## Abstract

Motivated by recent successful synthesis of transition metal dinitride TiN_{2}, the electronic structure and mechanical properties of the discovered TiN_{2} and other two family members (ZrN_{2} and HfN_{2}) have been thus fully investigated by using first-principles calculations to explore the possibilities and provide guidance for future experimental efforts. The incompressible nature of these tetragonal TMN_{2} (TM = Ti, Zr, and Hf) compounds has been demonstrated by the calculated elastic moduli, originating from the strong N-N covalent bonds that connect the TMN_{8} units. However, as compared with traditional *fcc* transition metal mononitride (TMN), the TMN_{2} possess a larger elastic anisotropy may impose certain limitations on possible applications. Further mechanical strength calculations show that tetragonal TMN_{2} exhibits a strong resistance against (100)[010] shear deformation prevents the indenter from making a deep imprint, whereas the peak stress values (below 12 GPa) of TMN_{2} along shear directions are much lower than those of TMN, showing their lower shear resistances than these known hard wear-resistant materials. The shear deformation of TMN_{2} at the atomic level during shear deformation can be attributed to the collapse of TMN_{8} units with breaking of TM-N bonds through the bonding evolution and electronic localization analyses.

## Introduction

Transition metal nitrides (TM_{x}N_{y}), synthesized under high-pressure and high-temperature conditions, represent a prominent class of materials exhibiting extreme usefulness in a wide variety of industrial applications^{1,2}. When it comes to their superior mechanical properties such as high hardness and elastic moduli, most of the early transition metal mononitrides (TMN), and in particular TiN and CrN are well known hard materials and are widely used in various industrial applications, such as cutting tools or wear-resistant coatings^{3,4}. Taking advantage of high-pressure techniques, two family members of hard nitrides (Zr_{3}N_{4} and Hf_{3}N_{4}) of the group IVB with TM_{3}N_{4} stoichiometry were successfully synthesized^{5}, opening a promising way to obtain other nitrides with N:TM > 1 under high nitrogen pressure. Compared to early transition metals, the noble metals (TM = Ru, Rh, Pd, Os, Ir, and Pt) were previously known to hardly form nitrides with high nitrogen content. Until 2004, a novel platinum nitride with ultra-high incompressibility was obtained under extreme conditions (50 GPa and 2000 K) by Gregoryanz *et al*.^{6} and was finally determined to crystallize in the pyrite structure with a stoichiometry of 1:2^{7,8}. Thereafter, there have been considerable researches to search for other transition metal dinitrides, and so far as we know, the OsN_{2}^{8,9}, IrN_{2}^{7,9,10}, PdN_{2}^{10}, RhN_{2}^{11}, and recently RuN_{2}^{12} have been experimentally obtained in a direct chemical reaction between platinum group elements and molecular fluid nitrogen at high pressures and temperatures. The follow-up studies of their structures and mechanical properties have stimulated significant in their potential applications. These works have been motivated by the design of intrinsic (super)hard materials proposed by Kaner *et al*.^{13,14} that the introduction of light and covalent-bond-forming elements, such as B, C, N, and O into the transition metal (TM) lattices with highly valence-electron density is expected to enhance the shear strength against plastic deformations.

