Solid State Communications 149 (2009) 1843–1848

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Solid State Communications

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Lattice dynamical properties of ScB2, TiB2, and VB2 compoundsE. Deligoz a,∗, K. Colakoglu b, Y.O. Ciftci ba Aksaray University, Department of Physics, 68100, Aksaray, Turkeyb Gazi University, Department of Physics, Teknikokullar, 06500, Ankara, Turkey

a r t i c l e i n f o

Article history:Received 28 March 2009Received in revised form26 May 2009Accepted 2 July 2009 by S. ScandoloAvailable online 15 July 2009

PACS:63.20.D-65.40.Ba71.15.Nc

Keywords:D. Heat capacityD. PhononsD. Thermodynamic properties

a b s t r a c t

We have investigated the structural and lattice dynamical properties of XB2(X = Sc,V, Ti) by usingfirst-principles total energy calculations. Specifically, the lattice parameters (a, c) of the stable phase,the bond lengths of X–B and B–B atoms, phonon dispersion curves and the corresponding density ofstates, and some thermodynamical quantities such as internal energy, entropy, heat capacity, and theirtemperature-dependent behaviours, are presented. The obtained results for structural parameters are ingood agreement with the available experimental and other theoretical studies.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Ultra-hard materials are used in many applications, from cut-ting and polishing tools to wear-resistant coatings [1]. Unfortu-nately, almost all ultra-hardmaterials (diamond, cubic BN, etc.) areexpensive because they either occur naturally in limited suppliesor have to be made at high pressure synthetically [2]. Therefore,intense research efforts have been carried out to design ultra-hardmaterials [1,2]. The transitionmetal diborides exhibit unique prop-erties such as high melting points, high hardness values, and ex-cellent oxidation resistance [1–7]. Therefore, they are promisingmaterials for new heat-resistant, corrosion-resistant, and wear-resistant alloys and coatings [8].ScB2, TiB2, and VB2 are members of the transition metal

diborides and they have the hexagonal AlB2 structure with thespace group P6/mmm (space number 191). There are three atomsin the unit cell, all of them on the special positions: X: (0, 0, 0), B:(1/3, 2/3, 1/2) and B: (2/3, 1/3, 1/2).A number of theoretical [9–23] and experimental [24,25]

studies exist in the literature dealing with structural, elastic, andelectronic properties of ScB2, TiB2, and VB2. Specifically, Shein andIvanovskii [9] reported the structural and elastic properties usingthe full-potential linearized augmented plane-wave (FP-LAPW)

∗ Corresponding author. Tel.: +90 382 2801232; fax: +90 382 2801246.E-mail address: edeligoz@yahoo.com (E. Deligoz).

0038-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2009.07.010

method with the generalized gradient approximation (GGA) forScB2, TiB2, and VB2. Oguchi [10] studied the systematic trendin lattice constants and heat of formation for these compounds.Vajeeston et al. [11] investigated electronic structure and groundstate properties of these transition metal diborides using theself-consistent tight-binding linear muffin-tin orbital method (TB-LMTO). The elastic properties, electronic structure, electroniccharge distribution and equation of states of titanium diboridewere studied by Milman and Warren [12], Perottoni et al. [13],and Camp et al. [14] by using the first-principles methods.Peng et al. [15] investigated the thermodynamical propertiesof TiB2 using ab initio plane-wave pseudopotential method.Ravindran et al. [16] studied electronic structure using full-potential linearized augmented plane-wave (FP-LAPW)method forTiB2. Wang et al. [17] predicted the elastic and thermodynamicalcharacteristics of TiB2 crystal through the method of densityfunctional theory within the generalized gradient approximation(GGA). Kaur et al. [18] applied the Rigid Ion Model (RIM) to studythe cohesive and thermal properties of TiB2. Munro [19] examinedthe physical, mechanical, and thermal properties of polycrystallineTiB2 and showed that these properties are significantly related tothe density and grain size of the used specimens.Structural, mechanical, elastic behaviour, electronic charge

density and band structure of VB2 were studied by Mahmudet al. [20] using the ab initio density functional method. Zhouet al. [21] performed first-principles study on the bonding nature,elastic property and hardness of VB2. The band structure and Fermi

1844 E. Deligoz et al. / Solid State Communications 149 (2009) 1843–1848

Table 1Calculated equilibrium lattice parameters (a, c in Å), the X–B and B–B bond distance (in Å) along with the available experimental and theoretical values for XB2 .

