First-principles study of the structural, electronic, and elastic properties of RRh_ $$3$$ B_ $$x$$...

First-principles Study of Structural, Electronic, Elastic, Phonon, and Thermodynamical Properties of the Niobium Carbide

Nikita Rathod1,2,a, S.D. Gupta2,b, S.K. Gupta2,c and Prafulla K. Jha2,d

1Department of Physics, Bhavnagar University, Bhavnagar-364022, India

2Vadodara Institute of Engineering, Vadodara- 391510, India

[email protected], [email protected],

[email protected], [email protected]

Key words: Carbide, Density Functional Theory, Phonon, Thermal Properties

Abstract. A detailed theoretical study of structural, electronic and vibrational properties of niobium carbide are carried out in rocksalt phase using the density functional theory implemented in ABINIT code. The calculated structural parameters like lattice constant and bulk modulus agree well with the available data. The Zener anisotropy factor (A), Poison's ratio (v), Young’s modulus (Y) and shear modulus (C’) are also presented. The electronic band structure and density of states are presented and discussed in light of bonding nature in NbC. The band structure indicates its metallic nature. The calculated phonon dispersion curves show that the NbC in rocksalt phase has all positive phonons throughout the Brillouin zone. The thermodynamical properties are also presented and discussed.

1. Introduction

Transition metal mono carbides and nitrides are scientifically interesting and promising materials for various industrial applications such as suitable metal gates in the new generation of complementary metal oxide semiconductor technology [1]. Presently scientific interest is to develop within the microelectronic industry for the use of transition metal mono carbides and nitrides especially niobium carbide (NbC) and niobium nitride (NbN) as an electrical conductivity barrier due to its high superconducting temperature (Tc ≈ 20K) [2]. The specific combination of thermal, electrical and mechanical properties such as high melting temperature, high hardness, high-temperature strength, good electrical and thermal conductivity of transition metal mono carbides posses a special attention. These properties are the expression of the interaction of the three different type of bonding; metallic, ionic and covalent which gives unusual combinations of bonding mechanism and manifests itself in their macroscopical properties. The over all bonding nature of mono carbides can be described as covalent of N and metal states, but there is also some ionic character with electron transfer from the metal to nitrogen atoms. In addition the metallic character is responsible for making transition metal carbides ultra hard [3-8]. These mono carbides are also used as high-temperature structural materials in the form of hard constituents in metal matrix composites [9–10], in adsorption, in catalytic properties and in surface chemical reactivity [11]. These properties of transition metal carbides have lead to technological applications in developing wear-resistant coatings for ball bearings and cutting machine tools [1].

Most of the unusual features of transition metal carbides (TMCs) in particular depends on occupancy of number of d-electrons and to reveal these properties, several first principles calculations have been conducted in the recent and past with regards to the bulk properties of these materials particularly for the properties categorizing for niobium carbide among refractory materials, which are usually associated with typical covalent character [13]. The rigorous understanding of the

Solid State Phenomena Vol. 171 (2011) pp 67-77Online available since 2011/May/17 at www.scientific.net© (2011) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/SSP.171.67

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 117.211.84.58-03/06/11,12:55:19)

http://www.scientific.net

http://www.scientific.net/feedback/121889


http://www.ttp.net

electronic and standard behaviour as well as structural properties of niobium carbide is representative and important members of the transition metal carbides.

Niobium carbide structurally crystallizes in a wide range of compositions with the chemical formula NbCx. In fact even its stoichiometry form has not been yet synthesized because of the energy

formation of the non-stoichiometric compound has the lowest even in the compositions range of 1≤ x

≤ 0.84 , niobium carbide should crystallizes in the cubic B1 (NaCl type) structure before it transforms

in to the hexagonal Nb2C structure at higher niobium content [14-16]. The electronic structure and bonding properties of TMC compounds have been studied theoretically by several authors in the recent past [17-19], using TB-LMTO [20] and plane wave pseudopotential method within the framework of density functional theory [21]. These studies reveal that the interaction of the TM-d and C-p state strongly influence the electronic structure and nature of bonding.