More recently, a new transition metal dinitride, TiN_{2}^{15}, was successfully synthesized at 73(3) GPa and 2400(40) K by choice of TiN and dense N_{2} as starting materials. The experiment revealed that this new dinitride adopts a tetragonal CuAl_{2}-type structure at high pressure, which is in agreement with previous theoretical prediction performed by Yu *et al*.^{16}. On decompression, the experiment found that this phase is recoverable to ambient conditions and possesses a high bulk modulus of 385(7) GPa comparable to those of PtN_{2} (372 GPa)^{6} and ReB_{2} (360 GPa)^{17}, much larger than that of TiN (288 GPa)^{18}. Therefore, this new tetragonal TiN_{2}, the first synthesized high-nitride phase in early transition metal nitrides, is expected to be a candidate as a potential superhard solid for wear- and scratch-resistant materials. However, this concept for the search of novel superhard materials failed in materials such as PtN_{2}^{19}, and ReB_{2}^{20}, and others^{21,22,23}, because plastic deformation occurs in shear at large strain at the atomic level, where electronic instabilities may occur upon bond breaking in the practical measurement of hardness. Meanwhile, the macroscopic behavior of a solid is strongly related to its elastic anisotropic properties, which can reveal, in some materials, an anisotropy degree decidedly non-negligible and in some cases so extreme to suggest the proximity of material instability. Accordingly, here, we have extended the mechanical behaviors of TiN_{2} and presented in detail the variations of the elastic moduli along the arbitrary directions. Moreover, the stress-strain relations and the underlying atomistic bond breaking processes under the applied strains were also systematically investigated to provide a deeper insight into mechanical properties and hardness of the newly discovered TiN_{2}. We have also applied this novel tetragonal structure to other two family members ZrN_{2} and HfN_{2} to explore the possibilities and provide guidance for future experimental efforts. We hope that the present findings will encourage further theoretical and experimental works on this class of material.

## Results and Discussion

The experiment has demonstrated that TiN_{2} crystallizes in the tetragonal CuAl_{2}-type structure with Ti and N atoms sitting at 4*a* and 8 *h* sites in a unit cell, as shown in Fig. 1(a). Polyhedral view of this tetragonal structure (Fig. 1(b)) reveals that TiN_{2} consists of the TiN_{8} face-sharing tetragonal antiprisms connected by N-N bonds and stacked along the *c*-axis, in contrast to the TMN_{6} octahedrons in the previous synthesized noble metals pernitrides^{7,8,9,10,11,12}. Through the full relaxations of both lattice constants and internal atomic coordination, the obtained equilibrium structure parameters for three TMN_{2} compounds are listed in Table 1, among which the calculated results for TiN_{2} compare well with the available experimental data^{15}. For ZrN_{2} and HfN_{2}, however, there are no available experimental data for comparison and the present results could provide useful information for further experimental or theoretical investigations. According to the recent experiment by Bhadram *et al*.^{15}, the pressure dependences of unit cell volume and lattice constants of TiN_{2} were calculated and plotted in Fig. 2, along with the experimental data^{15} and theoretical results of ZrN_{2} and HfN_{2}. First, one can see that the calculated results for TiN_{2} are in agreement with the experimental data under pressure, and the incompressibility of TiN_{2} (Fig. 2(a)) is almost identical to that of HfN_{2}, but larger than that of ZrN_{2}. Furthermore, from Fig. 2(b), it can be seen that the incompressibility along the *a*-axis is larger than that along the *c* axis for each TMN_{2} compound, indicating their clear elastic anisotropy. Second, the *E*-*V* data under pressures deduced from the Fig. 2(a) for each TMN_{2} are fitted to the third order Birch-Murnaghan equation of state (EOS)^{24}. The obtained the bulk modulus (*B*_{0}) and its pressure derivative (*B*_{0}′) for TiN_{2} are 276 GPa and 4.362 (see Table 1), which are lower than those of experimental data (385 GPa and 1.45), but consistent with the theoretical values (293 GPa and 3.7) predicted by Bhadram *et al*. using the same approach^{15}. The low value of *B*_{0}′ related to *B*_{0} in this discrepancy has been elucidated in this experimental work. Third, the fitted *B*_{0} values of TiN_{2} and HfN_{2} are nearly equivalent but larger than that of ZrN_{2}, which is in accord with the calculated compressibility of volume plotted in Fig. 2(a). Overall, the accuracy of the present calculations for TiN_{2} is made quite satisfactory with the experimental data in Table 1 and Fig. 2, which supplies the safeguard for the following studies.