Reference a c c/a X–B B–B

ScB2 Present (GGA PP) 3.117 3.407 1.09 2.478 1.799Theorya (GGA FP-LAPW) 3.114 3.512 1.13Theoryb (LDA FP-LAPW) 3.114 3.465 1.11Theoryc (LDA TB-LMTO) 3.154 3.523 1.11 2.533 1.821Experimentald 3.148 3.517 1.12

TiB2 Present (GGA PP) 3.024 3.154 1.04 2.352 1.745Theorya (GGA FP-LAPW) 3.006 3.212 1.07Theoryb (LDA FP-LAPW) 3.005 3.186 1.06Theoryc (LDA TB-LMTO) 3.070 3.262 1.06 2.409 1.443Theorye (GGA USP) 3.029 3.219 1.06(LDA USP) 2.990 3.152 1.05Theoryf (Hartree–Fock) 3.027 3.240 1.07Theoryg (GGA) 2.993 3.147 1.05Theoryh (GGA PP) 3.015 3.222 1.07Theoryi (LDA TB-LMTO) 3.070 3.262 1.06Theoryj (GGA PP) 3.030 3.232 1.06Theoryk 2.998 3.188 1.07 2.409 1.773Experimentald 3.038 3.239 1.07

VB2 Present (GGA PP) 3.002 2.999 0.99 2.291 1.733Theorya (GGA FP-LAPW) 2.970 3.029 1.02Theoryb (LDA FP-LAPW) 2.969 2.989 1.01Theoryc (LDA TB-LMTO) 2.983 3.047 1.01 2.297 1.722Theoryl (LCAO) 3.008 3.068 1.02 2.317 1.736Theorym (GGA) 2.992 3.028 1.01 2.297 1.728(LDA) 2.953 2.965 1.00 2.260 1.705Theoryn (GGA FP-LAPW) 3.007 3.048 1.01Theoryk 2.979 2.995 1.00 2.297 1.772Experimentald 2.998 3.005 1.00Experimentalo 2.997 3.056 1.02

a Ref. [9].b Ref. [10].c Ref. [11].d Ref. [24].e Ref. [12].f Ref. [13].g Ref. [14].h Ref. [15].i Ref. [16].j Ref. [17].k Ref. [23].l Ref. [20].m Ref. [21].n Ref. [22].o Ref. [25].

surface parameters were studied by Shein and Ivanovskii [22] byusing the self-consistent full-potential linearizedmuffin-tin orbital(FLMTO) method for VB2. Experimentally, the phonon density ofstates of TiB2 and VB2 were measured by Heid et al. [23] by usingthe technique of inelastic neutron scattering.In the present paper, we aim to investigate the phonon

dispersion relations and thermodynamical properties of ScB2, TiB2,and VB2 compounds in detail and interpret the salient results ofour calculations. The method of calculation is given in Section 2;the results are discussed in Section 3. Finally, the summary andconclusion are given in Section 4.

2. Method of calculation

Our calculations are based on the density functional formalismand generalized gradient approximation (GGA).Here we use thePerdew–Burke–Ernzerhof (PBE) functional [26] for the exchange-correlation energy as is implemented in the SIESTA code [27,28].This code calculates the total energies and atomic forces using alinear combination of atomic orbitals as the basis set. The basisset is based on the finite range pseudoatomic orbitals (PAOs) ofthe Sankey–Niklewsky type [29], generalized to include multiple-zeta decays. The interactions between electrons and core ionsare simulated with the separable Troullier–Martins [30] norm-conserving pseudopotentials.

