Pressure induced structural phase transformation in TiN: A first-principles studySoumya S. Bhat, Umesh V. Waghmare, and U. Ramamurty Citation: Journal of Applied Physics 113, 133507 (2013); doi: 10.1063/1.4798591 View online: http://dx.doi.org/10.1063/1.4798591 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/113/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Pressure induced structural phase transition in solid oxidizer KClO3: A first-principles study J. Chem. Phys. 138, 174701 (2013); 10.1063/1.4802722 High pressure phase transformation in uranium carbide: A first principle study AIP Conf. Proc. 1512, 78 (2013); 10.1063/1.4790919 First-principle investigations of structural stability of beryllium under high pressure J. Appl. Phys. 112, 023519 (2012); 10.1063/1.4739615 First-principles study of structural, electronic, and mechanical properties of the nanolaminate compound Ti 4GeC 3 under pressure J. Appl. Phys. 107, 123511 (2010); 10.1063/1.3446096 Structural stability of polymeric nitrogen: A first-principles investigation J. Chem. Phys. 132, 024502 (2010); 10.1063/1.3290954

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Pressure induced structural phase transformation in TiN:A first-principles study

Soumya S. Bhat,1 Umesh V. Waghmare,2 and U. Ramamurty1,a)

1Department of Materials Engineering, Indian Institute of Science, Bangalore 560012, India2Theoretical Science Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur,Bangalore 560064, India

(Received 30 January 2013; accepted 15 March 2013; published online 3 April 2013)

Titanium nitride (TiN), which is widely used for hard coatings, reportedly undergoes a pressure-induced

structural phase transformation, from a NaCl to a CsCl structure, at �7 GPa. In this paper, we use first-

principles calculations based on density functional theory with a generalized gradient approximation of

the exchange correlation energy to determine the structural stability of this transformation. Our results

show that the stress required for this structural transformation is substantially lower (by more than an

order of magnitude) when it is deviatoric in nature vis-�a-vis that under hydrostatic pressure. Local

stability of the structure is assessed with phonon dispersion determined at different pressures, and we

find that CsCl structure of TiN is expected to distort after the transformation. From the electronic

structure calculations, we estimate the electrical conductivity of TiN in the CsCl structure to be about

5 times of that in NaCl structure, which should be observable experimentally. VC 2013 AmericanInstitute of Physics. [http://dx.doi.org/10.1063/1.4798591]

I. INTRODUCTION

Transition metal mononitrides, which belong to the class

of refractory metal compounds, are typically very hard mate-

rials crystallizing in the rocksalt structure. These compounds

are both scientifically interesting and technologically impor-

tant as they show extreme and unique physical properties

such as high hardness, high melting points, and conductiv-

ities that are comparable with those of pure transition metals.

The nature of bonding in these systems is mixed: slightly

ionic and strongly covalent. The occurrence of an ionic like

structure (rocksalt) in combination with the hardness of

covalent materials makes them rather interesting.1 Titanium

nitride (TiN) is one of the most important2 among the transi-

tion metal mononitrides. This is due to its higher hardness as

well as shear strength relative to other nitrides, which is

associated with stronger covalent like bonds of weaker polar-

ity between Ti and N atoms.3 These interesting features of

TiN are extensively used in technological applications such

as hard and wear-resistant coatings on mechanical tools, dif-

fusion barriers in microelectronic components, and optical

and decorative coatings.4–7 Also, TiN finds various applica-

tions in semiconductor device technology due to its low elec-

trical resistivity, chemical and metallurgical stability, and

exceptional mechanical properties.8,9 More recently, it is

found that TiN is a useful biomaterial coating with highly

favorable biocompatibility.10 In addition, TiN and related

materials are being investigated to understand their potential

for thermoelectric applications,11,12 particularly at high oper-

ating temperatures where the refractory properties of the

nitrides offer a distinct advantage.13

Since TiN coatings are designed to work as coatings

under high pressure, it is important to understand the pressure

induced changes so to improve the component reliability and

to enhance service life. Further, the pressure-induced phases

may have very different properties as compared to the parent

material, and hence it is important to estimate the properties at

various strain levels. The study of the structure of matter at

high pressures has advanced steadily since the advent of the

diamond anvil cell (DAC). Modern third generation synchro-

tron facilities and the advances in quantum mechanical com-

putations14,15 have further facilitated it.

