.¼other theoretical.

a Ref. [2].b Ref. [3].c Ref. [12].d Ref. [11].e Ref. [10].f Ref. [9].g Ref. [7].h Ref. [6].i Ref. [1,4].j Ref. [8].

function of volume in their B1 (NaCl) and B2 (CsCl) phases. Theplots of total energy as a function of reduced volume are shown inFig. 1(a–d). It is worth mentioning that, from E–V curves it is clearthat the B1 phase is stable at ambient pressure which is consistentwith the literature [1–3]. The ground state properties, such asequilibrium lattice constant (a0), bulk modulus (B) and itspressure derivative (B0) of all the TMCs have been calculated intheir B1 and B2 phases. The calculated values of these propertiesare presented in Table 1 and compared with their experimental[1–3,6,7] and other theoretical [8–12] results. The calculatedvalues of lattice constants are in good agreement with themeasured values [2,3,6,7] and differ by less than 1% only. Ourresults are similar to earlier theoretical results, cited in the sametable, except the one reported in Ref. [10] in which the localdensity approximation (LDA) is used for total energy calculations.We could not compare the calculated values of bulk modulus forScC and YC with the measured values due to non-availability ofexperimental data. Nevertheless, these values are comparable tothose reported by Isaev et al. [11] and Vojvodic et al. [12]calculated using pseudo potential method and GGA. In the caseof VC and NbC, our calculated values of bulk modulus are in goodagreement with the range of measured data [1] and othertheoretical results [10–12]. As in the case of other binarycompounds, the calculated values of B0 are close to 4.0, and nomeasured or theoretical values are available for comparison. Thecomputed values of lattice energies for ScC and VC (listed inTable 1), agree well with theoretical values reported by Haglundet al. [8].

As shown in Fig. 1, our calculations suggest that there exists astructural phase transition from B1 (NaCl) to B2 (CsCl) phase for

Total energy (Ry) Transition pressure Pt (GPa) Volume collapse (%)

�1604.684 127.8 7.6

– – –

– – –

�1600.896j – –

– – –

– – –

�1604.608 – –

�6847.634 80.4 8.4

– – –

– – –

– – –

�6847.557 – –

�1974.894 – –

– – –

– – –

�1970.736j – –

– – –

– – –

�7717.234 – –

– – –

– – –

– – –

– – –

Table 2Calculated elastic properties of TMCs (TM¼Sc, Y, V and Nb) in B1 phase.

Solids C11(GPa) C12(GPa) C44(GPa)

ScC Pre. 310.41 77.01 62.31

YC Pre. 238.05 67.25 56.09

VC Pre. 648.24 156.88 209.99

O. theo. – – 155–192a

578.2b 147.2b 176.3b

NbC Pre. 651.13 125.47 161.32

O. theo. – – 153–205c

620d 200d 150d

667e 163e 161e

Pre.¼present and O. theo.¼other theoretical.

a Ref. [16].b Ref. [13].c Ref. [16,17].d Ref. [14].e Ref. [15].

P. Soni et al. / Journal of Physics and Chemistry of Solids 72 (2011) 810–816 813

group III TMCs (namely ScC and YC). B2 phase is considered asbody centered cubic (bcc) structure. This is a prediction for thefirst time, although Das et al. [10] have earlier suggested apossibility of B1–B2 structural phase transition for these com-pounds without any quantitative information. The results of thehigh pressure phase are listed in Table 1. The equation of statesfor ScC and YC are presented in Fig. 2. At high pressure, both ScCand YC transform from their initial B1 phase to B2 phase at 127.8and 80.4 GPa along with the volume collapse of 7.6 and 8.4%,respectively. It is interesting to note that the calculated transitionpressure decreases with the increase in size of TM atom. Forgroup V TMCs (namely VC and NbC), no structural phase transi-tion has been observed up to 400 and 360 GPa, respectively. Theresults of our present calculation of phase transition pressure inthese two group of TMCs seem to be very consistent with theirmeasured values of bulk modulus (See Table 1). The later pair (VCand NbC) is hardest amongst the four TMCs and hence leastdeformable. It can be noted that no experimental or theoreticalinformation regarding the high pressure structural phase transi-tion is available for these TMCs.