According to synthetic conditions of TiN_{2} proposed by Bhadram *et al*.^{15}, the thermodynamic feasibility of ZrN_{2} and HfN_{2} is evaluated through the formation enthalpy (energy) calculations. The formation enthalpy Δ*H*_{f} of each TMN_{2} with respect to the TMN and nitrogen at ambient conditions based on the reaction route: was quantified, where the *fcc* TMN phase and *α*-N_{2} phase are chosen as the reference phases. As listed in Table 1, the calculated formation enthalpies of three TMN_{2} dinitrides are all positive values, indicating that they are all metastable at ambient conditions. It is to be noted that the calculated formation enthalpies of ZrN_{2} (0.372 eV/atom) and HfN_{2} (0.328 eV/atom) are all close to that of TiN_{2} (0.398 eV/atom), which has been synthesized at 73(3) GPa and 2400(40) K by choice of TiN and dense N_{2} as starting materials. Thus, the syntheses of the ZrN_{2} and HfN_{2} could be expected at similar high pressure and temperatures conditions. The experiment has suggested that TiN_{2} can be quenchable to ambient conditions, and the dynamical stabilities of ZrN_{2} and HfN_{2} at 0 GPa have been thus carefully checked by the full phonon dispersions calculations using the 2 × 2 × 2 supercell method. Figure 3(a,b) show the phonon dispersion curves which confirm the dynamic stability of ZrN_{2} and HfN_{2} as there are no imaginary modes in the whole Brillouin zone. The lower frequencies of the phonon density of states are dominated by lattice dynamics of heavy TM atoms and higher frequencies by light N atoms.

The total and projected density of states (DOS) of each TMN_{2} at ambient pressure was plotted to further elaborate the electronic bonding feature, as shown in Fig. 4(a–c), respectively. All TMN_{2} compounds show metallic bonding because of finite value of DOS at the Fermi level (E_{F}), which originates mostly from the TM-*d* orbitals and the N-*p* orbitals. The major orbital occupancy in the energy range of −8–0 eV stems from the strong hybridized states of TM-*d* and N-*p* orbitals, as the usual cases in the most TM_{x}N_{y} compounds. The typical feature of the total DOS is the presence of a “pseudogap” (a sharp valley around the E_{F}), which is supposed the borderline between the bonding and antibonding states^{25,26,27}. For TiN_{2}, it is noteworthy that the bonding states are completely filled with the Fermi energy located exactly at the “pseudogap”. For ZrN_{2} and HfN_{2} (see Fig. 4(b,c)), it is found that the E_{F} shifts toward the higher energy and lies left at the pseudogap with a relative more electronic density of states [N(E_{F})]. It is known that for the most stable structure there is enough room to accommodate all its valence electrons into bonding states so as to bring the E_{F} to a valley position separating bonding and antibonding states (pseudogap) favorable for structural stability. Therefore, the TiN_{2} is energetically more favorable compared to the ZrN_{2} and HfN_{2} in the tetragonal phase. Figure 4(d–f) offers the calculated crystal overlap Hamilton population (COHP)^{28} for the TM-N and the N-N bonding inside TiN_{2}, ZrN_{2}, and HfN_{2}, respectively. For the TM-N combinations in all plots, there are only bonding states in the entire occupied regions, and antibonding states show up in the unoccupied crystal orbitals, well above the E_{F}. For the N-N combinations in TMN_{2}, the antibonding 1*π*_{g}^{*} states (starting near −3.5 eV) are almost completely occupied at the top of the conduction band, and a portion of the metallic nature can be ascribed to these states being occupied at the E_{F}. For TiN_{2}, this point has been addressed in a recent work by Yu *et al*.^{16}. As demonstrated in previous work^{29}, for the case of PtN_{2}, charge transferred from Pt to N (1.05 *e*) results in the full filling of antibonding 1*π*_{g}^{*} states of N_{2}^{4−} and leads to the elongation of N-N bonds. In a similar way, this mechanism is also applicable to the case of tetragonal TiN_{2} although antibonding states are not completely filled and there are differences in electronic and structural configurations, as suggested by Bhadram *et al*.^{15} and Yu *et al*.^{16}. Consequently, a charge balance of N_{2}^{4−} in these TMN_{2} is a good working hypothesis, and this leaves the TM atoms in TMN_{2} in a *d*^{0} configuration. In order to compare the “ionicity” of the three dinitrides, we also analyzed the charge density topology through the Bader charge analyses^{30}. The calculated charges of the three nitrides show decreasing trends from Hf ^{2.13}N_{2}^{−2.13} to Zr^{1.96} N_{2}^{−1.96} and Ti^{1.75}N_{2}^{−1.75}, indicating the relatively lower polarity of Ti-N bond. Meanwhile, it has been demonstrated^{31} that the shortening of the N-N bond is ascribed to the decrease in charge transfer from TM to N (*q*_{trans}) when one monitors the pernitrides from early to late TM elements. It can be seen that as the TM element moves from Hf through Zr to Ti, as *q*_{trans} from 2.13 *e* through 1.96 *e* to 1.75 *e*, and as *d*_{N-N} from 1.461 Å through 1.434 Å to 1.385 Å.