High pressure experiments on bulk TiN powder and theo-

retical calculations revealed that there exists a structural phase

transition from the relatively open NaCl (coordination-6)

to more dense CsCl structure (coordination-8) at elevated

pressures.16–19 Zhao et al.16 experimentally observed the

structural phase transformation in TiN powder at ambient

temperature between 7 and 11 GPa by inspecting a discontinu-

ity in the volume change with pressure. On the other hand,

Ahuja et al.,17 who studied the structural and elastic properties

of TiN using first-principles calculations, predicted a struc-

tural phase transition of TiN from NaCl to CsCl structure at a

pressure of 370 GPa. Liu et al.18 calculated enthalpies for

NaCl and CsCl phases of TiN estimating the transition pres-

sure, PT to be 364 GPa. The discrepancy of experimental

(PT� 7–11 GPa) and theoretical estimates (>300 GPa) makes

it imperative to carry out a thorough analysis.

Buerger20 illustrated, as early as in 1951, the mechanism of

NaCl to CsCl structural transformation with contraction along

[111] and dilatation perpendicular to it. He proposed that this

transition takes place rapidly in spite of large changes in the first

coordination, because no intermediate energy state exists

between those two structures.21 In recent years, considerable

effort has been invested in understanding the precise mechanism

of this structural phase transition.22–29 Watanabe et al.22 exam-

ined this in CsCl crystals by means of X-ray diffraction and op-

tical microscopy and found definite orientation relations of

[110]CsCljj[100]NaCl and [001]CsCljj[011]NaCl. They have shown

that the closest packing planes of both the structures are retaineda)Email: ramu@materials.iisc.ernet.in

0021-8979/2013/113(13)/133507/7/$30.00 VC 2013 American Institute of Physics113, 133507-1

JOURNAL OF APPLIED PHYSICS 113, 133507 (2013)

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in the transition and observed a uniaxial expansion (or shrink-

age) parallel to one of the 2-fold axes in the CsCl-type struc-

ture (or 4-fold axes in NaCl-type structure). They explained

the transition mechanism on the basis of two kinds of coopera-

tive movements of ions: intralayer rearrangement and inter-

layer translation. An analogous orientation relation was

disclosed by Blaschko et al.23 in RbI: [110]CsCljj[001]NaCl and

[001]CsCljj[110]NaCl and by Fraser and Kennedy24 in alkali

halides: [101]CsCljj[010]NaCl. A phenomenological model of

the reconstructive transformation between the NaCl and CsCl

structure types is given by Toledano et al.25 which consists of

two consecutive displacive mechanisms corresponding to the

NaCl to orthorhombic and the orthorhombic to CsCl transfor-

mations. They rationalized the multiplicity of observed coin-

ciding orientations in terms of the orientational domains,

transforming into one another by the cubic symmetry opera-

tions lost in the intermediate orthorhombic phases. Based on

first-principles calculations, a number of theoretical studies of

NaCl to CsCl transformation have been conducted27–30 which

refers to the direct NaCl-CsCl transformation that occurs in

alkali halides without assuming the intermediate pathways

between the two structures.

As mentioned earlier, the pressure required for the struc-

tural transformation of TiN predicted by the theoretical

results (>300 GPa) is much higher than the experimentally

observed pressure (�7 GPa). To fill this gap between the

theory and experiment, we study the structural phase trans-

formation of TiN using first-principles calculations based on

density functional theory (DFT). Through these calculations,

new transformation path from NaCl structure to CsCl struc-

ture has been established. Complete phonon spectra are

determined to investigate the local structural stability. We

provide a quantitative comparison of electronic structure of

TiN in NaCl structure with that of high pressure phase with

implication to electrical conduction properties.

II. COMPUTATIONAL DETAILS

Our calculations are based on first-principles DFT as

implemented in the Quantum Espresso package,31 with a gener-

alized gradient approximation (GGA)32 to exchange correlation

energy of electrons, and ultrasoft pseudopotentials33 to repre-

sent interaction between ionic cores and valence electrons.

Kohn-Sham wave functions are represented with a plane-wave

basis with an energy cutoff of 30 Ry and a charge density with

a cutoff of 240 Ry. A uniform mesh of 20� 20� 20 k-points

was used to sample Brillouin zone integration. Hellman-

Feynman forces on atoms and stresses on unit cell were used to

optimize crystal structures through minimization of total

energy. DFT linear response was used to determine dynamical

matrices on a mesh of wave vectors, which were Fourier inter-

polated to obtain full phonon dispersion.

