Point defects stabilise cubic Mo-N and Ta-N

The energy of formation as a function of the defect concentration (see figure 1) suggests that in the case of c-TaN, Ta vacancies are preferred over N vacancies up to the concentration $\sim 16 \%$ ($\text{T}{{\text{a}}_{0.69}}\text{N}$), which agrees well with the previously published predictions of Grumski et al [6], and Stampfl and Freeman [12]. Interestingly, also N vacancies up to the concentration $\sim 13 \%$ ($\text{Ta}{{\text{N}}_{0.75}}$) are found slightly more energetically favourable than the perfect TaN system. As intuitively expected, E_f of structures containing mixed Ta and N vacancies lie between E_f of structures containing only Ta or only N vacancies.

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Consistently with previous studies [12, 30], $\text{T}{{\text{a}}_{0.78}}\text{N}$ ($\sim 11 \%$ Ta vacancies) is the energetically most preferred structure; its formation energy  −1.05 eV at.⁻¹ lies about 0.17 eV at.⁻¹ lower than E_f of the perfect c-TaN. When restricted to only N vacancies, the energy of formation reaches its minimum for $\text{Ta}{{\text{N}}_{0.84}}$ ($\sim 8 \%$ N vacancies), however, this value is only $0.02~\text{eV}~\text{at}{{.}^{-1}}$ lower than E_f of the perfect structure.

The energy of formation can be considerably decreased even when keeping the $1:1$ metal-to-nitrogen stoichiometry, by introducing Schottky defects. Such a configuration is practically undetectable by conventional tools used for chemical analysis, e.g. energy-dispersive x-ray spectroscopy (EDX) or Rutherford backscattering spectroscopy (RBS). As discussed in the next section, structures containing Schottky defects yield smaller lattice constant (as estimated by, e.g. x-ray diffraction (XRD)) and hence may contribute to relatively large scatter of experimental data (see table 2). Finally, our calculations suggest that both interstitial and Frenkel pairs destabilize c-TaN, which is in agreement with the previous study of Stampfl and Freeman [12]. Interestingly, the formation energy of Frenkel pairs was found to be dependent on the distance between the missing and the interstitial atom sites. Assuming this distance is generally smaller than 2 Å–3 Å, we speculate that the interstitial atom attempts to fill the vacancy site, thus the structure finally ends up in its almost perfect configuration. However, considering the larger distance between vacancy and interstitial, the energy of formation remains almost constant. Hence only data points for the later case are plotted in figure 1 (analogically in the case of c-MoN).

Table 2. Comparison of lattice parameters (in Å) calculated in this study with corresponding experimental or theoretical values from the literature for selected compositions.

	TaN	$\text{T}{{\text{a}}_{0.97}}\text{N}$	$\text{T}{{\text{a}}_{0.94}}\text{N}$	$\text{T}{{\text{a}}_{0.87}}\text{N}$	$\text{T}{{\text{a}}_{0.75}}\text{N}$	$\text{Ta}{{\text{N}}_{0.97}}$
Present study	4.427	4.417	4.403	4.38	4.305	4.422
Experiment	4.42 [33], 4.427 [34], 4.336 [35]	—	—	4.361 [36]	—	—
LDA	4.353 [6], 4.386 [4], 4.397 [10]	—	—	—	—	—
GGA	4.424 [6], 4.415 [31], 4.414 [37],	4.41 [6]	4.40 [6]	—	4.30 [6]	4.42 [6]
	4.547 [38], 4.40 [1], 4.408 [39]	—	—	—	—	—

	MoN	$\text{Mo}{{\text{N}}_{0.75}}$	$\text{Mo}{{\text{N}}_{0.5}}$	—	—	—
Present study	4.364	4.282	4.211	—	—	—
Experiment	4.215–4.253 [40], 4.20–4.22 [41],	—	—	—	—	—
	4.20–4.27 [42]	—	—	—	—	—
LDA	4.280 [32], 4.41 [43], 4.285 [44],	4.217 [32]	4.162 [32]	—	—	—
	4.25 [10]	—	—	—	—	—
GGA	4.328 [39]	—	—	—	—	—

In regard to c-MoN, similar but more complex behaviour is predicted. Formation energy cannot be simply categorised according to the vacancy type, but instead seems to be governed by the ordering degree of the defects. Details will be discussed in section 3.5, here we focus on overall compositional trends. The E_f global minimum (−0.282 eV at.⁻¹) is obtained for $\sim \,4.69 \%$ fully disordered metal vacancies ($\text{M}{{\text{o}}_{0.91}}\text{N}$), nevertheless, a partially ordered structure with $\sim 15.63 \%$ of N vacancies ($\text{Mo}{{\text{N}}_{0.69}}$) exhibits a very close value of  −0.280 eV at.⁻¹. Furthermore, in contrast to c-TaN, not only Schottky defects, but also interstitials and Frenkel pairs may decrease the formation energy with respect to the perfect c-MoN. Namely, the energy of formation in the case of a N interstitial ($\text{Mo}{{\text{N}}_{1.03}}$) is comparable with that of $\text{Mo}{{\text{N}}_{0.84}}$ containing $\sim \,7 \%$ N vacancies. Slightly lower are found Mo and N Frenkel pairs, and the lowest E_f is predicted for Schottky defects yielding ${{E}_{\text{f}}}=-0.21$ eV at.⁻¹. We speculate that this behaviour can be ascribed to the fact that both metal and N vacancies are strongly favoured over the perfect structure (unlike the c-TaN case where the metal vacancies are significantly preferred over the N ones). Consequently, almost arbitrary combination of point defects at low concentration results in an increased stability of the Mo-N system.

