ZrN doped with 3d transition metals: a computational investigation of main physical properties for high-temperature structural applications

In the present study, ab initio density functional theory calculations were used to assess the effect of first-row transition metals (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn) on the stability of Zr0.5N0.5 nitrides. Specifically, the structural, mechanical, and electronic properties were studied to evaluate their applicability in high-temperature structural applications such as coating. The heat of formation for all X-doped Zr0.5N0.5 ternaries were found to be lower than that of the undoped Zr0.5N0.5. Specifically, Mn-doped Zr0.5N0.5 was observed to be the most thermodynamically stable structure, due to its lowest heat of formation. The density of states for both the undoped and doped Zr0.5N0.5 nitrides indicated full metallic behavior and observed that doping with 3d-transition metals reduce the density of states at the Fermi energy, thereby enhancing the electronic stability. Furthermore, mechanical stability was observed in these nitrides with increased melting temperatures expect for Zr0.5N0.5 doped Ti. Since Zr0.5N0.5 doped with X is thermodynamically, electronically, and mechanically stable, they are deemed suitable for high-temperature structural applications especially Zr0.5N0.5 doped Mn.


Introduction
Considerable attention has been directed towards transition-metal nitrides (TMNs) owing to their exceptional thermal, electrical, and mechanical properties.Among these, zirconium and zirconium-based compounds, including zirconium nitrides, are deemed significant in materials intended for extreme high-temperature environments.Furthermore, these materials have found industrial applications in wear-resistant parts, cutting tools, diffusion barriers, as well as decorative and optical coatings [1,2].Material hardness is directly linked to its incompressibility, elasticity, and resistance to deformation.In this regard, first-principles calculations based on density functional theory (DFT) [3,4] have emerged as a promising tool for designing superhard materials, involving the computation of key parameters such as bulk modulus (B) [5][6][7], Young's modulus (E), and shear modulus (G) [6,8,9].Materials with high values of bulk and Young's modulus demonstrate greater resistance to volume and linear compression, respectively.
Theoretical and experimental studies of transition metal nitrides (TMNs) involving elements such as Zr, Hf, V, Nb, Ta, and Ti have been ongoing for over a decade [9][10][11][12][13].It is worth noting that the TiN structure (in NaCl phase) is observed to be brittle [18] and susceptible to oxidation at elevated temperatures exceeding 600 °C, while high hardness is exhibited by the Zr 0.5 N 0.5 structure.Recent research conducted by Shuyin Yu et al [22] investigated finite-temperature properties of ZrN systems.The thermodynamic stability of various series of Zr 0.5 N 0.5 compounds at 0 K was observed, with Zr 0.5 N 0.5 structure (NaCl-phase, space group Fm3 ̅ m) as the most stable.A spectrum of bonding types, including metallic, covalent, or ionic bonding, is exhibited by transition metal (TM) nitrides, which can lead to the formation of meta-stable ternary solid solutions.This phenomenon arises from the hybridization of d and s-p orbitals in both the metals and nitrogen.Additionally, their unique combination of bonding types significantly contributes to the description of the metallic and high hardness characteristics found in Zr 0.5 N 0.5 compounds [14,17].
The hardness of a metal nitride is greatly influenced by its ductility, with this relationship being directly proportional to the shear modulus [23].Coatings of M-Al-N (where M = Ti, Cr, and Zr) have been extensively explored in previous research to enhance the oxidation resistance and mechanical properties of Zr 0.5 N 0.5 [22].Recent experimental investigations have revealed that the hardness of the Zr1-xAlxN solid solution (having an FCC NaCl-type structure) increases from 21 to 28 GPa as the aluminum content increases from x = 0 to 0.43.However, further addition of aluminum results in the hexagonal closed-packed (ZnS-type) AlN, lead to a decrease in hardness [24][25][26].First principles were utilized by Kim et al [27] to investigate the temperature and pressure-dependent elastic properties of ZrC and ZrN using Debye-Gruneisen theory for the first time.Their findings indicated the importance of volume and high-temperature entropy changes in determining the free energy of the solid solutions and that moderate temperatures approximately 1500 k to 2000 K) were needed to synthesize Zr(C1−xNx) solid solutions.