In the present contribution, we present the systematic first principles total energy calculation of structural, electronic, elastic, vibrational, and thermodynamical properties of one of the carbide, NbC from the group of transition metal carbides having potential applications due to its unique properties arising from the d-electron in transition metal ion. The physics and thermodynamical properties of several transition metal carbides and nitrides have been well reviewed in refs [9, 21]. Their electronic properties have been studied by means of both experimental and theoretical methods and the general trends as a function of composition of transition metal atom type are summarised in ref [22]. Experimental investigation of phonon spectra of NbC have been done by smith and co-workers [23-25], where pronounced anomalous of the phonon dispersion curves along the (100) and (110) directions for NbC and TaC found. Möstoller [26] carried out phonon calculation for transition metal nitrides and carbides in the framework of the dielectric response approximation using the Heine- Abarenkov pseudopotentials [27]. The model was not able to reproduce the observed anomalies of the phonon dispersion curves for HfC, TaC and NbC near the Brillouin zone boundaries. The influence of carbon vacancies on the first-order Raman spectrum of TiC was investigated in ref [28]. The measured spectrum for TiC, ZrC, NbC, TaC and HfC have been examined by means of phenomenological “double-shell model” [29]. Weber and co-worker [30-31] investigated phonon anomalies and high superconducting temperature Tc as well as electron-phonon coupling constants in transition metal carbides and nitrides using the strong coupling theory of superconductivity [32-33]. They concluded that the electron-phonon matrix elements are mainly determined by strong d-d overlap interaction. Ab-initio phonon calculation for a transition metal carbide (NbC) has been done by Saravsov [34] using the linear response technique in the frame work of LMTO. The agreement between theory and experiment was fairly good. The peculiarities of the spectrum along the [100] direction were well reproduced while anomalies lie along the high symmetry [110] were not so pronounced compared to the model calculation of Weber [28]. First-principles investigations of phonon spectrum for TiC [35] and ZrC [36] have been done in the framework of the supercell method, and the Hellmann–Feynman force theorem in conjunction with norm-conserving pseudopotentials. Besides, there was a considerable discrepancy between the calculated elastic constants, especially for C12 and the ones from ultrasonic experiments. The discrepancy was ascribed to the difference in slope of the experimental and theoretical phonon dispersion near the Brillouin zone centre which is presumably caused by lacking for the description of the long-range coulomb interaction.

The paper is organised as follows, the brief description of density functional theory along with the computational methodology is presented in section-II. We present theoretical results and discussion in section-III. The results are concluded in section-IV.

68 Latest Trends in Condensed Matter Physics



2. Methodology and Computational Details

The properties of materials determined by the interactions of electrons and nuclei and its description can be represented by well known Schrödinger equation. For all earlier calculations Born-Oppenheimer or adiabatic approximation was used, which separates the motions of the electrons from the nuclei from considering the static nuclei due to its heavy mass. However, to correlate the electron and nuclei mass together and find the properties with the total energy calculation using Schrödinger equation is being used. However, the Schrödinger equation is notanalytically solvable for every system and numerical approaches to obtaining the appropriate solutions for electronic structure have become invaluable to the determination of properties of materials. These computational efforts received a critical boost by Hohenberg and Kohn [37] and Kohn and Sham [38], reformulated the Schrödinger equation, which involves all the 3N spatial coordinates of N interacting electrons, into so called density functional theory (DFT), a theory only based on the electron density, which is a function only of space co-ordinates. The Kohn-Sham equations so obtained turns into a computationally manageable single particle effective potential via the exchange correlation functional. The local density [39-42], generalised-gradient [41-44] and hybrid [42-43] etc. approximations are being used for this purpose. There are different codes to implement the DFT to find the different kind of properties such as structural, energetic, electronic excitations, dynamical properties of materials. All the calculations reported in this work are based on the generalised gradient approximation of the density functional theory [44] implemented in the ABINIT code [45]. The ABINIT code employs first principles pseudo potentials and plane wave basis. The calculations are performed using Troulliers-Martins generalised gradient approximations pseudo potentials and Perdew, Burke and Enzerhoff (PBE) functionals for the exchange correlations functionals for the exchange and correlations energy [46]. The nature of the pseudopotentials ensures that they exhibit the same scattering properties (logarithmic derivatives) as the all electron potentials in the neighbourhood of the atomic eigenvalues, which is a measure of the pseudo potentials's performance. The smoothness ensures that they have rapid convergence in the calculated total energy of the system. This ensures that the rapid convergence of the convergence of the system properties, with respect to total energy and increase in the plane wave basis set. The electronic densities of states and wavefunctions used in the determination of the dynamical matrices are computed on a face-centred 8x8x8 point grid. These points are defined as in the same way as the Monkhorst-Pack special k-points [47]. The Fourier interpolation with specific treatment of the long range dipole-dipole interaction is employed to get the phonon bandstructure using the interatomic force constants. Technical details on the computation of responses to atomic displacements and homogeneous dielectric fields can be found in ref [48].

