Structural, electronic, and superconducting properties of carbides and nitrides of vanadium (V), niobium (Nb), and tantalum (Ta) (group V transition elements) have been studied by computing their electronic band structure characteristics. The electronic band structure calculations have been carried out based on the density functional theory (DFT) within the local density approximation (LDA) by using the tight binding linear muffin tin orbital method. The NaCl-type cubic structures of MN and MC (M=V, Nb, Ta) compounds have been confirmed from the electronic total energy minimum of these compounds. The ground state properties, such as equilibrium lattice constant (a0), bulk modulus (B), and Wigner–Seitz radius (S0) are determined and compared with available data. The electronic density of states reveals the metallic nature of the chosen materials. The electronic specific heat coefficient, Debye temperature, and superconducting transition temperature obtained from the band structure results are found to agree well with the earlier reported literature.
Transition metal carbides (TMC) and transition metal nitrides (TMN) are interesting materials in terms of their electronic, thermal, and superconducting properties. These types of materials are highly metallic in nature and found to have a high melting point. The occupancy of d-electrons from the transition metal (TM) ions gives the important properties, such as chemical stability, high corrosion resistivity along with superconductivity [1–6]. Both carbides and nitrides of the transition metals show the combination of three bonding nature, such as strong covalent metal–non-metal bonding, less ionic and metallic bonding [7, 8]. These materials have attracted considerable interest of research owing to their interesting scientific and technological applications [9–11].
Many authors have studied the TMC and TMN by experimental and computational methods in the recent years [12–15]. Several studies reveal the electronic, bonding, and elastic properties of these materials. The interaction between the d-band of the metal atom and p-band of the non-metal atom are discussed mainly through the first principles studies [16–19]. The electronic structure properties of Group V and VI metal nitrides are studied by Papaconstantopoulos et al. and reported results on their superconducting properties . Klein et al. reported the effect of carbon vacancies in the three metal monocarbides, namely NbC, TaC, and HfC . Srivastava et al. studied the high-pressure phase transition from B1 phase to B2 phase of both TMC and TMN based on the DFT approach [22, 23]. Cohesive and thermodynamic properties have been reported by Háglund et al.  and Guillermet et al.  for 3d and 4d transition metal carbides and nitrides based on the electronic band structure and CALPHAD studies. In the application of steel industry, the carbides and nitrides of Ti, Nb, Mo, and V form microstructural constituents for the high-strength low alloy (HSLA) steels along with elements, such as Ta, Zr, and Hf, to form carbides and nitrides that impart the strength to the HSLA steels .
In this study, we carry out the ab initio calculations for the rocksalt structure of the 5th group transition metal carbides and nitrides, and the results are compared with the other first principles calculations and also with the available experimental results. Electronic density of states reveals that the contribution of d-bands from 3d, 4d, and 5d states of transition metals across the group is dominant. The superconducting transition temperature has been calculated for the chosen TMC and TMN materials and compared with the earlier theoretical and experimental results available in the literature [13, 19].
2 Computational Details
The electronic structure and the total energy calculations of MN and MC (M=V, Nb, Ta) compounds have been investigated through the first principles calculations by using the computational scheme offered by Andersen’s tight-binding linear muffin-tin orbital method with atomic sphere approximation . In this method, the electronic structure calculation is based on the density functional theory (DFT) within the local density approximation (LDA) . Exchange and correlation contributions to both the atomic and crystalline potentials have been included through the von Barth–Hedin parameterisation scheme . The tetrahedron method of Brillouin zone integration has been used to calculate the density of states . All the compounds crystallise in the NaCl-type (B1 phase) face-centred cubic (FCC) structure with Fm-3m (no. 225) space group. A mesh of 12×12×12 and 14×14×14 has been taken in the irreducible wedge of Brillouin zone for TMC and TMN compounds, respectively. E and k convergence are also checked carefully. To find the equilibrium lattice constant, the total energies have been computed by reducing the crystal volume from 1.20V0 to 0.80V0, where V0 is the equilibrium volume. The computed electronic total energies for each of these compounds with respect to relative volume were fitted using the Murnaghan equation of state  to obtain the ground-state properties. The pressure (P) and bulk modulus (B) are obtained from the derivatives of the total energy.