Atomic pseudopotentials are generated separately for atomsSc, Ti, V, and B by using the 4s2 3d1, 4s2 3d2, 4s2 3d3 and2s2 2p1 atomic configurations, respectively. The cut-off radii for thepresent atomic pseudopotentials are taken as s: 2.84 au, p: 2.84 au,d: 2.54 au f: 2.54 au of Hf, s: 2.71 au p: 2.71 au d: 2.48 au f: 2.48 auof Ti, s: 2.66 au p: 2.66 au d: 2.50 au f: 2.50 au of V and 1.71 au forthe s, p, d and f channels of B.Atoms are allowed to relax until atomic forces were less than

0.005 eV Å−1. The cut-off energy are taken to be 400 Ryd. 792,1014, and 1183 k-points were found to be adequate to obtainingthe total energy to an accuracy of 1 meV/atom for ScB2, TiB2, andVB2, respectively.

3. Results and discussion

3.1. Structural properties

Firstly, we optimize the lattice constants to obtain a relaxedstructure. The initial lattice parameters are taken from previoustheoretical and experimental values as a starting point forgeometry optimization. The calculated lattice parameters andbond distances are listed in Table 1, along with the availableexperimental and theoretical data for comparison. It is seen thatthe present lattice parameter (a) is in good agreement with the

E. Deligoz et al. / Solid State Communications 149 (2009) 1843–1848 1845

25 TotalBSc

TotalBTi

TotalBV

20ScB2

TiB2

VB2

15

10

5

Γ ΓK A H L AM0

FR

EQ

UE

NC

Y [

TH

z]

Γ ΓK A H L AM

Γ ΓK A H L AM

FR

EQ

UE

NC

Y [

TH

z]

25

20

15

10

5

0

FR

EQ

UE

NC

Y [

TH

z]

25

30

20

15

10

5

0

6 8 10 12 14 16 18 20 22

FREQUENCY [THz]

4 24

FREQUENCY [THz]

6 8 10 12 14 16 18 20 224 2624 28

10 15 20

FREQUENCY [THz]

5 25

.20

.18

.16

.14

.12

.10

.08

.06

.04

.02

AM

PLI

TU

DE

[1/

Hz]

0

AM

PLI

TU

DE

[1/

Hz]

.20

.22

.18

.16

.14

.12

.10

.08

.06

.04

.02

0

.20

.22

.18

.16

.14

.12

.10

.08

.06

.04

.02

AM

PLI

TU

DE

[1/

Hz]

.24

0

Fig. 1. Calculated phonon dispersions, total and partial density of states for ScB2 , TiB2 , and VB2 .

theoretical experimental values, but lattice parameter (c) is lowerthan the experimental and other theoretical values [9–11] by% 2–3 for these compounds. This small difference may stemfrom the different density functional based electronic structuremethods. Also, it is clear from this table that, as we go along thesame row of the periodic table from Sc to V, the lattice constantsslightly decrease in accordance with the mass of the transitionmetal atoms.The bond lengths of ScB2, TiB2, and VB2 are consistent with the

theoretical data. The bond lengths decrease on going from ScB2 toVB2. In fact, it is known that the short bond length strongly affectssome physical properties, such as melting point, elastic constants,and hardness [31].