3. Results and Discussion

This section reports on the equilibrium lattice parameter, bulk moduli, electronic properties, elastic properties, mechanical properties and phonon properties. The lattice parameter of the NbC in RS structure is calculated corresponding to minimum total energy of the system at zero pressure. The Burch-Murnaghan equation applied to extract the lattice parameter with minimum of the total energy curve with respect to total unit cell volume of the system (Fig.1).

The fitting to the Murnanghan equation of state extract the bulk modulus and its derivative with the negligible error bar 10-7 GPa. The so obtained lattice constants is converged with the cut off energy (ecut) 50 Ha and mesh of 8x8x8 (Fig.3). The tolerance of ecut and mesh is 0.0001Ha and 0.000001 Ha respectively. The elastic constants of niobium mono carbide are calculated using the density functional perturbation theory and compared with other theoretical and experimental study shown in Table 1. The present table reveals that there is a good agreement between theoretical and

Solid State Phenomena Vol. 171 69



experimental elastic constants. It is noticeable that the uncertainty lies in measuring the elastic constant in neutron scattering experiment as high as 10% to 15% yet the experimental elastic constants in Table 1 have a very good agreement with the theoretically calculated elastic constants. The calculated elastic constant satisfies the elastic stability criteria for a cubic structure at equilibrium conditions are,

4.40 4.45 4.50 4.55 4.60 4.65 4.70

-11.054

-11.052

-11.050

-11.048

-11.046

-11.044

-11.042

4.47 Å (Exp)

T

ota

l e

ne

rgy

(H

a)

acell (Å)

4.504 Å (Theory)

Fig.1. Equilibrium lattice constants Vs total energy.

10 20 30 40 50 60 70

-11.05

-11.00

-10.95

-10.90

Eto

tal (H

a)

Ecut (Ha)

Cutoff energy= 50 Ha

Fig. 2. Cutoff energy Vs total energy.




4 6 8 10 12-11.0575

-11.0570

-11.0565

-11.0560

-11.0555

-11.0550

-11.0545

-11.0540

-11.0535

To

tal e

ne

rgy(H

a)

k-points Mesh(nXnXn)

k-points mesh (8x8x8)

Fig. 3. Mathfessel k-points Vs total energy.

C11 + 2C12 > 0, C44 > 0, and C11-C12 > 0 (1)

( )12110 23

1CCB += ,

1211

442

CC

CA

−= ,

+

−=

GB

GB

V

2

22

2

2

1 and

BG

GBY

3

9

+= (2)

The elastic compliance constants (Cij) obtained with DFPT, since the benefit of simple cubic symmetry C11, C12, and C44 are sufficient to describe the elastic property in the RS phase. The bulk modulus (B), Young modulus (Y) and anisotropy factor (A) are calculated using the formula shown in Eqn. (2) and as presented in Table 2. The calculated electronic bandstructure and density of states (DOS) in the high symmetry directions for first Brillouin zone using the calculated equilibrium lattice constants of the rocksalt NbC are presented in Fig. 2 Band structure indicates that the rocksalt NbC is metallic in nature. Table 1. Elastic constants Cij (GPa), Lattice constantsa a0 (Å) and Bulk Modulus B (GPa) for NbC in

rock-salt structure.