3 Results and Discussion
3.1 Structural Properties
The equilibrium lattice parameters and bulk modulus of the MN and MC (M=V, Nb, Ta) compounds have been computed by fitting the energy-volume curve which is shown in Figure 1a–c. From the Figure, it can be clearly observed that the transition metal nitrides (VN, NbN, TaN) are energetically more favourable than the transition metal carbides (VC, NbC, TaC). Calculated structural parameters of these compounds are reported in Table 1. The equilibrium lattice parameters from our results are found in good agreement with the earlier theoretical and experimental results.
|Compounds||a0 (a.u.)||B (GPa)|
aRef. , bRef. , cRef. , dRef. , eRef. , fRef. , gRef. , hRef. , iRef. , jRef. , kRef. , lRef. , mRef. , nRef. , oRef. , pRef. , qRef. , rRef. , sRef. , tRef. , uRef. , vRef. , wRef. , xRef. , yRef. , zRef. .
The bulk moduli of the compounds are calculated from the relation B= –VdP/dV, and these are also presented in Table 1. Experimental results for the bulk modulus are available only for NbN and NbC for comparison, while theoretical results are available to compare for all the compounds studied. It is observed that the metal nitrides, in general, show higher bulk modulus than the metal carbides.
3.2 Electronic Properties
Figure 2 shows the self-consistent energy band structures along the high symmetry directions of the MN and MC (M=V, Nb, Ta) from the calculated equilibrium lattice constant. The overall band profiles are similar to each other, and the overlapping of bands around the Fermi energy confirms the metallic nature of the TMC and TMN. The low-lying band arises from the contribution of the non-metal atom around –1 Ryd. The band arises just below the Fermi level mainly due to the 2p states of non-metal atom and the small contribution from the TM-p states. As we go from 3d to 5d transition metals in the fifth group with carbides and nitrides, the position of the d-band shifts towards the lower energy, that is, it occurs below the Fermi level. At higher energies above 1 Ryd, hybridisation occurs from the contribution of both TM-d states and the non-metal p states.
The total and partial density of states (DOS and PDOS) plots of the MN and MC (M=V, Nb, Ta) are presented in Figure 3 and the partial and total DOS values at the Fermi energy (Nl(EF) (l=0, 1, 2, 3), and N(EF)) are presented in Table 2. The finite value of DOS at EF reveals the metallic nature of the chosen TMC and TMN. The nitride materials TMN (VN, NbN, and TaN) show higher N(EF) values than those of the carbide TMC (VC, NbC, and TaC) materials. The DOS value at EF for the MN and MC (M=V, Nb, Ta) from our results agrees well with the other theoretical reports [5, 49]. The electronic-specific heat coefficient (γ) is calculated from the computed total DOS at Fermi level from the expression The calculated values are found to decrease from vanadium to tantalum in both of the nitride and carbide compounds, respectively, and agree well with available literature results given in Table 3.
|Nl (EF) (states/Ryd.-cell)||VN||NbN||TaN||VC||NbC||TaC|
|TaN||1.57||2.0 c||331.48||492.22i||0.495||8.64||8.9 c|
aRef. , bRef. , cRef. , dRef. , eRef. , fRef. , gRef. , hRef. , iRef. .
3.3 Thermal and Superconducting Properties
The Debye temperature (θD) is an important thermal property of the material that is calculated from the relation given by Moruzzi et al. 
where B is the bulk modulus evaluated at the equilibrium Wigner–Seitz sphere radius S0 and M is the atomic mass. Although Moruzzi et al.  verified the validity of this expression for elemental metallic solids, we assume its validity for the present alloys also by considering M to be the concentration average of the masses of the component atoms. The calculated values are given in Table 3, and it is maximum for both VN and VC, and further, it is found to decrease from V to Ta in both of their carbides and nitrides.
