Electronic, Thermal, and Superconducting Properties of Metal Nitrides (MN) and Metal Carbides (MC) (M=V, Nb, Ta) Compounds by First Principles Studies

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.

Figure 1:

Calculated electronic total energy versus primitive cell volume of (a) VC-VN, (b) NbC-NbN, (c) TaC-TaN.

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.

Figure 2:

Band structure plot of (a)-(i) VN, (ii) VC; (b)-(i) NbN, (ii) NbC; (c)-(i) TaN, (ii) TaC. Dotted lines represent the Fermi energy.

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 (N_l(E_F) (l=0, 1, 2, 3), and N(E_F)) are presented in Table 2. The finite value of DOS at E_F reveals the metallic nature of the chosen TMC and TMN. The nitride materials TMN (VN, NbN, and TaN) show higher N(E_F) values than those of the carbide TMC (VC, NbC, and TaC) materials. The DOS value at E_F 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 γ = 13π2kB2N(EF). 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.

Figure 3:

Total and partial density of states for MN and MC (M=V, Nb, Ta) compounds (a)-(i) VN, (ii) VC; (b)-(i) NbN, (ii) NbC; (c)-(i) TaN, (ii) TaC.

3.3 Thermal and Superconducting Properties

The Debye temperature (θ_D) is an important thermal property of the material that is calculated from the relation given by Moruzzi et al. [52]

(1)θD = 41.63(S0BM) (1)

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. [52] 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 [53] given by the following equation:

(2)Tc = θD1.45exp{− 1.04 (1 + λ)λ − μ∗ (1 + 0.62λ)} (2)

where θ_D is the Debye temperature, λ is the electron–phonon interaction constant and μ* is the electron–electron interaction constant. According to McMillan’s [53] strong coupling theory, the electron–phonon coupling constant λ_e–ph for a one-component system, that is, elemental solid, can be written as follows:

(3)λ = N (EF)〈I2〉M 〈ω2〉 (3)

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. [54], 〈ω²〉 is set to be equal to 0.5 θ_D². 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 [55] 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 [55], the spherically averaged part of the Hopfield parameter η= N (E_F)〈I²〉 can be written as follows: (in atomic Rydberg units)

(4)η = 2N (EF)∑l(l + 1)Ml, l + 12fl2l + 1fl + 12l + 3 (4)

where f_l is a relative partial state density,

(5)fl = Nl (EF)N (EF) (5)

and M_{l, l+1} is the electron-phonon matrix element. Gaspari and Gyorffy [55] derived an expression for M_{l, l+1} using the rigid muffin–tin approximation in terms of partial wave phase shifts. Glötzel et al. [56] and Skriver and Mertig using the LMTO [57] method, expressed this quantity in terms of the logarithm derivative D_l(E_F) 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:

(6)Ml, l + 1 = − ϕl(EF)ϕl + 1(EF)[(Dl(EF) − l)(Dl + 1(EF) + l + 2) + (EF − V(S))S2] (6)

where S is the sphere radius, V(S) is the one electron potential and ϕ_l(E_F) 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 [58] in equation (7) as follows:

(7)μ∗ = 0.26N (EF)1+N (EF) (7)

Where N (E_F) is the total density of states at E_F 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 [51]. 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.