NbC (Rock salt)

C11(GPa) C12(GPa) C44(GPa) a0 (Å) B(GPa)

Our Work

Other Theory

627.12a

620.0c, 640.0d

647.0 g

134.67 a

200.0 c 180d,108.9 g

164.26 a

150.0 c 140d,110.9g

4.51 a

4.43b 4.47c(Exp)

4.45 d ,4.476i

4.47j ,4.471k

4.45l

298.82 a

293 e 335b, 328e,302f,

332 d , 301i, 331k(Exp),

315m,302n(Exp)

aReference [our work], bReference [12], cReference [28], dReference [49],eReference [3], fReference [4], gReference [50], hReference [51], iReference [52], jReference [53], kReference [54], lReference [55], mReference [56]




Table 2. Anisotropy factor (A), Poisson’s ratio (v) and Young’s Modulus (Y) GPa for rock salt NbC.

NbC (Rock salt)

(A) (v) (Y) GPa

Our Work

0.667

0.2345

475.94

The levels at the symmetry points have been connected by smooth curve, crossing of the bands are observed, which is due to the relative magnitude of the effective mass. It is clear from Fig. 4 that the conduction band is formed due to the mixing of unbound 4-d states of Nb and 2p states of carbon. Furthermore, their hybridization with carbon 2p orbital forming the valence band is less strong and bands are in curve nature.

-1 0

0

1 0

2 0

3 0

L ΓΓΓΓ γγγγX

En

erg

y (

eV

)

Ef

E D O S

Fig. 4. The electronic band-structure and energy density of state calculated from the in full Brillouin zone.

Meanwhile it can be seen from density of states that the bonding and antibonding states are

primary composed of carbon 2p orbital and niobium 4d orbital. The electronic DOS suggests that the chemical bonds are mainly formed between carbon and niobium atoms, which are the same as in the early d-block transition metal carbides [57]. The density of states of rocksalt NbC lies mainly in four energy regions: (i) the lowest region stemming mainly from C-2s states; (ii) the region at the bottom of the valence-band complex, originating from 4d-Nb and 2p-C states; (iii) the region at the top of the valence-band complex due to the 4d-Nb states hybridizing with some 2p-C states; (iv) the energy region just above Fermi energy dominated by unoccupied 4d-Nb states. The electronic bandstructure and density of states are well matched with earlier reported theoretical work by Maksimov et.al [58]. The peak of the density corresponding to the 4d states of Nb site at 1.5 eV above the Fermi energy and 0.4 eV for Nb site. Here, it is also clear that s, p and d filled valence electrons are more populated at high symmetry points responsible for metallic nature.

In order to get an idea about the behaviour of phonons and role played by them in the stability in the rock salt NbC, we have calculated the complete phonon dispersion curves (PDC) alongwith phonon density of states in major symmetry directions of the BZ of NbC and plotted them in Fig. 5




As the cubic structure has two atoms in the primitive unit cell there are 6 phonon branches. Fig.5 reveals that the all phonon modes are positive in the phonon dispersion curves throughout the Brillouin zone. The positive frequencies reflect and confirm that the RS structure of NbC is dynamically stable. It is also seen from the Fig. 5 that the large difference in mass between anions and cations results in large gap and hence the divergence (≈256 cm-1) between the acoustical and optical phonon branches is very clear in stable RS-NbC.

The LO/TO splitting is zero and triply degenerate at the symmetry Γ- point (zone centre), suggesting metallic nature of NbC. The phonon gap can be easily seen in the phonon DOS plot. The sharp peaks in the phonon DOS correspond to the flat modes of the phonon dispersion curves which belong to both optic and acoustic branches. Similar to the dispersion curves, there is a gap between the optic and acoustic branches. We obtained precisely accurate ab-initio interatomic force constants and phonon band structure of NbC in RS phase based on variational density functional perturbation theory [44] implemented in ABINIT code [45].

