Trends in formation energies and elastic moduli of ternary and quaternary transition metal nitrides
The paper deals with characteristics of a wide range of ternary and quaternary metal nitrides (M = Ti, Zr, Hf, V, Nb or Ta) of various compositions obtained by ab initio calculations. We focus on the formation energies (Eform), bulk moduli (B), shear moduli (G) and a difference of B and G from the weighted average of B and G of binary metal nitrides (∆B and ∆G). We show numerous monotonous dependencies, and identify exceptions to them. For elastic moduli of M1M2N we find that ∆B decreases (down to −19 GPa) and ∆G increases (up to 20 GPa) with increasing difference between atomic radii of M1 and M2. In parallel, low ∆B and high ∆G correspond to high Eform and |Eform|, respectively. Eform of M1M2N increases with increasing difference between atomic radii and electronegativities of M1 and M2. The lowest Eform values were observed for Ta-containing compositions, and the difference between Eform of TaM1M2N and M1M2N is more significant for lower atomic radius and higher electronegativity of M1 and M2. Overall, we present trends which allow one to use fundamental arguments (such as atomic radii and electronegativities) to understand and predict which compositions form (nano)composites, which compositions form (stable) solid solutions, and which materials exhibit enhanced elastic moduli. The phenomena shown can be tested experimentally, and examined for even wider range of materials.
Thin films of binary early (IIIB–VIB) transition metal nitrides (MN), most often TiN or CrN, are of great importance as hard protective coatings (see e.g. Ref.  for a review). They exhibit high hardness, wear resistance, high melting point and high chemical and thermal stability. The MN structure is usually fcc (rock salt), with two partial exceptions. The first includes TaN and NbN which exist in both cubic and hexagonal modification. Although the hexagonal modification has lower energy, the cubic modification can be experimentally stabilized  and is frequently theoretically considered as well . The other exception is cubic-to-orthorhombic transition resulting from a low-temperature antiferromagnetic spin ordering in the case of CrN (in terms of unpaired spin density followed by VN ).
To further increase the hardness, wear resistance, thermal stability or oxidation resistance of the binary nitrides, other elements are incorporated. This leads to at least three classes of ternary (or higher) nitride phases (let alone multiphase nanocomposites). First, part of N can be replaced by C, to improve the shear modulus and consequently the hardness [4, 5]. Second, part of transition metal atoms can be replaced by Al, to further improve their mechanical and tribological properties and high-temperature performance [1, 6]. Third, two transition metal elements can be present in the metal sublattice, leading to M1M2N phases. Experimental examples include TiZrN [2, 7, 8], TiTaN , TiCrN , TiMN (M = Zr, W, Hf, Nb, Ta, Mo) , TaMN (M = Zr, W)  and TiAlTaN  (with cubic—not hexagonal—structure of the ternary Ta-containing compounds). Several of these materials were also studied theoretically, including TiZrN [8, 9] and Ti(Al)TaN [9, 10]. In addition to optimizing the mechanical properties, the ternary compositions allow one to tailor the color of TiVN, ZrVN or ZrCrN decorative coatings .
However, there are large gaps in the available knowledge. First, many M1M2N systems (e.g. V, Hf or Nb-based) have been studied (both experimentally and theoretically) very little or not at all. Second, many M1M2N solid solutions were empirically observed in a wide compositional range (see e.g. Ref. ), without an experimental test of their (meta)stability. Third, there even seems to be a disagreement concerning the stability of some of these materials. For example TiZrN solid solution is expected to be stable (due to the same valence electron configuration of Ti and Zr) in Ref. , it is largely metastable according to ab initio calculations in Ref. , it can be experimentally prepared according to Ref.  and it could not be prepared without simultaneous formation of binary TiN and ZrN in Ref. .