III. RESULTS AND DISCUSSION

A. Energetics

Lattice constants, a, bulk moduli, B, and critical pressures,

PT, for the two structures of TiN calculated using both GGA

and LDA (local-density approximation) are listed in Table I,

together with experimental literature data for the NaCl struc-

ture. Both a and B for TiN in NaCl structure are in good agree-

ment with experimental and theoretical results reported

earlier.16,36–38 For CsCl structure of TiN, there is no experi-

mental data for comparing our results. However, our calculated

values of a and B are in good agreement with the calculations

of Liu et al. (a� 2.64 A using GGA and� 2.58 A using LDA,

B� 254 GPa for GGA, and� 299 GPa for LDA, respec-

tively)18 and Srivastava et al. (a� 2.66 A using GGA).19

The total energy as a function of volume of both the

NaCl and CsCl structures of TiN is determined within both

GGA and LDA, and the GGA calculated results are shown in

Fig. 1(a). It indicates that TiN undergoes a structural phase

transition from NaCl to CsCl structure under volumetric

compression. The enthalpy as a function of pressure is illus-

trated in Fig. 1(b). The PT, which is given by the pressure at

which enthalpies of the two phases are the same, is calcu-

lated to be 347 GPa (it is 313 GPa in calculations performed

using LDA). Chauhan et al.39 also report PT to be 310 GPa in

their calculations using LDA. On the other hand, calculations

by Liu et al.18 indicate it to be 364 GPa and 322 GPa using

GGA and LDA, respectively, and Ahuja et al.17 using LDA

indicate it to be 370 GPa. Our estimate is in reasonable

agreement with these literature values.

B. Transformation path

First, we note that both the NaCl and CsCl structures

have the same stacking along their (111) planes [Ac Ba Cb].

However, they differ in their c/a ratios;ffiffiffi6p

andffiffiffiffiffiffiffiffi3=2

pfor

NaCl and CsCl structures, respectively. A path of transfor-

mation is obtained by performing a series of calculations of

NaCl structure with the choice of hexagonal unit cell

(c along [111], ahexagonal ¼ acubic=ffiffiffi2p

) by varying the c/a ra-

tio fromffiffiffi6p

toffiffiffiffiffiffiffiffi3=2

p(see Fig. 2(a)). Physically, this means

that a uniaxial compressive stress is applied along [111] of

NaCl structure, which transforms it to CsCl structure.

Variation of the total energy of TiN as a function of opti-

mized lattice parameter for different c/a ratios is displayed in

Fig. 2(b). From, it can be noted that the energy barrier for

the structural transformation in TiN is 1.8 eV. Interestingly,

the energy barrier for the reverse transformation (from CsCl

to NaCl structure) is only 2.07 meV, at c/a¼ 1.30 (inset

graph of Fig. 2(b)) which is substantially smaller than that

TABLE I. Calculated lattice constants, bulk moduli, and critical pressures of

transformation between the two structures of TiN using DFT calculations

based on GGA and LDA.

Present calculations Experimental results

GGA LDA

NaCl CsCl NaCl CsCl NaCl CsCl

a (A) 4.25 2.64 4.17 2.59 4.26a …

B (GPa) 277 262 322 305 288b …

Pc(GPa) 347 313 7c …

aReference 34.bReference 35.cReference 16

133507-2 Bhat, Waghmare, and Ramamurty J. Appl. Phys. 113, 133507 (2013)

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required for the forward transformation. This implies that the

reverse transformation can occur readily and the forward

transformation product is marginally stable.

Stress variations on the unit cell along x-direction, rxx

(¼ ryy), and along z-direction, rzz, as a function of the opti-

mized lattice parameter for c/a ratio varying from NaCl

(c/a¼ 2.45) to CsCl (c/a¼ 1.23) are shown in Figs. 3(a)

and 3(b), respectively. These figures indicate that the stress

variations are more or less symmetric, with peak values

occurring at c/a� 1.80, which is the average of the c/avalues of NaCl and CsCl. The principal stress tensor at

this c/a is

FIG. 2. (a) Energy as a function of lattice parameter a, for different c/a ratios varying from 1.00 to 3.00. (b) Minimum energy as a function of optimized lattice

parameter a, for different c/a ratios varying from 1.00 to 3.00.