0

100

200

300

400

500

600

700

Fre

qu

en

cy(c

m-1)

Γ Κ Χ Γ Γ Κ Χ Γ Γ Κ Χ Γ Γ Κ Χ Γ L Χ Χ Χ Χ W L

Fig.5 The Phonon dispersion curve and phonon density of state calculation in full Brillouin

zone.

A complete knowledge of the vibrational spectrum over the entire Brillouin zone (BZ) is required for the calculation of the thermodynamic function like the Helmholtz free energy (∆F), the internal energy (∆E), the entropy (S) and the constant-volume specific heat (Cv), at temperature (T). They are calculated using the following formulas within the harmonic approximation .

(3)




(4)

(5)

(6)

Where KB is the Boltzmann constant, n is the number of atoms per unit cell, N is the number of unit cells, ωmax is the highest phonon frequency and ω is the phonon frequency.

0 500 1000 1500 2000-160,000-140,000-120,000-100,000-80,000-60,000-40,000-20,000

020,000

F(J

/mo

l-C

)

(a)

0 500 1000 1500 20000

20000

40000

60000

80000

100000 (b)

E(J

/mo

l-C

)

0 500 1000 1500 20000

20

40

60

80

100

120

140

(c)

Temperature(K)

Sp

ecif

ic H

eat

(Cv)

0 500 1000 1500 20000

10

20

30

40

50 (d)

Temperature(K) Temperature(K)

S(J

/mo

l-C

-K)

Temperature(K)

Fig. 6. Phonon contribution to the Helmholtz free energy ∆F, internal energy ∆E, the entropy S and specific heat Cv at constant volume of Niobium carbide.

In Figure 6(a) we present the variation of free energy with temperature which reveals that the free energy decreases with increasing temperature. The computed temperature dependent internal energy is presented in Fig. 6(b) and opposite trend is observed. Fig. 6(c) presents the lattice specific heat at constant volume for NbC in RS phase and reveals that the specific heat behavior approaches to the Dulong and Petit limit at high temperature. The figure clearly shows that the CV does not follow the Debye’s specific heat formula at low temperature, which is due to the anharmonic

interaction. Fig. 6(d) shows the gradual increase of internal energy E∆ , and it approaches to the

same value at high temperature and increases almost linearly. The zero temperature values ∆F0 and




∆E0 (13.0 KJ/mol) do not vanish due to the zero point motion, and calculated from the symptotic

expressions of eqn. (3) and (4) [59].

4. Conclusions

In conclusion, we have presented the structural, electronic, elastic, phonon, and thermodynamical properties of niobium carbide in rock salt phase using density functional theory within the GGA approximation. There is a good agreement between the calculated lattice constant, bulk modulus and available experimental and theoretical ones. The zero pressure elastic constants and their related quantities such as, shear modulus, Young’s modulus, anisotropy factor etc. have been also calculated. The compound is highly anisotropic, stiffer and ductile in nature. The band structure shows that the NbC in RS phase behave metallic. Our phonon calculation based on density functional perturbation theory using linear response theory suggests that the RS-NbC is a dynamically stable phase as all the phonon modes in the phonon dispersion curves are positive. Within the harmonic approximation phonon properties are used to determine the thermodynamic properties such as Gibbs free energy, internal energy, entropy and constant-volume specific heat for NbC. The thermodynamic behaviour does not show any anomalous behaviour in the studied temperature region.

5. Acknowledgment

This work is supported by the University Grants Commission (UGC) and Department of Science and Technology (DST), Govt. of India. The computations are performed using PAWAN at the Department of Physics, Bhavnagar University, Bhavnagar under project grant of DST, Govt. of India.

References

[1] G .S. Upadhyaya, Nature and Properties of Refractory Carbides (New York: Nova Science Publishers), (Ed) 1996.

[2] Y. C. Yeo, Q. Lu, P. Ranade, H. Takeuchi, K. J Yang, I. Polishchuk, T. J. King, C. Hu, S. C. Song, H.F. Luan, D. L. Kwong, Electron cevice letters, Vol. 22 (2001), p. 227.