3.2. Phonon dispersion curves

Many physical properties of solids depend on their phononproperties such as specific heat, thermal expansion, heat conduc-tion, and electron–phonon coupling. Though several works werereported, as stated above, on the structural and electronic prop-erties of these compounds, neither experimental nor theoreticalstudies are exist in literature on the lattice dynamics of these com-pounds, except for density of states (DOS) of TiB2 and VB2 [23],to compare with the present results. Therefore, it is important to

report the results of the lattice dynamic7al calculations of thesecompounds. The present phonon dispersion curves without LO/TOsplitting and other related quantities are calculated by using thePHONON software [32]. This code is compatible with SIESTA anduses the ‘‘Direct Method’’ [33] and the Hellmann–Feynman forceson atoms for generating the phonon dispersion and the density ofstates (DOS). Its theoretical and applicational details can be foundin the PHONON manual and references therein.The optimized rhombohedral supercell with 36 atoms and

basis vectors are given as as = 2a + c , bs = 2b + c , cs =−2a − 2b + c , where a, b, c are the primitive translation vectorsof the hexagonal unit cell. [34]. The symmetry of the selectedrhombohedral supercell breaks the two fold symmetry axis, andlowers the P6/mmm space group to R3m. Here, the displacementamplitudes are taken as 0.03 Å, and the positive and negativeatomic displacements along x, y, and z axes are taken into account.TheHellmann–Feynman forces acting on the atoms in the supercellare evaluated to construct the force constants and dynamicalmatrix. Parlinski [34] used the samemethod and PHONON code forobtaining the correct phonon dispersion relations of the hexagonalMgB2.The calculated phonondispersion curves and the corresponding

one-phonon DOS for XB2 along the high-symmetry directions areillustrated in Fig. 1. The unit cell of XB2 contains three atoms,

1846 E. Deligoz et al. / Solid State Communications 149 (2009) 1843–1848

v=13.47 THz

v=13.56 THz

v=18.64 THz

v=24.11 THz

Fig. 2. E1u , A2u , B1g , and E2g phonon modes for ScB2 at the 0 point.

which give rise to a total of 9 phonon branches. It is seen fromFig. 1 that ScB2, TiB2, and VB2 have very similar dispersion curvesand the corresponding density of states. It is well known that themass difference between anions and cations strongly affects themaximumandminimumvalues of the acoustic and optic branches,and a clear gap is formed between them as in Fig. 1. Here, whilethemaximumvalue of the phonon frequencies for optical branchesfollows the order from larger to smaller as VB2→TiB2→ScB2, themagnitude of gap between the acoustic and optic branches followsthe order of ScB2 →TiB2 →VB2. Interesting features of opticalphonon modes are observed at 0 point. Optical phonon branchesare nearly flat at the 0 point. E2g mode (The highest frequencyoptical branch) which evolve from the doubly degenerate at the0 point for these compounds. Along the A–L and 0–M directionsthese branches are not degenerate.Besidesmany similarities, one can observe some distinctive dif-

ferences on the phonon dispersion curves of these compounds.Forexample, for TiB2 and VB2, a clear gap is formed between opticalbranches at 0 point, but this situation is not seen for ScB2.The calculated 0 point frequencies of that ScB2, TiB2, and VB2

are given Table 2 along with the available experimental data forcomparison. The 0 point frequencies of the E1u, A2u, and B1g are, ingeneral, favorable agreement with the experimental [23] ones. E2gis higher than experimental [23] values by 5% for VB2. The atomicmotions of gamma-point normalmodes are displayed in Fig. 2 onlyfor ScB2 to save space in journal. Graphical package Molekel [35,36] is used to analyze the individual normal modes. E1u and A2uphonon modes include atomic vibrations from both X (X = Sc, V)and B atoms. The situation is slightly different for TiB2. Here E1uphonon modes contain the vibrations from B atom; the A2u modescontain the vibrations from both Ti and B atom. B1g modes includethe atomic vibrations from B atoms for X (X=Sc, V) and includethe atomic vibrations from both Ti and B atoms for TiB2. Due tothe mass difference between X=(Sc, Ti, V) and B atoms, the E2gmodes contain only vibrations of B atom. B1g and E2g gamma-pointmodes involve atomic vibrations along the directions of z and y,respectively for these compounds. The other gamma-pointsmodesinvolve atomic vibrations along the directions as E1u:z, A2u:y forScB2, E1u:z, A2u:x for TiB2 and E1u:x, A2u:z for VB2.The figures on the right side of the phonon dispersion curves

show the corresponding total and partial density of phonon states

TotalBSc

TotalBTi

TotalBV

INT

ER

NA

L E

NE

RG

Y [m

eV/u

nit c

ell]