The calculated stable phases were then successfully synthesized.Nevertheless, despite considerable advancements in materials science technology, the use of these transition metal nitrides is constrained by their brittleness and susceptibility to oxidation thereby restricting their potential high-temperature structural applications, particularly in aerospace [28].Moreover, the absence of essential data concerning the thermodynamic stability and manufacturing parameters of zirconium materials have a significant obstacle to both technical advancement and the development of novel materials [27].
In the pursuit of identifying new materials, we use the first principle density functional theory technique to investigate the impact of first-row 3d transition metals (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn) on the structural, electronic, and mechanical properties of Zr 0.5 N 0.5 nitrides.This is due to the brittleness in ZrN nitrides.Our primary objective is to enhance the stability and ductility of Zr 0.5 N 0.5 compounds for hightemperature structural applications specifically in coating environments.It's worth noting that to the best of our knowledge, no previous studies have explored the doping effect of first-row transition metals X (at a concentration of 0.02%) on Zr 0.5 N 0.5 nitrides.
The selection of 3d-transition metals in Ru-based alloys is driven by their favorable mechanical [29] and magnetic attributes, particularly pronounced in Cr, Mn, and Fe.Additionally, these metals typically has higher melting points compared to main group metals [30] and exhibit relatively low densities [31].Elements such as Ni, Co, Mn, Cu, Ti, Fe, and V are acknowledged for their ability to enhance the dehydrogenation properties of magnesium hydride [32], improve thermodynamic stability [33], and augment room temperature ductility in various materials [34].The bonding characteristics in transition metals are influenced by the contributions of both outermost and inner shell electrons.When 3d-transition metals form alloys, nitrides, borides or carbides with other metals or elements, they yield durable materials highly sought for a variety of applications, including high-temperature structural, catalytic, opto-electronic, and spintronics applications [17-19, 29, 35-37].
In this paper, we study the structural, electronic and mechanical properties of undoped and doped Zr 0.5 N 0.5 with X in 0.02% atomic concentration for the first time using the first principles density functional theory technique.This study has not been conducted to date, thereby serving as the basis of our novelty and originality statement.Specifically, we study the lattice constants, heat of formation, density of states, elastic constants, elastic moduli, the anisotropic factor, melting temperatures, and Vickers hardness for both undoped and doped Zr0.5N0.5 nitrides.This comprehensive examination is essential for developing a deep understanding of the hardness and stability of Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X nitrides for coating in high-temperature structural applications.

Methodology
In this study, the ab-initio plane wave (PW) pseudopotential methods implemented in the CASTEP code [38,39] were utilized to compute the structural, electronic, and elastic properties.The CASTEP code, a plane-wave pseudopotential method providing an excellent balance between accuracy and computational efficiency, has been successfully applied to model a wide range of experimental phenomena [38].This simulation approach is remarkably ambitious as it aims to solely depend on quantum mechanics without incorporating any empirical data.In this study, CASTEP (material studio version 2020) within Density Functional Theory was used to model Zr 0.5 N 0.5 doped X (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) transition metal nitrides.The electronic exchange-correlation description was based on Hohenberg-Kohn-Sham density functional theory (DFT) within the framework of the generalized gradient approximation (GGA) [40].Specifically, the Perdew Burke Ernzenhof (PBE) functional within the GGA was employed as the exchange-correlation functional.For Zr 0.5 N 0.5 and Zr 0.5-m X m N 0.5 (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn) nitrides at a fixed doping concentration (m = 0.02).The core and valence interactions were represented by the ultra-soft pseudopotentials developed by Vanderbilt [41].
To model and optimize the unit cell of Zr 0.5 N 0.5 nitride using the experimental lattice parameter, 500 eV planewave energy cut-off was used with 15 × 15 × 15 k-mesh to ensure convergence in total energies and forces of the atoms.Following this, a 2 × 2 × 2 supercell consisting of 64 atoms was constructed from a relaxed unit cell of Zr 0.5 N 0.5 , which originally contained 8 atoms.To model the doped structure, Zr 0.48 X 0.02 N 0.5 , we adopted the supercell approach.Our calculations considered only one doping site: the substitution of a Zr atom located at fractional coordinates (0.5, 0.5, 0.25) with the transition metal dopant X, maintaining a fixed concentration of 0.02 at %.For visual reference, the unit cell and the supercell for Zr 0.5 N 0.5 , comprising 8 atoms and 64 atoms, are depicted in figures 1(a) and (b), respectively.Figure 1(c) illustrates the transition metal-doped nitride Zr 0.48 X 0.02 N 0.5 , where X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn.