For potential engineering applications, the elastic stabilities, incompressibility, and rigidity of three TMN_{2} dinitrides are determined from the calculated elastic constants by applying a set of given strains with a finite variation between −0.01 and +0.01. Table 2 summarizes the calculated single-crystal elastic constants *C*_{ij} and derived Hill elastic moduli as well as Poisson’s ratios of TMN_{2} dinitrides and compares them with those of typical hard substances TMN (TM = Ti, Zr, and Hf)^{32,33,34,35}. The calculated six independent elastic constants of TiN_{2} agree well with recent theoretical results^{16}, and the derived bulk moduli of three TMN_{2} dinitrides also accord well with those directly obtained from the fitting of the third-order Birch-Murnaghan EOS (see Table 1), demonstrating the reliability of the present calculations. The mechanical stabilities of three dinitrides satisfy the Born-Huang criterion^{36} for a tetragonal crystal [*C*_{11} > 0, *C*_{33} > 0, *C*_{44} > 0, *C*_{66} > 0, (*C*_{11} − *C*_{12}) > 0, (*C*_{11} + *C*_{33} − 2*C*_{13}) > 0, and 2(*C*_{11} + *C*_{12}) + *C*_{33} + 4*C*_{13} > 0], indicating their mechanically stable at ambient conditions. From Table 2, the high-incompressible nature of TMN_{2} is disclosed by the calculated bulk modulus (TiN_{2}: 276 GPa, ZrN_{2}: 250 GPa, HfN_{2}: 275 GPa), originating from the covalent TMN_{8} polyhedrons connected by the strong N-N covalent bonds in systems. Meanwhile, these values are comparable with the corresponding theoretical calculations and experimental data (in brackets) of typical hard transition metal mononitrides TMN, TiN: 278 GPa (288 GPa), ZrN: 250 GPa (215 GPa), HfN: 273 GPa (306 GPa). The critical values of the ratio of shear modulus *G* to bulk modulus *B* of about 0.57 separates brittle (*G*/*B* > 0.57) and ductile (*G*/*B* < 0.57) materials. For three TMN_{2} dinitrides, their *G*/*B* values (TiN_{2}: 0.707, ZrN_{2}: 0.608, HfN_{2}: 0.626) are all larger than 0.57, implying that they are intrinsically brittle. The theoretical Vickers hardness *H*_{v} of each TMN_{2} was estimated by using the Chen’s empirical model^{37}, *H*_{v} = 2(*k*^{2}*G*)^{0.585} − 3. The calculated hardness value for TiN_{2}, ZrN_{2}, and HfN_{2} is 26.1 GPa, 18.1 GPa, and 23.2 GPa, respectively, making them potentially interesting for applications as hard coating materials. By using the Bader atoms-in-molecules (AIM) method, the strong covalent nature of the N-N and TM-N bonds in TMN_{2} were quantitatively revealed by the evidences of local charge densities at their bond critical points (BCPs) with negative Laplacian values. The obtained who can measure the bond strength related to the mechanical behaviors located at N_{2} dumbbells and TM-N bonds decrease in the sequence of TiN_{2}: (2.324 *e*/Å^{3}) > ZrN_{2} (2.047 *e*/Å^{3}) ≈ HfN_{2}: (2.045 *e*/Å^{3}) and TiN_{2}: (0.457 *e*/Å^{3}) > HfN_{2}: (0.444 *e*/Å^{3}) > ZrN_{2} (0.428 *e*/Å^{3}), respectively. Therefore, compared to TiN_{2} and HfN_{2}, the ZrN_{2} exhibits the lowest moduli and hardness. Next we investigate the mechanical anisotropy of tetragonal TMN_{2} by calculating the orientation dependences of the Young’s modulus *E* and shear modulus *G* which can be determined from the elastic compliance constants *s*_{ij}^{38}. The computational details of elastic moduli-crystal orientation dependences conducted here are presented in the Supporting information section. Figure 5 illustrates the three-dimensional surface representation showing the variation of Young’s modulus with direction for each dinitride. Clearly, all three TMN_{2} dinitrides exhibit a well-pronounced elastic anisotropy due to their three-dimensional pictures show a large deviation from the spherical shape, which qualifies an isotropic medium. From Fig. 5(a–c), the calculated *E*_{max}/*E*_{min} ratio of the Young’s moduli for TiN_{2}, ZrN_{2}, and HfN_{2} is 2.115, 2.543, 2.766, respectively. The *E*_{max}/*E*_{min} ratios for TiN_{2} and ZrN_{2} are much larger than those of *fcc* TiN (1.148) and ZrN (1.539) proposed by Brik *et al*.^{39}, suggesting that the TMN_{2} with a larger elastic anisotropy may impose certain limitations on their possible applications. More specifically, the directional Young’s moduli along tensile axes within (001), (100), and specific planes are plotted in Fig. 6(a–c). For example, the variation of Young’s modulus in the (001) crystal plane for the quadrant of directions [*uvw*] between [100] (*θ* = 0°) and [010] (*θ* = 90°), the TiN_{2}/ZrN_{2}/HfN_{2} exhibits a maximum of *E*_{[110]} = 734/631/725 GPa and a minimum of *E*_{[100]} = *E*_{[010]} = 367/250/262 GPa, respectively. From Fig. 6(a–c), the ordering of Young’s modulus as a function of direction for three TMN_{2} dinitrides is *E*_{[110]} > *E*_{[001]} > *E*_{[111]} > *E*_{[011]} > *E*_{[100]}. Similarly, the orientation dependences of the shear modulus *G* were also conducted for shear on (001), (100), andplanes. From Fig. 6(d), the shear modulus of the TiN_{2} is independent of the shear stress from [100] to [010] directions within (001) basal plane, and the TiN_{2} possesses its minimum value for shear on [110] and its maximum value for shear on (100)[010] (*G*_{(100)[010]} = 343 GPa). The similar cases can be also found for ZrN_{2} and HfN_{2} in Fig. 6(e,f).