In this paper we partially fill the aforementioned gaps using ab initio calculations. They constitute a fast and reliable technique for scanning of properties of many different and well-defined compositions (contrary to not necessarily stoichiometric experimental compositions). Our first aim is to provide an overview of formation energies (Eform) and elastic moduli, namely bulk modulus (B) and shear modulus (G), for selected ternary and quaternary metal nitrides. The selected metal elements include Ti, Zr, Hf, V, Nb and Ta. In addition, we study the difference of B and G from the weighted average of B and G of the corresponding binary metal nitrides (∆B and ∆G). Note that there are engineering quantities which largely depend on these moduli, including indentation hardness (~G) and ductility (~B/G). Our second aim is to correlate the aforementioned quantities with known characteristics of the metal elements, namely atomic radii and electronegativities.
All calculations use the density functional theory [13, 14] as implemented in the Plane-wave self-consistent field (PWscf) code . Atom cores and inner electron shells were represented by the Vanderbilt-type ultrasoft pseudopotentials . Valence electrons were described using the Kohn–Sham equations, and the exchange and correlation (xc) term was treated using the Perdew–Burke–Ernzerhof (PBE) functional . Valence electron wavefunction was expanded using a plane-wave basis set with an energy cut-off of 30 Ry. The integration in the reciprocal space was performed using Marzari–Vanderbilt (“cold”) smearing , and a Monkhorst–Pack grid of k-points (12 × 12 × 12 in the case of 8-atom cell). All systems studied are nonmagnetic (see e.g. Ref. ), thus at even number of electrons in the simulation cell the spin polarization was not taken into account.
The materials considered include six binary systems (Ti, Zr, Hf, V, Nb and Ta nitride), 15 corresponding ternary systems (three compositions per each of them, i.e. 45 ternary compositions) and ten selected quaternary systems. In line with the aim of this work (i.e. to identify trends valid for all these materials rather than to study one material in detail), the calculations were performed using (only) 8-atom periodic cell in the case of binary metal nitrides, and 16-atom periodic cell in the case of ternary or quaternary metal nitrides. In the latter case the distribution of atoms in the metal sublattice, as well as the atomic coordinates, were optimized to minimize the total energy. Although even lower formation energies (and consequently slightly different properties) may be, in principle, obtained for a special arrangement of atoms  in a larger metal sublattice (particularly so-called “special quasirandom structure” or SQS—a phenomenon which was reported for TiAlN [20, 21, 22, 23], and which may be possibly expected for some other MAlN materials), this is obviously beyond the scope of the present study. Moreover, a comparison with available literature provided in “Results and Discussion” section does not support validity of this phenomenon in the M1M2N case. Instead, the comparison shows that our results are in full qualitative agreement with literature (in fact Eform is even below literature values in some cases), at much smaller quantitative differences compared to the differences reported for TiAlN  (possibly due to the much more similar valence electron configuration of M1 and M2 compared to M and Al).
Deformation tensor used for calculations on tetragonal (1–6) and cubic (1, 4 and 5) cell
e1 = e2 = x
(C11 + C12)x2 + O[x3]
e1 = e2 = x
(C11 + C12 + 2C33 − 4C13)x2 + O[x3]
e3 = −x(2 + x)/(1 + x)2
e3 = x
C33x2/2 + O[x3]
e1 = [(1 + x)/(1 − x)]1/2 − 1
(C11 − C12)x2 + O[x4]
e2 = [(1 − x)/(1 + x)]1/2 − 1
e4 = e5 = x
C44x2 + O[x4]
e3 = x2/4
e6 = x
C66x2/2 + O[x4]
e1 = e2 = (1 + x2/4)1/2 − 1
A polycrystalline (isotropic) material is defined by B and G only, while the other elastic moduli, such as Young’s modulus (E) and Poisson’s ratio (ν), can be calculated using the relationships E = 9BG/(3B + G) and ν = E/2G − 1.
The electronic density of states (EDOS), i.e. the energies of Kohn–Sham states, was obtained at the previously optimized a0 value using the same PWscf code.