In addition, the BFGS algorithm was used in optimizing the geometry of the Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X nitrides.In addition, the maximum SCF cycles was set to 100 which were sufficient to converge the Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X nitrides.To ensure convergence of the calculations for total energies and forces acting on the atoms, a plane wave cut-off energy of 500 eV was uniformly used for all structures.Furthermore, the smearing width was set to 0.1 eV for Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X structures in a non-magnetic state.In addition, were also did a spin polarization calculation for all the Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X nitrides, were a smearing width of 0.001 eV was used to ensure the accuracy of the magnetic moments.All the spin-polarized calculations were set in a Ferromagnetic state.The calculations for the set of k-points were conducted within the first Brillouin zone integration, using Monkhorst and Pack's scheme [42].A k-spacing of 0.027 and a 6 × 6 × 6 k-point mesh were used when calculating the elastic constants and the density of states.During the optimization of structural parameters, including lattice constants and atomic positions, the forces and stress tensors were minimised.The convergence criteria of 1 × 10 -5 eV for the total energy per atom, 0.03 eV/Å for the maximum force, and 0.001 for atomic displacements was utilized for both the unit cells and the supercell nitrides.
The heat of formation of Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X was calculated using the equations (1) and (2) below: and , ( ) E Zr0.5N0.5 refers to the total energies of the Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X nitrides the total energy of undoped system E E E , and g are the total energies of Zr, N and X ( X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn) per atom in their ground state, and lastly a, b and c shows the number of atoms for individual element in the unit cell and supercell [43][44][45].The total energies of all the studied structures are obtained from the CASTEP code.
To determine the three independent elastic constants (C 11 , C 12 , and C 44 ), the stress-strain method was applied.The Voigt, Reuss, and Hill averages were applied to calculate bulk (B), shear (G), and Young's (E) moduli [46][47][48].It is essential to note that the primary focus of this work is to study the influence of transition metals X doping on Zr 0.5 N 0.5 , with no consideration given to porosity or defects.The sodium chloride structure (space group Fm 3m) was used to simulate Zr 0.5 N 0.5 compounds with an experimental lattice parameter of = 4.5675 Å [49].The fractional coordinates for N (0, 0, 0) and Zr (0, 0, 0.5) were determined from materialsproject.org.

Results and discussion
3.1.Stability of the pure Zr 0.5 N 0.5 and the transition metal X doped Zr 0.5 N 0.5 nitrides at 0 K Table 1 presents the computed heat of formation and equilibrium lattice constants for both undoped Zr 0.5 N 0.5 and first-row transition metal (X) doped Zr 0.5 N 0.5 .It is noteworthy that the unit cells for bulk Zr 0.5 N 0.5 (8 atoms) and the supercell for undoped and X-doped Zr 0.5 N 0.5 (64 atoms) share the same lattice constant of 4.59 Å.This uniformity results from the construction of the 2 × 2 × 2 Zr 0.5 N 0.5 supercell, which was derived from the optimized unit cell.The computed lattice constants for the studied Zr 0.5 N 0.5 and Zr 0.48 × 0.02 N 0.5 (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) align well with both theoretical (4.59 Å, 4.53 Å, and 4.62 Å) and experimental (4.63 Å) data, as depicted in table 1.These results affirm the accuracy and reliability of our density functional theory calculations.
The structural stability of a crystal is closely associated with its heat of formation (ΔH f ).The heat of formation for pure and doped Zr 0.5 N 0.5 were computed.Data on the heat of formation (ΔH f ) are provided in Table 1, with the pure Zr 0.5 N 0.5 displaying a value of −3.271 eV/atom, indicating thermodynamic stability.The influence of first-row transition metals X on the heat of formation of pure Zr 0.5 N 0.5 was explored.The ΔHf for all doped Zr0.5N0.5 (Zr 0.48 × 0.02 N 0.5 ) was notably lower than the original pure Zr 0.5 N 0.5 value of ΔH f = −3.271eV/atom.The Zr 0.48 Cr 0.02 N 0.5 exhibited a higher ΔH f of −5.312 eV/atom, implying that Cr-doped Zr 0.5 N 0.5 is less stable compared to all the other studied transition metal-doped nitrides.The data in table 1 clearly indicate that Zr 0.48 Mn 0.02 N 0.5 is the most stable nitride, with the lowest heat of formation of −5.557 eV/atom.This is because the increase in heat of formation is attributed to the atomic number and weight of the 3d-metal series, resulting in a decrease in thermodynamic stability.This observation is consistent with other theoretical data [43,54].The decrease in heat of formation for all the studied Zr 0.48 × 0.02 N 0.5 nitrides follows the order: Table 1.Computed heat of formation (eV/atom) and lattice constants (Å) of the pure Z 0.5 N 0.5 and the X-doped Z 0.5 N 0.5 (X = Sc,Ti,V,Cr,Mn,Fe,Co,Ni,Cu,Zn) compounds in the sodium chloride (NaCl) rocksalt structure displayed in atomic concentrations.Mn < Sc < Ti < V < Zn < Cu < Fe < Ni < Co < Cr.These results show that the introduction of first-row transition metals significantly reduces the ΔH f of the Zr 0.5 N 0.5 nitrides and therefore, thermodynamic stability is enhanced.