To determine the electronic and structural stabilities as well as the ideal strengths of three TMN_{2} compounds, the stress-strain relations upon tension and shear for tetragonal TMN_{2} phase are calculated in some main crystallographic directions through projection of a 12-atom unit cell onto the corresponding crystal axes with one axis parallel to the strain direction for tension deformation, or with one axis parallel to the slip direction and another axis perpendicular to the slip plane for shear deformation. The schematic of tensile/shear deformation and the ideal strengths deduced from the stress-strain curves for three TMN_{2} compounds are shown in Fig. 7. From Fig. 7(a–c), one can see that the calculated tensile strengths show a similar anisotropy for all three compounds. It shows that all three TMN_{2} have strong stress responses in the [110] directions (TiN_{2}: 74.14 GPa, ZrN_{2}: 64.95 GPa, HfN_{2}: 69.59 GPa) that accord well with their largest directional Young’s moduli (see Fig. 6), which measure the resistance against uniaxial tensions. However, the weakest tensile strength along [011] with the peak tensile stresses below 20 GPa for TMN_{2} (TiN_{2}: 19.15 GPa, ZrN_{2}: 14.88 GPa, HfN_{2}: 16.53 GPa) is much lower than those of 40 GPa for PtN_{2}^{19} and 31.1 GPa for TiN^{40} along the [100] directions. The anisotropy ratio of tensile strength (*σ*_{max}:*σ*_{min}) for TiN_{2} (3.87) is smaller than those of ZrN_{2} (4.36) and HfN_{2} (4.21). Meanwhile, the shear strengths upon large strains for three TMN_{2} are presented in Fig. 7(d–f) in order to further examine the shear deformation where plastic deformation proceeds irreversibly on the atomic scale. First, the highest shear strength for TMN_{2} is found under the (100)[010] direction compare well with their largest shear modulus orientation in the (100) principal shear plane shown in Fig. 6(d–f). Second, the values of the ideal shear strength *τ* and shear strain *γ* of the weakest system is TiN_{2}: (, *γ* = 0.166), ZrN_{2}: (, *γ* = 0.188), and HfN_{2}: (, *γ* = 0.177), which is basically lower than that^{41} of TiN: (, *γ* = 0.21), ZrN: (, *γ* = 0.17), and HfN: (, *γ* = 0.15), respectively, showing their lower shear resistance or hardness than these known hard wear-resistant materials. Third, the lowest shear strength of TMN_{2} is lower than the lowest tensile strength. This means the failure mode in tetragonal TMN_{2} phase is dominated by the shear type.