Part a shows calculated characteristics of binary metal nitrides: lattice constant (a0; line 1), bulk modulus calculated using Birch equation or elastic tensor (B; lines 2 and 3, respectively), shear modulus (G; line 4), Young’s modulus (E; line 5), Poisson’s ratio (ν; line 6) and elastic tensor (Cij; lines 7–9)
B (GPa) − Birch eq.
B (GPa) − Cij
M radius [from a0] (Å)
M radius  (Å)
M elneg.: Allred–Rochow
M elneg.: Pauling
Results and discussion
Table 2a shows selected calculated characteristics of binary metal nitrides. The purpose of these calculations was to validate the technique used, and to provide raw data (total energies, B and G) used in “Ternary nitrides - formation energies” Section (and “Quaternary nitrides” Section) for calculations of Eform, and in “Ternary nitrides: mechanical properties” Section for calculations of ∆B and ∆G. The other elastic moduli are added for completeness, and to facilitate a comparison with literature. The table allows us to conclude that the calculated data on binary metal nitrides are fully consistent with available literature—see e.g. Ref.  (and Refs. therein) for a recent review. Comparison of the results obtained for IVB and VB metal nitrides confirms known trends [3, 24] (which are however valid for nonmagnetic materials only ), namely increasing B, C11, C12, ν and decreasing G, E, C44 with increasing number of valence electrons per atom. Comparison of the second and third line reveals that the B values obtained from the Birch equation (used below) and from Cij are practically same (average difference 3 GPa—or 1 %—only, and always in the same direction).
Table 2b summarizes selected characteristics of the corresponding metal elements. Also these quantities serve as input data for the following sections. The first characteristic is an (effective) radius of metal elements, calculated as a difference of a0/2 and nitrogen radius (taken as 0.65 Å ). Literature atomic radii are provided for comparison. While both sets of metal radii exhibit the same trends, the differences between the radii calculated from a0 are significantly below the differences between the literature radii. The other characteristic is electronegativity according to Allred and Rochow and according to Pauling. The following figures use the former one (XAR), while the latter one is provided for comparison (despite the large difference especially in the case of Nb, the trends shown below are qualitatively same for both electronegativities).
Ternary nitrides—formation energies
Figure 1 shows Eform of ternary metal nitrides from the 15 systems considered (M1,2 = Ti, Zr, Hf, V, Nb or Ta). The compositions include that M0.251M0.752N, M0.501M0.502N and M0.751M0.252N. It can be seen that in most cases the absolute value |Eform| is the largest for the M0.501M0.502N composition. The (relatively) most significant exception in terms of absolute differences is HfVN, where Eform of Hf0.25V0.75N is 3.8 meV/at. (5 %) larger compared to Eform of Hf0.50V0.50N. The most significant exception in terms of relative (percentage) differences is HfTaN, the only system which exhibits more than one extreme and spans (at, however, very close proximity of zero) both positive and negative Eform values. For simplicity, in the following figures we show only Eform of compositions M0.501M0.502N.
The formation energies of M0.501M0.502N materials are summarized in the complementary Fig. 2. The left part of the figure shows stable solid solutions, starting with TaTiN. The right part of the figure shows metastable solid solutions, starting with VZrN and TiZrN. There is not only qualitative but also good quantitative agreement with available literature, particularly for the most often studied systems TiZrN and TaTiN. Ti0.50Zr0.50N has largely positive Eform of 96 meV/at. in Ref. , approx. 73 meV/at. in Ref.  and 75 meV/at. in our case. TaxTi1−xN has negative Eform of down to (depending on code used) −50 meV/at. at x = 0.5 in Ref. , approx. −10 meV/at. at x = 0.5 and −30 meV/at. at x = 0.75 in Ref. , −49 meV/at. at x = 0.4 and 0.6 in Ref.  and −40 meV/at. at x = 0.5 in our case. Ref.  is the only one where the most negative Eform of TaxTi1−xN was not reported at or close to x = 0.5. Apart from Ref.  (which seems to be contradicted by more recent Ref.  published by the same laboratory), the largest spread of Eform was found for Ti0.50Zr0.50N where our Eform is in full agreement with Ref. , but more than 20 meV/at. lower compared to Ref.  (possibly because of energetically more favorable distribution of metal atoms in their sublattice in our case). Overall, the Figs. 1 and 2 predict which experimentally prepared Mx1M1−x2N materials will decompose to M1N and M2N upon annealing (a temperature dependence of Gibbs energy—see Ref.  for an example—has to be taken into account in the case of annealing up to very high temperatures). This is one of predictions of our calculations which can be, e.g. for VZrN, tested experimentally.