The superconducting transition temperature is calculated by using McMillan’s formula  given by the following equation:
where θD is the Debye temperature, λ is the electron–phonon interaction constant and μ* is the electron–electron interaction constant. According to McMillan’s  strong coupling theory, the electron–phonon coupling constant λe–ph for a one-component system, that is, elemental solid, can be written as follows:
Where M is the atomic mass, 〈ω2〉 is the average squared phonon frequency and 〈I2〉 is the square of the electron–phonon matrix averaged over the Fermi surface. Following the work of Papaconstantopoulous et al. , 〈ω2〉 is set to be equal to 0.5 θD2. For the estimation of the electron–phonon coupling constant, phonon frequency is the essential parameter rather than the electronic properties of the metals. Therefore, such electronic structures and phonon calculations together with the simplified rigid muffin–tin approximation are very useful and efficient tools for studying the superconducting properties of a material. Gaspari and Gyorffy  constructed a theory to calculate the quantity 〈I2〉 on the assumption that the additional scattering of an electron caused by the displacement of an atom (ion) is dominated by the change in the local potential. Within the rigid muffin–tin approximation used by Gaspari and Gyorffy , the spherically averaged part of the Hopfield parameter η= N (EF)〈I2〉 can be written as follows: (in atomic Rydberg units)
where fl is a relative partial state density,
and Ml, l+1 is the electron-phonon matrix element. Gaspari and Gyorffy  derived an expression for Ml, l+1 using the rigid muffin–tin approximation in terms of partial wave phase shifts. Glötzel et al.  and Skriver and Mertig using the LMTO  method, expressed this quantity in terms of the logarithm derivative Dl(EF) of the radial solution at the sphere boundary, obtained from the gradient of the potential and the radial solutions that can be expressed as follows:
where S is the sphere radius, V(S) is the one electron potential and ϕl(EF) the sphere-boundary amplitude of the l partial wave evaluated at the Fermi level. The electron–electron interaction constant μ* is obtained from the empirical relation  in equation (7) as follows:
Where N (EF) is the total density of states at EF taken from the band structure results.
The values of the superconducting transition temperature are obtained for the chosen compounds along with the other parameters such as Debye temperature, electron–phonon interaction parameter given in the Table 3. The deviation observed between the present results and those of the earlier reported theoretical results [3, 13, 24] may be attributed to the different approximations used in the calculation techniques. The values of electronic-specific heat coefficient and Debye temperature show a decreasing trend from vanadium to tantalum in both of the nitrides and carbides in the present studies. The electron–phonon coupling constant is computed from the Hopfield parameter and used for the calculation of superconducting transition temperature of the materials under study by using the McMillan formula . The calculated superconducting transition temperature values show that both the TMC and TMN are strongly coupled superconductors and are in good agreement with the earlier theoretical results. There are no experimental results available for comparison in the literature hitherto.
The electronic and superconducting properties in carbides and nitrides of group V transition elements have been studied by using the TB-LMTO method. The ground-state structural properties have been obtained from the electronic total energy minimum and are found to be in good agreement with the experimental results in the literature. From the electronic total energy versus volume curve, it has been found that the transition metal nitrides are energetically more favourable than the transition metal carbides. Tantalum nitrides and carbides show higher bulk modulus than the other compounds studied. The band structure and DOS plots at Fermi level confirm the metallic nature of all the compounds. The DOS studies further confirm that the d states of the metal atom and p states of the non-metal atom yield major contribution at the Fermi energy in all the materials. The electronic-specific heat coefficient calculated for all the materials are found in reasonable agreement with the other theoretical estimates of the literature, while there are no experimental reports available for comparison. Among the materials studied, NbN is found to have maximum value of superconducting transition temperature and it is attributed to the maximum value of electron–phonon interaction parameter (λ) of the material in comparison with all others, and this trend of variation is in agreement with that of the corresponding literature results.
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