INT

ER

NA

L E

NE

RG

Y [m

eV/u

nit c

ell]

0 200 400 600 800 1000

TEMPERATURE [K]

1200 1400 1600 1800 2000

1600

1400

1200

1000

800

600

400

200 400 600 800 1000 1200 1400 1600 1800

TEMPERATURE [K]

0 2000

200 400 600 800 1000 1200 1400 1600 1800

TEMPERATURE [K]

0 2000

1600

1400

1200

1000

800

600

400

1400

1200

1000

800

600

400

200

INT

ER

NA

L E

NE

RG

Y [m

eV/u

nit c

ell]

1600

0

Fig. 3. Temperature dependence of internal energy for XB2 .

Table 2Calculated 0 point frequencies in meV for ScB2 , TiB2 , and VB2 .

E1u A2u B1g E2g

ScB2 Present 55.72 56.11 77.08 99.72TiB2 Present 65.11 66.21 67.52 111.85

Experimentala 65.5 66.4 70.0 112.8VB2 Present 60.69 62.69 65.24 120.22

Experimentala 60.6 62.1 69.6 114.9a Ref. [23].

for each compound. While the main contribution to acousticphonons results from the transition metals sublattice, the high-frequency phonons stem from the boron ions. This is expectedbecause the boron atom is lighter than transition metal atoms,which leads to but not very strong electron–phonon interactions.The covalent character of the B–B bonding is also decisive for thehigh frequency of phonons involving the boron atoms.

E. Deligoz et al. / Solid State Communications 149 (2009) 1843–1848 1847

TotalBSc

TotalBTi

TotalBV

TEMPERATURE [K]

200

0 200 400 600 800 1000 1200 1400 1600 1800 2000

TEMPERATURE [K]

0 200 400 600 800 1000 1200 1400 1600 1800 2000

TEMPERATURE [K]

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

-200

-400

-600

-800

-1000

-1200

-1400

FR

EE

EN

ER

GY

[meV

/uni

t cel

l]

-1600

200

0

-200

-400

-600

-800

-1000

-1200

-1400

FR

EE

EN

ER

GY

[meV

/uni

t cel

l]

-1600

0

200

-200

-400

-600

-800

-1000

-1200

-1400

FR

EE

EN

ER

GY

[meV

/uni

t cel

l]

400

Fig. 4. Temperature dependence of free energy for XB2 .

3.3. Temperature dependence of the thermodynamical quantities

As is known, the first-principles phonon calculations are limitedto T = 0 K, and the thermodynamic properties of the crystalscan be determined in detail by phonons. The quasi-harmonicapproximation can describe such properties quite satisfactorily. Inthe Gibbs free energy expression G(T , P) = F(T , V )+ PV , the freeenergy F(T , V ) of a crystal is the sum of the ground state energyE(V ) and the phonon free energy Fphon(T , V ). The temperature Tappears in F(T , V ) via the phonon term. Themore explicit forms ofthe internal energy, free energy, entropy and heat capacity in theharmonic approximation per primitive unit cell are given [32] asfollows:

E =12r∫∞

0g(ω)(h̄ω) coth

(h̄ω2kBT

)dω, (1)

F = rkBT∫∞

0g(ω) ln

[2 sinh

(h̄ω2kBT

)]dω, (2)

Total18

16

14

12

10

8

6

4

2

BSc

TotalBTi

TotalBV

TEMPERATURE [K]

0 200 400 600 800 1000 1200 1400 1600 1800 2000

TEMPERATURE [K]

0 200 400 600 800 1000 1200 1400 1600 1800 2000

TEMPERATURE [K]

0

EN

TR

OP

Y [k

_b/u

nit c

ell]

EN

TR

OP

Y [k

_b/u

nit c

ell]

18

16

14

12

10

8

6

4

2

0

EN

TR

OP

Y [k

_b/u

nit c

ell]

18

16

14

12

10

8

6

4

2

00 200 400 600 800 1000 1200 1400 1600 1800 2000

Fig. 5. Temperature dependence of entropy for XB2 .