Electronic properties
The physical properties of materials have influence on their electronic structure, making the analysis of electronic properties a valuable method for predicting various characteristics of new materials.In particular, the Density of States (DOS), which is the number of electronic states available for occupation by electrons at specific energy levels, is closely associated with the type of bonding present and, thus, the stability of a material [33,55,56].Moreover, the electronic stability of materials can be linked to the quantity of accumulated electrons, represented as N(E f ), at the Fermi energy level.Compounds with greater stability typically demonstrate a reduced accumulation of electrons at the Fermi energy [19,[57][58][59].
In figures 2 (a)-(k) illustrates the total density of states (TDOS) and the partial density of states (PDOS) for both pure and doped Zr 0.5 N 0.5 nitrides at 0 K. Across all the pure and doped Zr 0.5 N 0.5 nitrides studied, a common characteristic emerges, where there is no gap separating the valence and conduction bands, hence they possess a metallic behavior.This is attributed to the partial filling of Zr (4d) and transition metal X (3d) states, leading to strong hybridization of the 4d states of Zr, transition metal X, and N 2p orbitals.
It is noted that the overall features of the total Density of States (DOS) in Zr 0.5 N 0.5 doped with X exhibit similar characteristics, as all show high peaks in the valence band compared to the conduction band.The introduction of dopants results in a general shift of the valence band to lower energy levels, as evidenced in figures 2(b)-(k).Specifically, states that were originally between −13 eV and −12 eV shift to lower energies, now ranging between −16 eV and −13 eV.Bands previously situated between −5 eV and −1.9 eV have shifted to the valence band region, between −8 eV and −3 eV.
The introduction of X-doping significantly reduces the density of states in the valence band, conduction band and around the Fermi level.The Partial Density of States (PDOS) depicted in figure 1 of the supplementary files illustrates that the 2p-N states make a greater contribution to the electronic density in the valence band (within the energy range of −8 eV to −4 eV) compared to the 4d-Zr and 3d-X states.Conversely, the 4d-Zr states exhibit the highest contribution at the Fermi energy and in the conduction band (within the energy range of 0 eV to 4 eV) across all the studied Zr 0.5 N 0.5 doped with X nitrides.
To assess the stability of the nitrides, the Partial Density of States (PDOS) near the Fermi energy level was crucial to consider.In both doped and undoped Zr 0.5 N 0.5 , states near the Fermi levels mainly arise from Zr 4d states; it is found that the PDOS of all the studied Zr 0.5 N 0.5 doped X nitrides are significantly lower near the Fermi level compared to the PDOS of undoped Zr 0.5 N 0.5 .This may be attributable to 3d-X transition metals and 4d-Zr and 2p-N orbital mixing at the Fermi energy level leading to strong hybridization.It is noted that Mn, Cr, V, and Ti doped Zr 0.5 N 0.5 are more electronically stable due to the lowest density of states at the Fermi energy level compared to Sc, Fe, Co, Ni, Cu, and Zn doped Zr 0.5 N 0.5 .This observation is in accord with our previous theoretical work [33].These results suggest that the introduction of dopants into Zr 0.5 N 0.5 enhances the electronic stability of the nitride, aligning with the conclusions drawn from the preceding stability analysis based on heat of formation.The doping of transition metals represents a potential avenue for controlling the electronic properties of the nitrides [52,53].