To further illustrate the atomistic deformation mechanism and the origin of the intriguing bond-breaking pattern of such novel materials in engineering applications, take TiN_{2} for example, we further investigate the variations of bond lengths and electronic structures as a function of applied strain along (110)directions. As presented in Fig. 8 where there are two types bond lengths [the Ti-N (2.201 Å) and N-N (1.385 Å) bond length is denoted as *d*1 and *d*2, respectively] in TiN_{2} at equilibrium state. Under increasing shear strains, the N-N lengths denoted as *d*2 remain nearly invariant (*d*2 = 1.385 Å at *γ* = 0 and *d*2 = 1.380 Å at the critical shear strain of *γ* = 0.166). The Ti-N length indicated as *d*1 in TiN_{8} building block is split from one bond distance to eight different bond distances denoted as *ln* (*n* = 1, 2, …) (see the inset in Fig. 8). In Fig. 8, the Ti-N bond lengths indicated as *l*1, *l*4, *l*5, *l*7, and *l*8 decrease in the whole studied shear strain range, on the contrary, the *l*2, *l*3, and *l*6 bonds in TiN_{8} polyhedrons increase conformably at each strain. Especially, the stretched Ti-N bonds denoted as *l*6 increases sharply and breaks at the critical shear strain of *γ* = 0.166, which limits the achievable strengths of TiN_{2}. Such a bond-breaking can also be clearly seen from the selected Electronic Localization Function (ELF)^{42,43} distributions of TiN_{2} on (110) plane before and after shear instability. At equilibrium state (*γ* = 0, see Fig. 9(a)), a certain electron localization can be seen in the region between adjacent N and Ti atoms indicative of ionic bonding, whereas the electron localization located between N-Ti (*l*6 bonds) atoms decreases gradually upon the incremental shear strains [(110)direction] from Fig. 9(c,d). For ELF at strain of *γ* = 0.188 presented in Fig. 9(d), where no electron localized at *l*6 bonds and results in the breaking of this bond. Therefore, the shear-induced structural deformation for tetragonal TiN_{2} can be attributed to the collapse of TiN_{8} polyhedrons by simultaneously breaking of *l*6 bonds, and this is also the case for other two family members of ZrN_{2} and HfN_{2}.