S = rkB

∫∞

0g(ω)

{(h̄ω2kBT

)[coth

(h̄ω2kBT

)− 1

]− ln

[1− exp

(h̄ω2kBT

)]}dω, (3)

C = rkB

∫∞

0g(ω)

(h̄ωkBT

)2 exp(h̄ωkBT

)[exp

(h̄ωkBT

)− 1

]2 dω, (4)

where r is the number of degree of freedom in the unit cell, h̄ isthe Planck constant, kB is the Boltzmann constant, and T is thetemperature.The above formulae are used to plot the temperature depen-

dence of the internal energy (E), free energy (F ), entropy (S), andheat capacity at constant volume (Cv) for each compound and theirconstituent atoms, by using the data obtained from SIESTA andPHONON software.The calculated internal energies for XB2 as a function of

temperature are depicted in Fig. 3. It is seen that, above 300 K, thetotal internal energies increase almost linearly with temperature.

1848 E. Deligoz et al. / Solid State Communications 149 (2009) 1843–1848

TotalBSc

TotalBTi

TotalBV

TEMPERATURE [K]

0 200 400 600 800 1000 1200 1400 1600 1800 2000

TEMPERATURE [K]

9

8

7

6

5

4

3

2

1

HE

AT

CA

PA

CIT

Y [k

_B/u

nit c

ell]

0

HE

AT

CA

PA

CIT

Y [k

_B/u

nit c

ell]

0 200 400 600 800 1000 1200 1400 1600 1800 2000

9

8

7

6

5

4

3

2

1

0

TEMPERATURE [K]

HE

AT

CA

PA

CIT

Y [k

_B/u

nit c

ell]

9

8

7

6

5

4

3

2

1

00 200 400 600 800 1000 1200 1400 1600 1800 2000

Fig. 6. Temperature dependence of heat capacity for XB2 .

The obviously, the contributions to internal energy from B isdominant than transition metal atoms. At high temperatures theinternal energy tends to kBT behaviour.Fig. 4 shows the variations of phonon free energy with

temperature for the same compounds. The overall profiles ofall plots are similar and the phonon free energy decreases withincreasing temperature.The variations of entropy with temperature for XB2 are given in

Fig. 5 for the same temperature range. The total and partial entropygraphs exhibit a similar trend and the contributions to internalenergy from transitionmetal atoms are dominant thanboron atom.The contributions to the total heat capacity from the lattice

vibrations of XB2 are illustrated in Fig. 6. It is realized from figuresthat when T < 700 K, Cv increases very quickly with temperature;

when T > 700 K, Cv increases slowly with temperature andit almost approaches a constant called the Dulong–Petit limit athigher temperatures. It is seen that while the contribution tothe transition metal atoms (Sc, Ti, V) to the heat capacity curvesdecreases on going fromSc to V, the contribution to the boron atomincreases. The temperature is limited to 2000 K to minimize thepotential influence of anharmonicity in all graphs.

4. Summary and conclusion

The structural, vibrational, and thermodynamical propertiesfor ScB2, TiB2, and VB2 are presented by using the first-principles calculations implemented in SIESTA within the GGAapproximation. The results, generally, are in agreement with theavailable experimental and theoretical data. The original aspectsof the present calculations concern the phonon dispersion curvesand thermodynamical results at various temperatures,which is notconsidered previously. ScB2, TiB2, and VB2 exhibit similar, but notvery strong electron–phonon interactions.

Acknowledgment

Thiswork is supported byGazi University Research-Project Unitunder Project No: 05/2008-42.

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