Furthermore, the spin-polarization density of states were considered.In this section, the electronic and magnetic properties of the Total and Partial Density of States for (a) Zr 0.5 N 0.5 , (b) Z 0.48 Sc 0.02 N 0.5 , (c) Zr 0.48 Ti 0.02 N 0.5 , (d) Zr 0.48 V 0.02 N 0.5 , (e) Zr 0.48 Cr 0.02 N 0.5 , (f) Zr 0.48 Mn 0.02 N 0.5 , (g) Zr 0.48 Fe 0.02 N 0.5 , (h) Zr 0.48 Co 0.02 N 0.5 , (i) Zr 0.48 Ni 0.02 N 0.5 , (j) Zr 0.48 Cu 0.02 N 0.5 , and (k) Zr 0.48 Zn 0.02 N 0.5 compounds are discussed in a spin-polarized setting.The Total and partial DOS with the magnetic moments are found in the supplementary materials.It was observed that ZrN doped with Mn (0.068 μ B ), Fe (0.052 μ B ), and Co (0.040 μ B ) have the highest magnetic moments.Conversely, other undoped and doped transition metal nitrides possess zero or negligible moments such as Sc (0.005 μ B ), Ti (0.002 μ B ), V (0.002 μ B ), Cr (0.003 μ B ), Ni (0.000 μ B ), Cu (0.002 μ B ), and Zn (0.000 μ B ).The trend of decreasing magnetic moments in Zr 0.5 N 0.5 doped with X transition metals is as follows: Therefore, the magnetic properties of undoped Zr 0.5 N 0.5 nitride are increased by doping with Mn and Co.It is noted that Zr 0.5 N 0.5 doped with Fe, Mn, and Co exhibit slightly asymmetric spin up and down Total Density of States (TDOS), particularly in the −3 eV energy range, as shown in figures 2 and 3 (in supplementary files), resulting in non-zero magnetic moments.Conversely, undoped and doped Zr 0.5 N 0.5 with X = Sc, Ti, V, Cr, Ni, Cu, and Zn elements exhibit symmetric spin up and down states with no net spin polarization.This symmetric balance in the up and down spin states cancels the magnetic moments associated with the electron spin, explaining the zero or negligible magnetic moments in the nitrides.These observations are consistent with our previous studies [17-19, 37, 60-62].

Elastic properties
The investigation of a solid's elastic properties is crucial for understanding the material's mechanical behavior.To gain deeper insights into the impact of first-row transition metals on the stability of the Zr 0.5 N 0.5 nitride, calculations were performed on the elastic properties of both pure and doped Zr 0.5 N 0.5 nitrides.Elastic constants play a vital role in describing a crystal's mechanical resistance when subjected to external stresses.These constants can be computed by applying a small strain to the unit cell and measuring the resulting variation in total energy [53,63].The elastic strain was determined using the following relationship (equation ( 3)): Where ∆E is the total energy difference between the deformed and initial unit cell, V 0 is the original cell volume, = ( ) C i j to , 1 6 ij are the elastic constants and e ore i j is the strain.It is widely recognized that a cubic structure possesses three independent elastic constants, denoted as C 11 , C 12 , and C 44 .Table 2 shows the calculated elastic constants for both pure and first-row transition metal-doped Zr 0.5 N 0.5 , with the relevant theoretical and experimental data for the pure Zr 0.5 N 0.5 nitride.In the field of mechanics, for a crystal to be considered mechanically stable, its elastic constants must satisfy the Born-Huang stability criteria [64,65].The mechanical stability criterion for a cubic system was shown in equation (4): As presented in table 2, the results for the elastic constants (C ij ) for all the studied nitrides meet the conditions, confirming the mechanical stability of both the pure and doped materials, which agree with the heat of formation results (table 1) and electronic properties.Therefore, these materials, are thermodynamic, electronic and mechanically stable.The elastic constant C 11 provides insights into the resistance to compression along the x-direction under applied stress [67].Table 2 reveals that the elastic constant C 11 for Mn-doped Zr 0.5 N 0.5 (510.7 GPa) is notably higher compared to the other X-doped Zr 0.5 N 0.5 and the undoped structure.Zr 0.48 Mn 0.02 N 0.5 (510 GPa) and Zr 0.48 Ti 0.02 N 0.5 (481.1 GPa) exhibit high and low compressibility resistance along the x-axis, respectively.Furthermore, it is well-established that the elastic constant C 44 is an important parameter that indirectly determines a solid's indentation hardness [68].In the undoped Zr 0.5 N 0.5 , the value of C 44 (134.5 GPa) surpasses that of the doped Zr 0.5 N 0.5 .Moreover, the calculated C 44 for Zr 0.48 Sc 0.02 N 0.5 is 125 GPa, indicating a stronger ability to resist shear distortion in the (100) plane.