The elastic constants play an important role for the determi-nation of the mechanical properties and provide important

Fig. 2. Equation of states (a) ScC and (b) YC.

information concerning the nature of the interatomic forces. Inparticular, they provide information on the stability and stiffnessof materials. In the present work, the second order elasticconstants have been calculated and are given in Table 2. It isobserved that our calculated elastic constants satisfy the stabilitycriteria. This is indicative of the stability of these compounds in B1

phase. To the best of our knowledge, there is no experimental ortheoretical data on elastic properties of ScC and YC reported so farwhereas for VC and NbC very little information is available[13–17]. It can be seen from Table 2 that the calculated valuesof elastic constants are in agreement with the reported theoreticaldata [13–17]. For group V TMCs, the values of elastic constants arenear about twice as compared to group III TMCs. This fact,therefore suggests that the nature of bonding in ScC and YC ispredominantly ionic while the same in VC and NbC is predomi-nantly covalent. The effect of pressure on these SOECs is essential,especially for understanding interatomic interactions, mechanicalstability and phase transition mechanisms. We further investigatethe behavior of SOECs of these compounds at high pressure bycomputing them at various pressures as shown in Fig. 3. Asexpected, all the three elastic constants show a linear variationwith pressure. It is noticed that C11 is more sensitive to thechange of pressure as compared to C12 and C44 because C11

represents elasticity in length and a longitudinal strain producesa change in C11 while C12 and C44 are related to the elasticity inshape, which are shear constants. A transverse strain causes achange in shape without a change in volume. Therefore, C12 andC44 are less sensitive to pressure as compared to C11. The Young0smodulus (E) describes the materials response to linear strain anda large value of E implies the stiffness of the material. The shearmodulus (G) represents resistance to plastic deformation. ThePoisson0s ratio (n) is frequently measured for polycrystallinematerials when investigating their hardness. It takes the valuebetween �1 to 0.5. Another important parameter is the elasticanisotropic factor (A), which gives a measure of the anisotropy ofthe elastic wave velocity in a crystal, which is unity for acompletely isotropic material and the deviation from unitymeasures the degree of elastic anisotropy. We have calculatedall the above mechanical properties for these four TMC com-pounds in B1 phase, as given in Table 3. The variation of theseelastic moduli with the pressure is shown in Fig. 4. All thethree elastic moduli (Young0s modulus, bulk modulus and Shearmodulus) increase linearly with increasing pressure for allthe compounds. To the best of our knowledge, no experimentalor theoretical data for the comparison of above propertiesare available in literature for ScC and YC. The ductile andbrittle behavior of materials can be explained from empirical

Fig. 3. Variation of elastic constants with pressure for (a) ScC (b) YC (c) VC and (d) NbC.

Table 3Calculated Young’s modulus (E), shear modulus (G), anisotropic factor (A),

Poisson’s ratio (n), B/G, Cauchy pressure (C12–C44) for TMCs (TM¼Sc, Y, V and

Nb) in B1 phase.

Solids E(GPa) G(GPa) A n B/G C12–C44(GPa)

ScC Pre. 205.46 80.33 0.53 0.28 1.92 14.68

YC Pre. 169.09 66.41 0.65 0.27 1.86 11.16

VC Pre. 544.30 223.60 0.85 0.21 1.43 �53.11

O. theo. 518.5a 215.5a 0.81a 0.23a – –

NbC Pre. 483.77 196.35 0.61 0.23 1.53 �35.84

O. theo. 440.83b 171.67b 0.71b 0.28b – –

484.33c 192.79c 0.64c 0.26c – –

Pre.¼present and O. theo.¼other theoretical.

a Ref. [13].b Ref. [14].c Ref. [15].

P. Soni et al. / Journal of Physics and Chemistry of Solids 72 (2011) 810–816814

relationship proposed by Pugh [30]. According to him, if the ratioof B/G is greater than 1.75, the material will be ductile otherwisebrittle. From Table 3, we can notice that for ScC and YC, the ratioof B/G41.75 and Cauchy0s pressure (C12–C44) is positive, hencethey are expected to be ductile in nature, whereas the brittlenature of VC and NbC can be correlated to their B/Go1.75 andnegative Cauchy0s pressure.

