1. Introduction

The transition metal (TM) nitrides and carbides compounds have attracted considerable interest because of their novel mechanical, electrical, and chemical properties, they are usually tailored by the selection of specific deposition conditions [1] and [2], or modified by post-growth ion treatment [3], [4] and [5]. These compounds have high melting points and extreme hardness [6]. Moreover, these compounds are machinability and thermal conductivities. These combination properties cause these materials to be potential candidates as hard coating and thin films for electronic devices [7], [8] and [9], and for a variety of high-temperature structural applications [10] and [11].

Nitrides and carbides of TMs, such as TiN, TiC, ZrC and HfN, often crystallize with the NaCl-type structure, where the metallic atoms form the face-centered cubic sublattice, and nonmetallic atoms occupy interstitial positions. The unusual properties for TM nitrides and carbides mentioned above reveal that the atomic bonding between metallic and nonmetallic atom is very strong, and has a covalent nature. In fact, TiC has metallic conductivity character. However, some nitrides, such as ScN, appear to be semiconductor, although there are some disputes for ScN binary compound [12]. A large body of theoretical and experimental studies had been conducted for binary TM nitrides and carbides compounds [13]. Most of these studies focused on the physical, electronic, thermodynamical properties and phonon spectra. Weber studied the phonon dispersion relations of ZrC, NbC, HfC TaC and UC, and concluded that the novel physical properties of these compounds were attributed mainly to strong covalent metal–nonmetal bonding [14]. Gusev and Zyryanova studied the equilibrium phase diagram of HfC_y (0.59≤y<1.00) considering the magnetic susceptibility [15]. The phonon dispersion curves for TiN and ZrN had been measured experimentally by inelastic neutron scattering [16] and [17]. Christensen et al. reported an anomalous behavior in the measured phonon spectrum of isoelectronic HfN [18]. High-pressure phonon spectra of HfN, ZrN and NbN had been obtained by Raman-scattering measurements [19].

Meanwhile, many ab initio electronic structure calculations on TM carbides and nitrides in the NaCl-type structure have been carried out. The density of states (DOS) of TiX (X=C, N, O) was studied by means of the linear combination of atomic orbitals and the augmented plane wave (APW) methods, and the role of covalent metal–nonmetal bond in structural stabilization of these compounds has been discussed [20] and [21]. Full potential linear muffin-tin orbital method had been used to study the electronic structure, DOS, and optical properties of monocarbides, -nitrides, and oxides of Ti and Zr [22] and [23]. Electronic band structure of NbC was investigated by means of APW, full potential augmented plane wave, and pseudopotential methods [24], [25] and [26]. Mechanical properties such as elastic constants, hardness, Young's and shear moduli for XN (X=Hf, Zr, Nb) binary compounds were examined by using ab initio pseudopotentials and the linear response of the unperturbed system to homogeneous strains [27]. Therefore, many experimental and theoretical works mainly focused on binary TM carbides and nitrides compounds. However, very few studies have been performed on ternary TM carbides and nitrides compounds. Experimental studies had been conducted to study the phase equilibria of Ti–C–N, Zr–C–N and Hf–C–N systems [28]. Zhukov et al. had investigated the electronic band structures and cohesive energies for TiCN ternary compounds by using linear muffin-tin orbital in the atomic-spheres approximation [29]. Moreover, the structural stability and elastic stiffness of TiCN alloys had been investigated by using the ab initio pseudopotential total-energy method [30]. Zaoui et al. observed a charge transfer from the metallic atoms to the nonmetallic atoms (C, N), where the quantity of this transfer is larger in nitrides than in carbides [31]. Former studies for TM carbides and nitrides revealed that intermediated compositions of the ternary systems are harder than the corresponding binary compounds [32]. To learn more about the nature of these binary and ternary compounds, a detailed theoretical investigation into the mechanical and electronic behavior must be conducted. In the present work, we describe the mechanical and electronic properties of TiC_xN_1−x, Zr_xNb_1−xC and HfC_xN_1−x alloys. Such study is not only important for other binary compounds but also valuable for other intermediate compositions of the ternary system. The rest of paper is organized as follows: in Section 2, we briefly describe the computational method used in the present work. Results and discussion will be presented in Section 3. A summary of the work will be given in Section 4.

2. Computational details

The calculations were performed by using a first-principles pseudopotential plane wave method based on the density-functional theory [33]. The electronic exchange–correlation interactions were treated by the generalized gradient approximation (GGA) within the scheme due to Perdew–Burke–Ernzerhof (PBE) [34]. The Vanderbilt-type ultrasoft pseudopotentials were employed to model the ion–electron interactions [35]. The integrations over the Brillouin zone (BZ) were replaced by discrete summation over a special set of k points using the Monkhorst–Pack scheme [36]. The energy cutoff of the plane-wave basis was chosen as 500 eV. Large sets of Monkhorst–Pack meshes of 10×10×10 for all the systems were used to sample the BZ. The chosen plane-wave cutoff energy and the numbers of k points were carefully checked to ensure good convergence of the computed energies. The convergence tolerance in the self-consistent field calculations is less than 5×10⁻⁷ eV/atom. For a given external hydrostatic pressure, lattice constants and the internal coordinates of the atoms were fully relaxed until the atomic forces had converged to less than 0.01 eV/Å. The elastic constants were calculated by the finite-strain technique [37]. TiC_xN_1−x and HfC_xN_1−x anionic ternary alloys were simulated by substituting the nonmetal C and N atoms in the unit cell with the variation of the concentrations in the NaCl-type structure with eight Ti (or Hf) sites and eight sites for C and N. In addition, for constructing Zr_xNb_1−xC, we adopted the scheme proposed by Zaoui et al. [31].

3. Results and discussion

3.1. Structural properties

Firstly, we analyzed the structural properties of binary TM carbides and nitrides. The calculated lattice constants a, bulk modulus B for these binary compounds at zero pressure were summarized in Table 1, together with the available experimental and theoretical data. It should be noted that our calculated a and B are in good agreement with experimental values, except lattice constants of Hf based binary compounds are slightly larger than the experimental value. These ensure the reliability of the present first-principles computations. The calculated a and B as a function of the concentration of these alloys are presented in Fig. 1(a) and (b). As can be seen in Fig. 1(a), the variation of a is approximately linear with concentration, thus obeying Vegard's law. The bulk modulus versus concentration is plotted in Fig. 1(b), we observe a linear decreasing behavior of B with the concentration of ternary alloys.