Since it has been experimentally reported that the MoN and TaN stoichiometry sensitively depends on the deposition conditions [8, 18, 19], we analyse the role of chemical potentials. The energy of formation E_f is considered as a function of two independent variables, ${{\mu}_{\text{N}}}$ and ${{\mu}_{\text{Mo}}}$ in the case of c-MoN, ${{\mu}_{\text{N}}}$ and ${{\mu}_{\text{Ta}}}$ in the case of c-TaN. The most stable structures, i.e. the structures with the lowest E_f, are depicted in figure 2 as a function of the respective chemical potentials. The unshaded regions correspond to unstable structures, i.e. structures yielding a positive value of E_f. In order to provide guidance for future experimental studies, the N chemical potential has also been recalculated to N₂ pressure at 300 K and 800 K, using equations (7) and (8).

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The stability regions of metal and N vacancies are almost of equal size in the Mo-N case, suggesting that the actual formation of either metal, or nitrogen deficient structure very sensitively depends on the experimental conditions. Four off-stoichiometric structures $\text{M}{{\text{o}}_{0.84}}\text{N}$, $\text{M}{{\text{o}}_{0.91}}\text{N}$, $\text{Mo}{{\text{N}}_{0.69}}$ and $\text{Mo}{{\text{N}}_{0.44}}$ are predicted to be stable. Interestingly, keeping the Mo-rich conditions while arbitrarily varying ${{\mu}_{\text{N}}}$ results in stabilizing only N deficient structures and vice versa, i.e. solely Mo deficient compounds are predicted in the case of N-rich conditions and arbitrary values of ${{\mu}_{\text{Mo}}}$. Furthermore, while nitrogen vacancies tend to order, the opposite trend is shown for their metal counterparts.

In contrast, metal deficient structures, namely $\text{T}{{\text{a}}_{0.75}}\text{N}$, $\text{T}{{\text{a}}_{0.78}}\text{N}$, $\text{T}{{\text{a}}_{0.81}}\text{N}$ and $\text{T}{{\text{a}}_{0.84}}\text{N}$, dominate the Ta-N phase diagram showing that the (partially) ordered metal vacancies are strongly favoured. We note that $\text{T}{{\text{a}}_{0.75}}\text{N}$ represents a cell with fully ordered vacancies (1 Ta missing in every conventional cubic 8-atom cell), and hence is the most ordered among all defected supercells in this study (see equation (2)). Underpinning our previous results, $\text{T}{{\text{a}}_{0.78}}\text{N}$ is predicted to be stable in a remarkably wide range of ${{\mu}_{\text{N}}}$ and ${{\mu}_{\text{Ta}}}$ values. Our calculations further reveal an array of disordered highly nitrogen deficient compositions: $\text{Ta}{{\text{N}}_{0.78}}$, $\text{Ta}{{\text{N}}_{0.69}}$ and $\text{Ta}{{\text{N}}_{0.63}}$, followed by $\text{Ta}{{\text{N}}_{0.56}}$ and $\text{Ta}{{\text{N}}_{0.5}}$, while the atomic ratio N/Ta decreases as approaching the strongly N-poor conditions.

Importantly, neither of the two systems, Mo-N and Ta-N, exhibits a stability region for perfectly stoichiometric metal-to-nitrogen cubic structures. This is in excellent agreement with previous studies showing that the stoichiometric structures are hexagonal δ-MoN (space group of P6₃mc, no. 186) [8] and π-TaN (CoSn type structure with space group of $P\bar{6}2m$, no. 189) [6, 31].

The negative value of vacancy formation energy, ${{E}_{\text{f},\text{vac}}}$, evaluated according to equation (9), means that energy is released when forming a vacancy and therefore that the configuration with the vacancy is energetically preferred to a perfect structure. The trends shown in figure 3 are a different evaluation of the calculations already discussed in the previous sections. It is important to realise, however, that E_f and ${{E}_{\text{f},\text{vac}}}$ represent different phenomena. E_f describes chemical stability with a variable number of atoms in the supercell (e.g. evaluation of E_f of a structure with (N  −  1) atoms and one vacancy takes into consideration the chemical potential of only (N  −  1) species, see equation (3)), hence refer to the formation of perfect or defected state from a reservoir of particles. On the other hand, vacancy formation energy, ${{E}_{\text{f},\text{vac}}}$, considers always N species (see equation (9)), and hence describes formation of the defected states from the perfect structure.