Figure 3a shows the relationship between Eform of M0.51M0.52N and the ratio of M1 and M2 radii (r1 and r2; the ratio is shown with the higher radius in numerator as rhigher/rlower) calculated using a0 (see Table 2). It can be seen that on the average the Eform increases with increasing rhigher/rlower. The materials TaTiN and TaVN have to be considered as exceptionally stable in this sense, as their Eform values are below the dependence formed by the other 13 materials (possibly due to stronger covalent features upon Ta incorporation, see “Quaternary nitrides” Section for details). This dependence can be considered as linear (correlation >0.9) or slightly faster than linear (mainly due to VZrN). One of the quantitative conclusions which the figure leads to is that close r1 and r2 guarantee low Eform (stable materials), e.g. Eform < 14 meV/at. for rhigher/rlower < 1.06.
While the simple argument that the solubility is supported by the same valence electron configuration [9, 29] is far from a full story (see the positive Eform of TiZrN despite the same d2s2 configuration of Ti and Zr), we examine the effect of electronic properties and structure in a broader sense in Figs. 3b and 4. Figure 3b shows that Eform grows not only with increasing difference between sizes of metallic atoms, but also with increasing difference between their electronegativities (XAR). Again, the lowest (TaTiN and TaVN) and highest (VZrN and in this case also TiZrN) Eform values are below and above, respectively, the almost linear dependence formed by the other materials. The similarity of dependencies of Eform on r (Fig. 3a) and XAR (Fig. 3b) is not surprising as XAR is a function of electron-nucleus distance. The dependence in Fig. 3a is clearer probably due to the fact that it uses the actual r (based on true lattice constants), while that in Fig. 3b uses theoretical XAR calculated from ideal(ized) radii. The figure indicates that very close electronegativities guarantee low Eform (stable materials), e.g. Eform < 14 meV/at. for XAR difference ≤10−2.
Figure 4 compares Eform of materials M0.51M0.52N with the correlation of EDOS of the corresponding binary nitrides M1N and M2N. The EDOSs are shown as an inset in Fig. 4. Only the highest occupied bands (formed predominantly of nitrogen p and metal d orbitals) between left side of the inset and the Fermi energy were taken into account. A straightforward calculation of the correlations is possible only for materials where M1 and M2 have the same number of valence electrons (the triplets connected by dashed lines). The figure shows for both IVB and VB ternary nitrides that they are actually stabilized by non-correlating EDOSs (distant energies of electronic states) of the corresponding binary nitrides which the ternary nitrides are formed of.
Note that the materials (systems) which (i) have the lowest Eform values (see Fig. 2) and (ii) which have Eform below the dependencies formed by the other materials (see Fig. 3) are formed by the heaviest metal element considered, Ta, and the lightest metal elements considered, Ti and V. A hypothesis which generalizes this finding is examined in “Quaternary nitrides” Section.
Ternary nitrides—mechanical properties
∆B (Fig. 5a) is mostly negative, between −19 and +3 GPa. This increases the application potential of two-phase materials M1N + M2N (assuming that they can be either deposited or prepared by annealing—for a direct relationship between ∆B and Eform see below).
∆B decreases with increasing ratio of metal radii, rhigher/rlower. This is particularly pronounced for rhigher/rlower above approximately 1.08 where the decrease of ∆B is practically linear. On the other hand, ∆B is close to zero (±3–4 GPa) for rhigher/rlower < 1.08.