[3] T. Amrious, B. Bouhafs, S. Bresson, B. Khelifa, C. Methieu, Physica B, Vol. 325 (2005), p. 46.

[4] A. F. Gullermet, H. Jaglund, G. Grimvall, M. Smith, Phys. Rev. B, Vol. 45 (1992), p. 11557.

[5] K. Schwarz, Solid State Mater. Sci, Vol. 13 (1987), p. 211.

[6] P. Blaha, K. Schwarz, Int. J. Quantum chem. Vol. 23(1983), p.1535.

[7] M. Gupta, A. J. Freeman, Phys. Rev. B, Vol. 14 (1976), p. 5202.

[8] P. M. Klein, D. A. Papaconstantopoulos and L. L. Boyer, Phys. Rev. B, Vol. 22 (1980), p. 1946.

[9] L. E. Toth 1971 Transition Metal Carbides and Nitrides (New York: Academic).

[10] S. T. Oyama, The Chemistry of Transition Metal Carbides and Nitrides (Glasgow: Blackie Academic and Professional) 1996.

[11] S. V. Didziulis, K. D. Butcher and S. S. Perry, Inorg. Chem, Vol. 42 (2003), p. 7766.

[12] K. B. Joshi and U. Paliwal, Phys. Scr, Vol. 80 (2009) p. 055601.

[13] R. B. Levy, M. Boudart, science, Vol. 181, (1973), p. 547.




[14] H. P. Liermann a,, A.K. Singh, M. Somayazulu and S.K. Saxena, Int. Journal of Refractory Metals & Hard Materials, Vol. 25 (2007), p. 386.

[15] H. W. Hugosson, O. Eriksson, U. Jansson, B. Johansson, Phys. Rev. B, Vol. 63 (2001), p. 134108.

[16] H. W. Hugosson, U. Jansson, B. Johansson and O. Eriksson, Chem Phys Lett Vol. 50 (2001), p. 333.

[17] J. C. Grossman, A. Mizel, M. Cote, M.L. Cohen, and S.G. Louie, Phys. Rev. B, Vol. 60 (1999), p. 6343.

[18] S. T. Oyama, Catalysis Today, Vol. 15 (1992), p. 179.

[19] J. Häglund, A. Fernández, Guillermet, Phys. Rev. B, Vol. 48 (1993), p. 11685.

[20] B. N. Onwuagba, V. Sundararajan, Physica solidi status(b), Vol. 190 (1995), p. 393.

[21] E. K. Storms, The Refractory Carbides, Academic, New York, 1967.

[22] V. A. Gubanov, A. L. Ivanovsky, and V. P. Zhukov, Electronic Structure of Refractory Carbides and Nitrides, Cambridge University Press, Cambridge, 1994.

[23] H. G. Smith and W. Gläzer, Phys. Rev. Lett, Vol. 25 (1970), p. 1611.

[24] H. G Smith, Phys. Rev. Lett, Vol. 29 (1972), p. 353.

[25] H. G Smith and W Gläzer, in Proceedings of the International Conference on Phonons, Rennes, France, July 1971, edited by M. A. Nusimovici Flammaron Sciences, Paris, 1971.

[26] M. Mostoller, Phys. Rev. B, Vol. 5 (1972), p. 1260.

[27] V. Heine and I.A.barenkov, Philos. Mag, Vol. 9 (1964), p. 451.

[28] I. A. Barenkov and V. Heine, ibid, Vol. 12 (1965), p. 529.

[29] M. V. Klein, J. A. Holy and W. S Williams, Phys. Rev. B, Vol. 17 (1978), p. 1546.

[30] W. Weber, H. Bilz and U. Schröder, Phys. Rev. Lett, Vol. 28 (1972), p. 600

[31] W. Weber, Phys. Rev. B, Vol. 8 (1973), p.5093.

[32] W. L. McMillan, Phys. Rev, Vol. 167 (1968), p. 331.

[33] G. M. Eliashberg, Z. Eksp and T. Fiz, Vol. 38 (1960) p. 966. [Sov. Phys. JETP, Vol. 11 (1960). p. 696]

[34] S. Yu. Savrasov, Phys. Rev. B, Vol. 54 (1996). p. 16470.