FIG. 3. Stresses along (a) x (equal to that along y-direction) and (b) z-directions, as a function of optimized lattice parameter a, for different c/a ratios varying

from 1.00 to 3.00.

FIG. 1. (a) Total energy as a function of volume and (b) enthalpy as function of pressure for NaCl and CsCl structures of TiN.

133507-3 Bhat, Waghmare, and Ramamurty J. Appl. Phys. 113, 133507 (2013)

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r ¼�20:99 0:0 0:0

0:0 �20:99 0:00:0 0:0 42:16

24

35 GPa;

which represents a cylindrical state of stress on the unit cell.

Note that the hydrostatic component of the stress or the

mean stress, rm (¼rii/3) is only 0.062 GPa, whereas the devi-

ator component, r0ij, is

r0ij ¼�21:05 0:0 0:0

0:0 �21:05 0:00:0 0:0 42:09

24

35 GPa:

This indicates that the transformation is a predominantly

shear driven process and the volumetric change associated

with it is negligible. Note also that the magnitudes of these

stresses are substantially smaller than that observed for the

FIG. 4. Phonon dispersion curves of TiN for (a) NaCl and (b) CsCl structures.

FIG. 5. (a) Distorted structure-1 and (b)

its phonon dispersion curve.

FIG. 6. (a) Distorted structure-2 and (b)

its phonon dispersion curve.

133507-4 Bhat, Waghmare, and Ramamurty J. Appl. Phys. 113, 133507 (2013)

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transformation under hydrostatic pressure. They are closer,

although still higher, to that estimated from the experiment,

wherein the samples are likely to experience both deviatoric

and hydrostatic stresses simultaneously.

C. Structural stability

To investigate local structural stability of the phase formed,

phonon dispersions of both the structures were determined,

which are displayed in Figs. 4(a) and 4(b) for TiN in NaCl and

CsCl structures, respectively, while former displays no unstable

modes, TiN in CsCl structure exhibits imaginary frequencies

(unstable modes). When a structure exhibits imaginary frequen-

cies, it is at a local maximum with respect to the atomic dis-

placements corresponding to an unstable mode. This implies

that the product of the phase transformation is unstable.

To explore the CsCl based structures with lower energy,

we allowed the structure to distort using eigenvectors of

unstable modes at point M. The ideal CsCl structure of TiN

is distorted by doubling the cell and displacing the N atoms

along z-direction (distorted structure-1) (Fig. 5(a)). The pho-

non dispersion of this structure clearly shows the removal of

FIG. 7. Phonon dispersion curve of TiN in CsCl structure at pressures of (a) 0, (b) 14, (c) 36, (d) 63, and (e) 99 GPa.

133507-5 Bhat, Waghmare, and Ramamurty J. Appl. Phys. 113, 133507 (2013)

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instability at M point, but new unstable modes appear at

other points (Fig. 5(b)). The above structure is again doubled

along y-direction and atoms are displaced along z-direction

(distorted structure-2) (Fig. 6(a)). As can be seen from the

phonon dispersion plots (Fig. 6(b)), relatively strong unstable

modes appear at points R and X. Hence, the structure still

remains unstable. Also, the resultant structure has Ti with

6-fold coordination, in a way similar to NaCl structure.

To investigate whether the observed CsCl structure of

TiN could possibly be stabilized as a function of pressure, we

have determined phonon dispersion for TiN in CsCl structure

at different pressures and the same are illustrated in Fig. 7.

Our calculations show that the instabilities become weaker

with increase in pressure and the structure becomes stable at

P¼ 99 GPa (note here the reference of pressure is P¼ 0 GPa

at the CsCl structure at a¼ 2.64 A). Hence, we establish that

the cubic CsCl structure becomes stable effectively only at

pressures above 446 GPa or a stress state given as following:

r ¼�20:99 0:0 0:0

0:0 �20:99 0:00:0 0:0 42:16

24

35� 99

¼�119:99 0:0 0:0

0:0 �119:99 0:00:0 0:0 �56:84

24

35 GPa

D. Electronic structure

Electronic structure and density of electronic states

(DOS) of TiN in the NaCl and CsCl structures obtained

within GGA are shown in Figs. 5 and 6, respectively. Our

band structure and DOS of NaCl structure (Figs. 8(a) and

9(a)) are quite similar to those in Refs. 17, 36, and 38. For

both NaCl and CsCl structures, the lowest states in the va-

lence bands are due to N 2s electrons. The states below

Fermi level, Ef, are dominated by hybridized Ti 3d and N 2pstates. The electronic states near Ef are dominated by Ti 3dstates, thus the metallic properties are dominated by the

d-states originating from Ti atom.40 As can be seen in Fig. 8,

fewer bands cross Ef in NaCl structure as compared to those

in the CsCl structure, consistent with lower DOS at Ef as

shown in Fig. 9.