The tetragonal shear elastic factor linked to shear modulus along 〈011〉 direction on {011} plane, can represent the elastic stability of a cubic crystal as shown in equation ( 5) Positive shear elastic factor indicates elastic stability whereas, negative shear elastic factor implies elastic instability [45].Table 1 shows the shear elastic factor for Zr 0.5 N 0.5 and Zr 0.5 N 0.5 doped X nitrides.The introduction of dopants enhances the stability factor C′ for pure Zr 0.5 N 0.5 nitride.Notably, for all the considered dopants, Mn-doped Zr 0.5 N 0.5 exhibits the highest stability factor C' along with the lowest C 12 value (95.5 GPa).These findings indicate that Zr 0.48 Mn 0.02 N 0.5 displays greater elasticity and stability compared to all the considered X-doped nitrides.
To delve deeper into the mechanical properties of pure and doped Zr 0.5 N 0.5 , we investigate polycrystalline elastic moduli, including the shear modulus (G), bulk modulus (B), and Young's modulus (E).According to the Hill theory [46], the shear modulus of polycrystalline materials can be expressed in equation ( 6): Where Gv and G R are Voigt shear modulus and Reuss shear modulus, respectively.Gv and G R are respectively calculated as follows [46, 47, The bulk modulus B and Young's modulus E [70] can be deduced by the following formula The calculated values for bulk, shear, and Young's modulus are summarized in table 3. The bulk modulus of pure Zr 0.5 N 0.5 (248.6 GPa) is slightly greater than that of the doped Zr 0.5 N 0.5 , which ranges from 233.1 GPa (Mndoped) to 247.8 GPa (Sc-doped).As the bulk modulus (B) is a measure of resistance to volume change under applied pressure [71], a comparison of the calculated bulk modulus for the doped systems indicates that Scdoped and Mn-doped Zr0.5N0.5 have the highest and lowest resistance to volume change, respectively.Additionally, a significant increase in shear moduli was observed with the introduction of dopants into the Zr 0.5 N 0.5 nitride, as shown in table 3. Shear modulus is generally related to a material's resistance to plastic deformation [72].A higher shear modulus indicates stronger resistance to reversible deformation under shear stress.The calculated shear modulus for pure Zr 0.5 N 0.5 (125.7 GPa) is notably smaller than the G values for the doped nitrides, ranging from 141.6 GPa (Ti-based) to 150 GPa (Cr-based) in Zr 0.5 N 0.5 .Clearly, resistance to plastic deformation for undoped Zr 0.5 N 0.5 nitride is enhanced by first-row transition metal doping.
Young's modulus (E) measures a material's stiffness [70], where greater stiffness is indicated by a larger Young's modulus.The results in table 3 demonstrate that the Young's modulus for all the X-doped Zr 0.5 N 0.5 materials is lower than that of pure Zr 0.5 N 0.5 , which is evidently stiffer.The deviations between the Young's modulus for pure Zr 0.5 N 0.5 and the lowest and highest E values for the doped systems are 42% and 54%, respectively.However, when comparing the stiffness of the doped Zr 0.5 N 0.5 materials, higher stiffness is exhibited by Fe-doped Zr 0.5 N 0.5 (173.2GPa), whereas lower stiffness is displayed by Mn-doped Zr 0.5 N 0.5 (134.3GPa).
Pugh [71] proposed a criterion based on the bulk/shear modulus ratio (B/G) that is commonly used to differentiate between ductile and brittle materials.Pugh suggested that if B/G exceeds the critical value of 1.75, the material is ductile, and if it is less than 1.75, it is brittle.The values in table 3 show that pure Zr 0.5 N 0.5 changed from ductile (B/G = 1.98) to brittle when doped with the first-row transition metals.The B/G ratios for all the doped Zr 0.5 N 0.5 materials are less than 1.75, with the Ti-doped system displaying a higher B/G ratio of 1.72, which is closer to the critical value of 1.75.Hence, the Zr 0.5 N 0.5 doped X transition metals does not improve the ductility of the undoped Zr 0.5 N 0.5 .