## Conclusions

To conclude, the structural, electronic, and mechanical properties as well as the ideal strengths of the recent synthesized tetragonal TiN_{2} and two family members, yet-to-be-synthesized ZrN_{2} and HfN_{2} have been systematically studied by using first-principles calculations. Phonon dispersion and formation enthalpies calculations suggest that three tetragonal TMN_{2} are all dynamically stable at ambient condition and can be synthesizable at readily attainable pressures. The high-incompressible of TMN_{2} is associated with the strong N-N covalent bonding in N_{2} dumbbells and polar covalent bonding between TM and N atoms in TMN_{8} building blocks. However, as compared with known *fcc* TMN, all these tetragonal TMN_{2} exhibit a much larger elastic anisotropy and substantially lower shear strength, which may impose certain limitations on their possible applications. Detailed analyses of the deformed atomic structures under shear strain reveal that the lattice instability of TMN_{2} is due to the collapse of TMN_{8} polyhedrons by simultaneously breaking of TM-N bonds which limits their achievable strength.

## Methods

All first-principles plane wave calculations were performed using the VASP code^{44} in the framework of density functional theory with the generalized-gradient approximation (GGA) proposed by Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional^{45,46}. The electron-ion interaction was described by the frozen-core all-electron projector augmented wave (PAW) method^{47}, which called for a *d*-electron as valence states. The integration in the Brillouin zone for all transition metals dinitrides was employed using the Monkhorst-Pack scheme^{48} (8 × 8 × 6), an energy cutoff of 600 eV for the plane-wave expansions, and a tetrahedron method with Blöch corrections for energy calculations and Gaussian smearing for the stress calculations. The conjugate gradient method was used for the relaxation of structural parameters. Phonon frequencies were calculated using direct supercell^{49}, which uses the forces obtained by the Hellmann-Feynaman theorem. Chemical bonding analyses were performed by means of the crystal orbital Hamilton population (COHP) method as implemented in the LOBSTER code^{50,51}. The independent elastic constants were determined from evaluation of stress tensor generated small strain and bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio were thus estimated by the Voigt-Reuss-Hill approximation. The stress-strain relationships were calculated by incrementally deforming the model cell in the direction of the applied strain, and simultaneously relaxing the cell basis vectors conjugated to the applied strain, as well as the positions of atoms inside the cell, at each step.

## Additional Information

**How to cite this article**: Zhang, M. *et al*. Electronic bonding analyses and mechanical strengths of incompressible tetragonal transition metal dinitrides TMN_{2} (TM = Ti, Zr, and Hf). *Sci. Rep*. **6**, 36911; doi: 10.1038/srep36911 (2016).

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## Acknowledgements

This work was financially supported by the Natural Science Foundation of China (No. 11204007), Natural Science Basic Research plan in Shaanxi Province of China (Grant Nos 2016JM1016 and 2016JM1026), Education Committee Natural Science Foundation in Shaanxi Province of China (Grant No. 16JK1049), and Baoji University of Arts and Sciences Key Research (Grant No. ZK16068). The authors thank the computing facilities at High Performance Computing Center of Baoji University of Arts and Sciences.

## Author information

## Affiliations

### College of Physics and Optoelectronic Technology, Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, China

- Meiguang Zhang
- & Baobing Zheng

### College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China

- Ke Cheng

### College of Chemistry and Chemical Engineering, Baoji University of Arts and Sciences, Baoji 721013, China

- Haiyan Yan

### School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China

- Qun Wei

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### Contributions

M.G.Z. and K.C. initiated the project; M.G.Z., H.Y.Y. and B.B.Z. performed theoretical calculations; M.G.Z., K.C., H.Y.Y., B.B.Z. and Q.W. analyzed results and wrote the manuscript text.

### Competing interests

The authors declare no competing financial interests.

## Corresponding authors

Correspondence to Meiguang Zhang or Qun Wei.

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