The curvature of bands near Ef for TiN in CsCl structure

is found to be higher than that in NaCl structure gives rise to

lower effective mass. The effective mass, calculated from the

curvature of a band near Fermi level, for NaCl and CsCl type

structures is 1.21 me and 0.56 me, respectively, where

me¼ 9.11� 10�31 kg is the free electron rest mass. The den-

sity of states is used to estimate the number of electrons, n,

near Fermi level.41 They are 2.09� 1027 and 4.71� 1027 m�3

for NaCl and CsCl structures of TiN, respectively. Thus,

TiN’s electrical conductivity in the CsCl structure can be

FIG. 8. Electronic structure of TiN in (a) NaCl and (b) CsCl structures. The Fermi level is set at zero energy.

FIG. 9. Calculated DOS of TiN in (a) NaCl and (b) CsCl structures. The Fermi level is set at zero energy and marked by a vertical line.

133507-6 Bhat, Waghmare, and Ramamurty J. Appl. Phys. 113, 133507 (2013)

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expected to be 4.8 times higher of that in NaCl structure, if the

relaxation time s is assumed to be the same for the two struc-

tures of TiN.

IV. SUMMARY

We have studied the high pressure behavior of TiN and

examined in detail the structural phase transition from NaCl

to CsCl structure. The lattice constant and bulk modulus for

the NaCl structure are in good agreement with the experi-

ments and theoretical results of others where for the CsCl

structure of TiN the computed values match well with the

other theoretical calculations. The new transition path reveals

that when stress applied is deviatoric in nature, TiN undergoes

a structural transition from NaCl to CsCl structure at much

lower stress compared to hydrostatic pressure. Structural sta-

bility of TiN has been investigated using phonon calculations

which disclose the instability of the CsCl phase as formed af-

ter the transformation. Structural distortions of the ideal CsCl

structure of TiN do not eliminate its instabilities. On examin-

ing the possible stability of CsCl structure of TiN using pho-

non dispersion curves at different pressures, we predict that it

can be stabilized by an additional pressure (to that of 347 GPa

required for transforming TiN from NaCl to CsCl structure) of

99 GPa. Relative electrical conductivities of both the struc-

tures of TiN are estimated using electronic structure and den-

sity of states, that of TiN in CsCl structure is expected to be

about 5 times higher than that for NaCl structure, which could

be useful in experimental characterization of the NaCl to CsCl

structural phase transformation.

1D. L. Price and B. R. Cooper, Phys. Rev. B 39, 4945 (1989).2H. O. Pierson, Handbook of Refractory Carbides and Nitrides: Properties,Characteristics, Processing and Applications (Noyes Publications, New

Jersey, 1996), p. 193.3R. F. Zhang, S. H. Sheng, and S. Veprek, Appl. Phys. Lett. 94, 121903 (2009).4L. Hultman, U. Helmersson, S. A. Barnett, J.-E. Sundgren, and J. E.

Greene, J. Appl. Phys. 61, 552 (1987).5P. Patsalas, C. Charitidis, S. Logothetidis, C. A. Dimitriadis, and

O. Valassiades, J. Appl. Phys. 86, 5296 (1999).6M. Birkholz, K.-E. Ehwald, P. Kulse, J. Drews, M. Fr€ohlich, U. Haak,

M. Kaynak, E. Matthus, K. Schulz, and D. Wolansky, Adv. Funct. Mater.

21, 1652 (2011).7M. S. R. N. Kiran, M. G. Krishna, and K. A. Padmanabhan, Int. J.

Nanomanufacturing 2, 420 (2008).8M. Wittmer, B. Studer, and H. Melchior, J. Appl. Phys. 52, 5722 (1981).9R. C. Glass, L. M. Spellman, S. Tanaka, and R. F. Davis, J. Vac. Sci.