Table 3. Calculated bulk (B), shear (G) and Young's modulus (E) in GPa, B/C 44 ratio, B/G ratio and Poisson's ratio (υ).The density (ρ cal ), elastic anisotropy factor (A) and melting temperature (MT) and and Vickers hardness (H V ) in GPa for undoped and doped Zr 0.5 N 0.5 with X (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) -transtion metals.The calculated values are compared with the available theoretical and experimental data.Poisson's ratio (u) provides more information about the characteristics of the bonding nature between the atoms than any of the other elastic constants.The values of the Poisson's ratio for covalent material are small (u = 0.1), whereas for ionic materials a typical value is 0.25 [75].As listed in table 3, the Poisson's ratios for the pure Zr 0.5 N 0.5 (u = 0.28) and doped Zr 0.5 N 0.5 (u range: 0.24-0.26)are very close to 0.25, which means that the ionic bonding is dominant in the studied pure and doped nitrides.In addition, Poisson's ratio is able to characterize the bonding forces in crystals [76].The Poisson's ratio in lower limit and upper limit for central force solids are u = 0.25 and 0.5.

Phases
The computed Poisson's ratios for the pure Zr 0.5 N 0.5 doped and the eight X-doped nitrides (excluding Mn and Cr-doped Zr 0.5 N 0.5 with υ = 0.24) are within the range u   0.25 0.5.The results indicate that all the studied structures have interatomic forces that are central except for the Mn and Cr-doped systems.Lastly, Poisson's ratio can evaluate ductility or brittleness of the materials according to Frantsevich [75].If υ > 0.3 then the material is regarded as ductile, otherwise as brittle (υ < 0.3).In table 3, the values of υ for all the X-doped Zr 0.5 N 0.5 are below 0.3 indicating that the materials are brittle.Comparison of B/G and Poisson's ratios for the doped nitrides was made, the trends given by both the Poisson's and B/G ratios are similar as shown in figures 3(a) and (b).Both ratios predicted the brittleness of the materials, the direct relation between the B/G and Poisson's ratio correspond to an observation by Popoola et al [77].
Density (r incm cal 3 ) of the alloys determines the heaviness or lightness of the material.Heavy and light alloys are defined as materials with higher and lower densities respectively.Lightweight materials are more suitable for use in high temperature environment (aerospace) than heavy materials.The density will be calculated as follows in equation ( 11) Where M W is the average molecular weight of the total elements, Vol is the volume of the cell measured in Å ,

3
A 0 is the Avogadro's number in atom/mole (6.022 X 10 23 ) and N is the number of atoms.Densities for the pure and doped Zr 0.5 N 0.5 are shown in table 3. The calculated density r cal for pure Zr 0.5 N 0.5 (7.2 g cm −3 ) agrees very well with the previous theoretical (7.10 g cm −3 ) [66] and experimental (7.32 g cm −3 ) [78] results.The density drop was noted when Zr 0.5 N 0.5 was doped with Sc and Ti, on the other hand an increase was observed for Mn, Ni, Cu and Zn-doped nitrides.The pure and V, Cr, Fe, Co-doped nitrides have the same density (7.2 g cm −3 ).The Zr 0.48 Sc 0.02 N 0.5 (r cal = 6.9 g cm −3 ) was identified as the lightest weight material compared to all the studied nitrides.
Elastic anisotropy plays an important role in engineering science since it is highly correlated with the possibility to induce micro-cracks in materials [79].The anisotropic factor (A) provides a measure of the degree of anisotropy, and can be defined as A represents the ratio of the two extreme elastic-shear coefficients.For most materials, A > 1; but for many bcc elements and for many compounds, A < 1 therefore these materials are anisotropic.Except accidently, one never expects isotropic behavior, A = 1.For a completely isotropic material the value for A = 1, while any value smaller or larger than 1 measure the degree of elastic anisotropy [76,80].The calculated A values for the pure and X-doped Zr 0.5 N 0.5 are listed in table 3. It is noticed that an anisotropic factor (A) less than 1 is observed in all the studied nitrides, indicating that they can be regarded as being elastically anisotropic.Furthermore, it is worth noting that higher anisotropic behavior (0.68) is exhibited by Ti-doped Zr 0.5 N 0.5 compared to the other X-doped nitrides.