[35] P. T. Jochym, K. Parlinski and M. Sternik, Eur. Phys. J. B, Vol. 10 (1999) p. 9

[36] P. T. Jochym, K. Parlinski, M. Sternik, Eur. Phys. J. B, Vol. 15 (2000), p. 265.

[37] W. Kohn and. L. J. Sham, Phys. Rev. A, Vol. 140 (1965), p. 1133.

[38] J. P. Perdew and A. Zunger Phys. Rev. B, Vol. 23 (1981), p. 5048.

[39] J. P. Perdew and Y Wang, Phys. Rev. B, Vol. 45 (1992), p. 13244.

[40] S. H. Vosko, Wilk and M.Nusiar Can. J. Phys, Vol. 58 (1980), p.1200.

[41] J. P. Perdew, K. Burke and M. Ernzerhof , Phys. Rev. Lett, Vol. 77 (1996), p. 3865.

[42] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, Phys. Rev. B, Vol. 46 (1992), p. 6671.




[43] A.D. Becke, Phys. Rev. A Vol. 38 (1988) p. 3098, C. Lee, W Yang and R.G Parr, Phys. Rev. B, Vol. 37 (1988) p. 85

[44] P. Hohenberg, and W. Kohn, Phys. Rev B, Vol. 136 (1964), p. 3864.

[45] The ABINIT Code is a Common Project of Universite Catholique de Louvain,Corning Incorporated and Other Contributors <www.abinit.org>

[46] J. P. Perdew, K. Burke, M Erzerhoff, Phys. Rev. Letter, Vol. 77 (1996), p. 3865.

[47] H. J Monkhorst and J. D Pack, Phys. Rev B, Vol. 13 (1976), p. 5188.

[48] X. Gonze, Phys. Rev B, Vol. 55 (1997), p. 10337.

[49] J. Chen, L. L. Boyer, H. Karakauer, Phys. Rev B, Vol. 37 (1988), p. 7.

[50] A. Singh, M. Aynyas, S. P. Sanyal, Cent. Eur. J. Phys, Vol. 7 (2009), p. 102.

[51] E. I. Esav, S. I. Simak, I.A. Abrikosov, R. Ahuja, Y. K. Vekilov, M. I. Katsnelson, A. I. Lichtenstein and B. Johnsson, Journal of applied physics, Vol. 101 (2007), p. 123519.

[52] R. Colella and B. W. Batterman, Phys. Rev. B, Vol. 1 (1970), p. 3913.

[53] A. M. Nartowski, I. P. Parkin, M. MacKenzie, A. J. Craven, and I. MacLeod, J. Mater. Chem. Vol. 9 (1999), p. 1275.

[54] D. J. Singh and B. M. Klein, Phys. Rev. B, Vol. 46 (1992), p. 14969.

[55] D. J. Siegel, L. G. Hector and J. B. Adams, Phys. Rev. B, Vol. 67 (2003), p. 092105.

[56] V. A. Gubanov, A. L. Ivanovsky and V. P. Zhukov, Electronic Structure of Refractory Carbides and Nitrides Cambridge University Press, Cambridge (1994).

[57] R. B. Levy, M. Boudart, Science Vol. 181 (1973), p. 547.

[58] E. G. Maksimov, S.V. Ebert, M. V. Magneistkay and A.E. Karakozov, J. Exp and Theo. Phys., Vol. 105 (2007), p. 642.

[59] C. Lee and X. Gonze, Phys. Rev B, Vol. 51 (1995), p. 8610.




Latest Trends in Condensed Matter Physics doi:10.4028/www.scientific.net/SSP.171 First-Principles Study of Structural, Electronic, Elastic, Phonon, andThermodynamical Properties of the Niobium Carbide doi:10.4028/www.scientific.net/SSP.171.67



First-principles study of the structural, electronic, and elastic properties of RRh_ $$3$$ B_ $$x$$...

Documents

Transcript of First-principles study of the structural, electronic, and elastic properties of RRh_ $$3$$ B_ $$x$$...