Technol. A 10, 1625 (1992).10M. S. R. N. Kiran, M. G. Krishna, and K. A. Padmanabhan, Solid State

Commun. 151, 561 (2011).

11M. Zebarjadi, Z. Bian, R. Singh, A. Shakourie, R. Wortman, V. Rawat,

and T. Sands, J. Electron. Mater. 38, 960 (2009).12V. Rawat, Y. Koh, D. Cahill, and T. Sands, J. Appl. Phys. 105, 024909

(2009).13B. Saha, J. Acharya, T. D. Sands, and U. Waghmare, J. Appl. Phys.107,

033715 (2010).14M. T. Yin and M. L. Cohen, Phys. Rev. Lett. 45, 1004 (1980).15M. C. Payne, M. P. Teter, D. C. Alan, T. A. Arias, and J. D. Joannopoulos,

Rev. Mod. Phys. 64, 1045 (1992).16J.-G. Zhao, L.-X. Yang, Y. Yu, S.-J. You, R.-C. Yu, F.-Y. Li, L.-C.

Chen, C.-Q. Jin, X.-D. Li, Y.-C. Li, and J. Liu, Chin. Phys. Lett. 22, 1199

(2005).17R. Ahuja, O. Errikson, J. M. Wills, and B. Johansson, Phys. Rev. B 53,

3072 (1996).18K. Liu, X.-L. Zhou, H.-H. Chen, and L.-Y. Lu, J. Therm. Anal. Calorim.

110, 973 (2012).19A. Srivastava, M. Chauhan, and R. K. Singh, Phys. Status Solidi B 248,

2793 (2011).20M. J. Buerger, Phase Transformations in Solids (Wiley, New York, 1951),

p. 183.21M. J. Buerger, Fortschr. Mineral. 39, 9 (1961).22M. Watanabe, M. Tokonami, and N. Morimoto, Acta Cryst. A 33, 294

(1977).23O. Blaschko, G. Ernst, G. Quittner, G. Pepy, and M. Roth, Phys. Rev. B

20, 1157 (1979).24W. L. Fraser and S. W. Kennedy, Acta Crystallogr., Sect. B: Struct.

Crystallogr. Cryst. Chem. 28, 3101 (1972).25P. Toledano, K. Knorr, L. Ehm, and W. Depmeier, Phys. Rev. B 67,

144106 (2003).26V. P. Dmitriev, Yu. M. Gufan, and P. Toledano, Phys. Rev. 44, 7248

(1991).27C. E. Sims, G. D. Barrera, N. L. Allan, and W. C. Mackrodt, Phys. Rev.

57, 11164 (1998).28M. A. Blanco, A. Costales, A. Mart�ınPend�as, and V. Luana, Phys. Rev. B

62, 12028 (2000).29A. M. Pend�as, V. Lua~na, J. M. Recio, M. Fl�orez, E. Francisco, M. A.

Blanco, and L. N. Kantorovich, Phys. Rev. B 49, 3066 (1994).30A. T. Asvini meenaatci, R. Rajeswarapalanichamy, and K. Iyakutti, Solid

State Sci. 19, 36–44 (2013).31See http://www.pwscf.org for information about the code used in this

study.32J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 80, 891 (1998).33D. Vanderbilt, Phys. Rev. B. 41, 7892 (1990).34W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals

and Alloys 2 (Pergamon Press, Oxford, 1967).35V. A. Guvanov, Al. L. Ivanovsky, and V. P. Zhukov, Electronic Structures

of Refractory Carbides and Nitrides (Cambridge University Press,

Cambridge, 1994).36M. Marlo and V. Milman, Phys. Rev. B 62, 2899 (2000).37B. D. Fulcher, X. Y. Cui, B. Delley, and C. Stampfl, Phys. Rev. B 85,

184106 (2012).38M. G. Brik and C.-G. Ma, Comput. Mater. Sci. 51, 380–388 (2012).39R. Chauhan, S. Singh, and R. K. Singh, Cent. Eur. J. Phys. 6, 277 (2008).40J.-I. Jang, M. J. Lance, S. Wen, and G. M. Pharr, Appl. Phys. Lett. 86,

131907 (2005).41N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks/Cole,

Canada, 1976).

133507-7 Bhat, Waghmare, and Ramamurty J. Appl. Phys. 113, 133507 (2013)

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