Melting temperature (T m ) is widely used as an indicator of atomic strength between intermetallic compounds.In order to evaluate the potential of high-temperature structural application of the nitrides, the melting temperatures of the pure and first-row transition metal (X) doped Zr 0.5 N 0.5 are estimated based on the calculated elastic moduli C ij [81][82][83][84].The T m for a cubic system is estimated by the following relation.
Where the symbol Mbar stands for Megabar.Our calculated melting temperatures are listed in table 3. The values of the Tm for all the studied Zr 0.48 X 0.02 N 0.5 (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) are higher than the value for the pure Zr 0.5 N 0.5 .In all cases, the presence of dopants had been noted to enhance the melting temperature.The decrease in the melting temperatures for all the studied Zr 0.48 X 0.02 N 0.5 follow the order Mn > Ni > Cu > Cr > Fe > Sc > Zn > Co > V > Ti.The elastic stability factor C' can be correlated with melting temperature as shown in figures 4(a) and (b) respectively.Both the graph in figure 4 and table 3 data show that Zr 0.48 Mn 0.02 N 0.5 possesses both the highest melting temperature and elastic stability factor.In fact, our results indicate that Zr 0.48 Mn 0.02 N 0.5 is more elastically stable and has high melting temperature compared to all the studied X-doped Zr 0.5 N 0.5 .
The elastic constants enable us to evaluate some macroscopic properties of the materials such as hardness of materials.To estimate the values of Vickers hardness (H v ) the empirical formula for hardness prediction proposed by Chen et al [85] was used as follows: Where K is the ratio of shear modulus to bulk modulus (G H /B H ). The Vickers hardness (Hv) of the pure and first-row transition metal (X) doped Zr 0.5 N 0.5 are predicted by using equation (14) and the results are listed in table 3. It is found that the value of Hv for the pure Zr 0.5 N 0.5 (18.0 GPa) agrees well with the available experimental data (18.1 GPa).In order to further understand the effect of the first-row transition metals X on pure Zr 0.5 N 0.5, the Vickers hardness of the X-doped Zr 0.5 N 0.5 are investigated and reported in table 3. Our Hv values of Cr-doped Mn-doped Zr 0.5 N 0.5 are 18.5 GPa and 18.9 GPa respectively, which are higher than the value for the pure Zr 0.5 N 0.5 (18.0 GPa).On the other hand, it is noted that a decrease in Vickers hardness is produced by the seven dopants (Sc, Ti, V, Co, Ni, Cu, and Zn).However, the Hv values of both Fe-doped Zr 0.5 N 0.5 (18.0 GPa) and pure Zr 0.5 N 0.5 (18.0 GPa) are the same.The order of increase in Vickers hardness for all the studied Zr 0.48 X 0.02 N 0.5 is as follows: Ti < V < Sc < Cu < Ni < Zn < Co < Fe < Cr < Mn.It is concluded that Zr 0.48 Mn 0.02 N 0.5 is the hardest material compared to all the doped and undoped materials studied in this paper.

Conclusion
Through first-principles calculations, the structural, electronic, and mechanical properties of undoped and firstrow transition metal X doped Zr 0.5 N 0.5 (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) nitrides were investigated.Our focus was to evaluating their structural, mechanical, and electronic characteristics, particularly in relation to their potential use in high-temperature structural applications such as coating.It was found that the heat of formation for all X-doped Zr 0.5 N 0.5 ternaries was consistently lower compared to the undoped Zr 0.5 N 0.5 , indicating increased stability.Specifically, it was observed that Mn-doped Zr 0.5 N 0.5 exhibited the highest thermodynamic stability, primarily due to its lowest heat of formation.Analysis of the density of states revealed that both undoped and doped Zr 0.5 N 0.5 nitrides displayed full metallic behavior, with doping leading to a reduction of states at the Fermi energy level, thus enhancing electronic stability.Additionally, these nitrides demonstrated mechanical stability with high melting temperatures.Given their favorable thermodynamic, electronic, and mechanical properties, the X-doped Zr 0.5 N 0.5 nitrides are considered suitable candidates for high-temperature structural applications especially Mn-doped Zr 0.5 N 0.5 .given to the high-performance computing (HPC).Gratitude is extended to Dr Mohammed Khenfouch for proofreading this work.

Table 2 .
Calculated, elastic constants (C 11 , C 12 and C 44 ) and elastic stability factor for undoped and doped Zr 0.5 N 0.5 with X-transition metals.The calculated values are compared with the available theoretical and experimental data.