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Both c-MoN and c-TaN exhibit a remarkably broad range of compositions with negative ${{E}_{\text{f},\text{vac}}}$, confirming that off-stoichiometric compounds are strongly favoured at 0 K (see figure 1). The c-TaN phase yields mostly positive (though relatively small) formation energies for N vacancies which is in line with the computational results reported by Grumski et al [6]. In contrast to c-TaN, values of ${{E}_{\text{f},\text{vac}}}$ of c-MoN lie in a surprisingly wide range (here from $-6~\text{eV}/\text{defect}$ to $2~\text{eV}/\text{defect}$). Additionally, comparison of metal and N (partially) ordered and disordered configurations reveals substantial differences in the case of c-MoN, while the degree of order seems not to play any significant role for c-TaN. These predictions underline the qualitatively different incorporation of vacancies in both c-MoN and c-TaN.

Cubic lattice parameter, a was evaluated as

$$\begin{eqnarray}&&a=\sqrt[3]{{{V}_{0}}},\end{eqnarray} \tag{ 10 }$$

where V₀ is the equilibrium volume corresponding to a conventional cubic cell. Our structural models are based on $2\times 2\times 2$ supercells constructed from the conventional cubic cells, therefore we take V₀  =  V/8 where V is volume of a fully relaxed supercell with or without defect(s). It is important to note that due to supercell-size effect, the relaxed defected supercells do not have, in general, cubic shape anymore. However, this is a consequence of the atomistic approach using only a few dozens of atoms; the defected materials will macroscopically still possess cubic symmetry. Evaluation using equation (10) hence effectively averages these local distortions and restores the macroscopically cubic material.

Our calculations for c-MoN and c-TaN illustrate strong decrease of a as the vacancy concentration increases (see figure 4). Comparable trends were predicted using the first principles for c-MoN by Lowther [32] and for c-TaN by Stampfl and Freeman [12], and Grumski et al [6]. Table 2 lists the calculated lattice parameters for several selected compositions for which experimental or ab initio values were previously published in the literature, hence corroborating our predictions.

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To provide a quantitative estimation of the lattice parameters, we fitted the calculated values for defect concentrations below x  <  0.12, with linear functions of the defect concentration, x, (see figure 4):

$$\begin{eqnarray}&&a(x)=a(0)-\beta x.\end{eqnarray} \tag{ 11 }$$

Here, a(0) is the lattice parameter of perfect (defect-free) TaN or MoN, respectively. The fitted β values for MoN are $0.009\,28$, $0.006\,27$ and $0.007\,46$ Å for Mo vacancies, N vacancies and Schottky defects (Mo+N vacancies), respectively. In the TaN case, β takes values of $0.00747$, $0.00358$, and $0.00573$ Å for Ta vacancies, N vacancies and Schottky defects (Ta+N vacancies), respectively. In general, metal deficient structures show much greater lattice parameter (and hence volume) decrease upon vacancy introduction than their nitrogen deficient counterparts. It is a well-established fact that the GGA and the local density approximation (LDA) exchange-correlation functionals tend to over- and underestimate, respectively, the predicted lattice constants with respect to experimental values. This has been explicitly shown by Grumski et al [6] to be valid in the case of c-TaN. Consistently with this conclusion, our predictions listed in table 2 are slightly higher than the experimental results and the values calculated using the LDA.

Finally, lattice parameters corresponding to the other types of point defects investigated in the present study were calculated. As a consequence of introducing one interstitial atom into c-MoN or c-TaN, the lattice expands to accommodate the additional atom. Interestingly, similar lattice expansion is predicted also for Frenkel pairs. On the other hand, Schottky defects lead to a decrease of the lattice constant. This can be intuitively understood by the fact that Schottky defects are complexes of metal and N vacancies, which both lead to a lattice parameter decrease (see figure 4).

A spectrum of structures ranging from fully disordered to (partially) ordered has been considered in this work, which allows the results to be discussed with respect to the structural degree of order. We note that the data presented in figure 5 describe the case of N-rich and Mo-rich or N-rich and Ta-rich conditions, respectively. Our analysis clearly demonstrates that the key difference between c-MoN and c-TaN lies in their opposite preference for ordering/disordering vacancies. While the metal vacancies in c-MoN favour rather disordered configurations (differing from ordered and partially ordered by 0–0.15 eV at.⁻¹), the partially ordered metal vacancies in c-TaN exhibit lower values of formation energy up to the concentration $\sim \,14 \%$ ($\text{T}{{\text{a}}_{0.72}}\text{N}$). Concerning N vacancies in c-TaN, the same trend as for their metal counterparts is observed. Finally, an interesting behaviour is predicted for N vacancies in c-MoN. Generally, disordered configuration is preferred, although an ordered arrangement is favoured between 16%–22% of N vacancies ($\text{Mo}{{\text{N}}_{0.69}}$–$\text{Mo}{{\text{N}}_{0.44}}$). We also conclude that the impact of the vacancy degree of order on the stability significantly decreases with increasing non-stoichiometry in both cases, as demonstrated by the energy of formation being almost independent of the order parameter (equation (2)).

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