B values themselves (Fig. 5b), provided for completeness, do not constitute any significant dependence on rhigher/rlower. Instead, Fig. 5b mostly follows the aforementioned relationship which predicts higher B for higher density of valence electrons. Three highest B values were found for VB materials TaVN, NbTaN (which may be however difficult to prepare in cubic form) and NbVN, and two lowest B values were found for IVB materials TiZrN and HfZrN. However, due to the non-zero ∆B values, this relationship (well pronounced for binary materials—see Table 2 and Refs. [3, 24]) is much less pronounced for ternary materials: TaTiN (IVB–VB) has almost as high bulk modulus as NbVN (VB), and HfTiN (IVB) has higher bulk modulus than VZrN (IVB–VB).
∆G (Fig. 6a) is positive, between 0 and 20 GPa. This increases (i) the application potential of solid solutions M0.51M0.52N (assuming their stability) when it comes to hardness (proportional to G) and (ii) the application potential of two-phase materials M1N + M2N (assuming that they can be either deposited or prepared by annealing) when it comes to ductility (proportional to B/G).
∆G increases with increasing ratio of metal radii, rhigher/rlower. On the one hand, contrary to ∆B (see the previous figure), this trend takes place already at low values of the rhigher/rlower ratio. On the other hand, at high values of the rhigher/rlower the trend becomes less clear mainly because of high ∆G of materials TaTiN and TaVN. Thus, the same materials which exhibit the highest stability of the solid solutions exhibit also unusually enhanced G of these solid solutions.
G values themselves (Fig. 6b), provided for completeness, exhibit slight but insignificant increase with increasing rhigher/rlower (mostly due to low G of NbTaN). The highest G (i.e. probably the highest hardness) among all solid solutions exhibits HfTiN. The highest G among stable solid solutions (negative Eform) exhibits TaTiN.
Figure 7a shows the relationship between ∆B and Eform. It can be seen that ∆B decreases with increasing Eform (linear correlation >0.9). This is one of our findings which can be tested for a wider range of materials. It is particularly clear for positive or close-to-zero Eform (linear correlation 0.94). On the other hand, materials with the lowest Eform—TaTiN and TaVN—exhibit ∆B “only” close to zero, i.e. below the rest of the linear dependence. This phenomenon constitutes a fortunate situation from an application point of view: stable solid solutions have approximately the same B as the constituent binary nitrides (in parallel to the advantages discussed in the Introduction), while in the case of metastable solid solutions their decomposition (due to annealing) is predicted to improve their B.
Figure 7b shows the relationship between ∆G and Eform. In this case there is no single monotonous trend. Instead, the figure shows that (i) close-to-zero or slightly positive Eform leads to relatively low ∆G, while (ii) the highest values of ∆G were obtained at extremal values of Eform, (namely for TaTiN and TaVN—largely negative Eform, and for HfVN and VZrN—largely positive Eform). From an application point of view (and if we let alone the other reasons for individual ternary nitrides—see the Introduction) the figure has two main consequences. First, it confirms (see also Fig. 6) that TaTiN and TaVN are particularly interesting for applications where high hardness (i.e. high G) is of importance. Second, in the case of using (especially) HfVN and VZrN for applications where high ductility (i.e. low G) is of importance, their preparation in the form of two-phase materials is recommended.
Indeed, both the B/G ratio and the Cauchy pressure are almost equivalent (linear correlation >0.95) when predicting the ductility. The datapoints corresponding to binary and ternary materials form a single dependence.
The ductility monotonously increases with increasing number of valence electron per (metal) atom (and thus with an increasing amount of metallic bonding). This is in agreement with trends shown previously for other materials, namely in Ref.  for binary nitrides and in Refs [31, 32] for MMoN and MWN (M = Ti, V) ternary nitrides.
- (3)The B/G ratio and in most cases (except TaVN and NbVN) also the Cauchy pressure of materials M0.51M0.52N are between properties of the constituent binary nitrides. However, similarly as for B and G, the B/G ratio and the Cauchy pressure can be far from the average of properties of the corresponding binary nitrides (not shown as a separate figure).