We have derived the average sound velocities and Debyetemperatures in B1 phase by using the calculated elastic con-stants, for all the TMCs and presented them in Table 4. It is

noticed that the increasing trend of Debye temperatures isdirectly related to the increasing trend in elastic constants. Wehave also examined the variation of yD with pressure andpresented in Fig. 5. It can be seen that yD increases as pressureincreases for all the compounds. Unfortunately, in the absence ofany measured data related to the sound velocities in the litera-ture, they could not be compared for these compounds. Futureexperimental work will testify our calculated results.

We have also calculated the electronic specific heat coefficient(g) by using an approximate model. The specific heat capacity [31]of materials is given as the sum of the electronic and latticecontribution

C ¼ gTþAT3ð12Þ

where gT represents the electronic contribution and AT3 repre-sents the lattice contribution, with g as the electronic specific heatcoefficient. For non-interacting electrons, the electronic specificheat coefficient (g) is proportional to the total DOS at the Fermilevel N(EF) and is given by

g¼P2k2

BNðEFÞ

3ð13Þ

The calculated values of (g) are reported in Table 4. Due to thelack of experimental data, a comparison of our results could notbe possible. It is well known that the theoretically calculatedvalue of g is less than the experimental value because of theneglect of the electron phonon interaction. By using the

Fig. 4. Variation of elastic moduli with pressure for (a) ScC (b) YC (c) VC and (d) NbC.

Table 4Calculated density (r), longitudinal (vl), transverse (vt), average (vm) acoustic

wave velocities, Debye temperature (yD) and specific heat coefficient (g) of TMC

(TM¼Sc, Y, V and Nb) in the B1 structure.

Solids r*103

(kg/m3)

nl

(ms�1)

nt

(ms�1)

nm

(ms�1)yD(K) g

(mJ/mol K2)

ScC Pre. 3.68 8510.9 4776.7 5311.3 665.0 3.40

YC Pre. 5.09 6491.8 3649.2 4057.3 467.6 3.93

VC Pre. 5.82 10,317.0 6206.5 6858.3 967.3 2.68

O. theo. 5.64a 9844.2a 5882.8a 6454.8a 901.0a –

NbC Pre. 7.71 8600.0 5118.9 5662.4 739.6 1.75

O. theo. 7.80b 8562.2b 4722.4b 5258.2b 689.7b –

7.85c 8697.5c 5013.0c 5561.6c 731.1c –

Pre.¼present and O. theo.¼other theoretical.

a Ref. [13].b Ref. [14].c Ref. [15].

P. Soni et al. / Journal of Physics and Chemistry of Solids 72 (2011) 810–816 815

expression gexp¼gtheo (1þl), one can have a rough estimate ofthe value of l, the electron phonon mass enhancement parameter.

Fig. 5. Debye temperature as a function of pressure for (a) ScC (b) YC (c) VC and

(d) NbC.

4. Conclusion

The FP-LAPW method has been used to systematically investigatethe structural, elastic and thermal properties of four TMC (TM¼Sc, Y,

P. Soni et al. / Journal of Physics and Chemistry of Solids 72 (2011) 810–816816

V and Nb) compounds. The calculated ground state properties in B1

phase are in good agreement with the experimental data and othertheoretical results. The pressure induced structural phase transitionfrom B1 to B2 is predicted at 127.8 and 80.4 GPa for ScC and YC,respectively, while no phase transition has been found for VC andNbC. Our calculated elastic constants obey the mechanical stabilityconditions for cubic crystals. We have found Group III TMCs (ScC andYC) are ductile in nature while group V TMCs (VC and NbC) are brittle.Some related mechanical properties such as Young0s modulus (E),shear modulus (G), Poisson0s ratio (u) and anisotropic ratio (A), Debyetemperature (yD) and specific heat (g) have also been calculated. Theelastic and thermal properties for ScC and YC are reported for the firsttime in this work.

Acknowledgments

The authors are thankful to S. S. Chouhan for computationalassistance and useful discussion. The authors are thankful toMPCST, Bhopal and UGC, New Delhi for their financial support tothis project. SPS is thankful to MPCST, Bhopal and CSIR and UGC(SAP), New Delhi for financial assistances.

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