Electronic bonding analyses and mechanical strengths of incompressible tetragonal transition metal dinitrides TMN2 (TM = Ti, Zr, and Hf)

Motivated by recent successful synthesis of transition metal dinitride TiN2, the electronic structure and mechanical properties of the discovered TiN2 and other two family members (ZrN2 and HfN2) 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 TMN2 (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 TMN8 units. However, as compared with traditional fcc transition metal mononitride (TMN), the TMN2 possess a larger elastic anisotropy may impose certain limitations on possible applications. Further mechanical strength calculations show that tetragonal TMN2 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 TMN2 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 TMN2 at the atomic level during shear deformation can be attributed to the collapse of TMN8 units with breaking of TM-N bonds through the bonding evolution and electronic localization analyses.

, 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 1 College of Physics and Optoelectronic Technology, Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, China. 2 College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China. 3 College of Chemistry and Chemical Engineering, Baoji University of Arts and Sciences, Baoji 721013, China. 4  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 4a 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  (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 1π 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  , 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 − 2C 13 ) > 0, and 2(C 11 + C 12 ) + C 33 + 4C 13 > 0], indicating their mechanically stable at ambient conditions. From Table 2 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 ,   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 (110) specific planes are plotted in Fig. 6(a-c).  Fig. 6(e,f).

Compounds
Source 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 (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  , γ = 0.177), which is basically lower than that 41  , 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)[111]directions. As presented in Fig. 8    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 l1, l4, l5, l7, and l8 decrease in the whole studied shear strain range, on the contrary, the l2, l3, and l6 bonds in TiN 8 polyhedrons increase conformably at each strain. Especially, the stretched Ti-N bonds denoted as l6 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 (l6 bonds) atoms decreases gradually upon the incremental shear strains [(110)[111]direction] from Fig. 9(c,d). For ELF at strain of γ = 0.188 presented in Fig. 9(d), where no electron localized at l6 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 l6 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.