In particular, mostly negative ∆B and mostly positive ∆G (see the previous figures in this section) lead to lower B/G (lower ductility) compared to the average of the constituent binary nitrides for most of the solid solutions (including all largely metastable solid solutions with high Eform). For example, the Fig. 8 predicts that a two-phase mixture of VN and ZrN (stable configuration) is more ductile than V0.5Zr0.5N solid solution (metastable configuration according to Fig. 2).
However, there are exceptions, most clearly HfZrN, where close-to-zero ∆B and ∆G lead to practically same ductility of the two-phase mixture and the solid solution.
Indeed, the Ta (the heaviest element considered) incorporation in all cases decreases the formation energy.
The difference in Eform is almost negligible if both M1 and M2 = Zr, Hf or Nb, while it is significant if M1 or M2 = Ti or V (the lightest elements considered). This is consistent with the low Eform of TaTiN and TaVN.
Figure 10a shows the dependence of ∆Eform = Eform(Ta0.25M0.3751M0.3752N) − Eform(M0.51M0.52N) on the radii of the constituent elements. Despite the high |∆Eform| of TaVZrN (which is an artifact of the extremely high Eform of VZrN) it can be seen that ∆Eform generally decreases (|∆Eform| increases) with increasing ratio of (i) Ta radius and (ii) average radius of M1 and M2. Similarly, Fig. 10b shows that ∆Eform generally increases (|∆Eform| decreases) with increasing ratio of (i) Ta electronegativity and (ii) average electronegativity of M1 and M2. Higher importance of the discussed phenomenon at closer Ta and M electronegativities is consistent with the aforementioned statement concerning directional covalent-like bonding between Ta and M upon Ta incorporation. Collectively, these findings constitute a formalized version of the phenomenon discussed above: Ti and V are not only the lightest metals considered but also, more importantly, metal elements with the smallest radius and (together with tantalum itself) highest electronegativity (see Table 2).
We systematically studied characteristics of ternary and quaternary metal nitrides (M = Ti, Zr, Hf, V, Nb or Ta) of various compositions by ab initio calculations. The modeling approach allowed us to obtain their formation energies and elastic moduli. Moreover, we found numerous trends for these quantities.
Eform of M1M2N increases with increasing difference between atomic radii and electronegativities of M1 and M2. The lowest Eform values were observed for Ta-containing compositions, and the difference between Eform of TaM1M2N and M1M2N is more significant for lower atomic radius and higher electronegativity of M1 and M2.
For elastic moduli of M1M2N we found that ∆B decreases (down to −19 GPa) and ∆G increases (up to 20 GPa) with increasing difference between atomic radii of M1 and M2. Mostly (i) negative values of ∆B and (ii) positive values of ∆G lead to mostly lower ductility (predicted using the B/G ratio) of solid solutions compared to the corresponding two-phase mixtures. In parallel, low ∆B corresponds to high Eform and high ∆G corresponds to high |Eform|.
Consequently, from an application point of view we predict that decomposition of the metastable solid solutions (mainly VZrN, TiZrN, HfVN and NbVN) or direct preparation of the corresponding two-phase materials improves their B and ductility (~B/G) and decreases their hardness (~G). On the other hand, we identified stable solid solutions (mainly TaTiN and TaVN) with higher hardness (~G) and approximately same B compared to the corresponding two-phase materials.
The presented results allow one to understand and predict which materials form (stable) solid solutions, and to tailor characteristics of hard, ductile, wear-resistant and/or oxidation-resistant ternary metal nitrides of required elastic, electronic or adhesive characteristics for various protective, decorative, tribological or other advanced applications. The phenomena shown can be tested experimentally, and examined for a wider range of materials.
This work was supported in part by the Grant Agency of the Czech Republic GACR under Project No. P108/12/0393, and by the European Regional Development Fund under Project “NTIS—New Technologies for Information Society”, European Centre of Excellence, CZ.1.05/1.1.00/02.0090. Computational resources were provided by Metacentrum Czech Republic.
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