Structure-strength relations of distinct MoN phases from first-principles calculations

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I. INTRODUCTION
Transition metal (TM) nitrides constitute a remarkable class of materials, which have drawn considerable interest and attention because of their excellent physical properties, such as low electrical resistivity, high melting point, superconductivity, and high strength and hardness [1][2][3][4][5][6][7][8][9][10]. This class of materials has a wide range of scientific and industrial applications ranging from wear-resistant coatings to cutting tools. The synthesis of bulk and high-quality TM nitrides, however, presents a technical challenge since direct reactions of transition metals with nitrogen gas is unfavorable. In general, it is more difficult to break the strong N#N triple bond and form a stable compound with TM elements compared to that of the corresponding oxides, borides, and carbides. Among various TM nitrides, MoN is one of the most promising hard materials with excellent mechanical and electronic properties [6,10]. Intense experimental and theoretical studies have been performed in the past few years [10][11][12][13][14][15]. MoN has been reported to crystallize in a variety of phases depending on the synthesis conditions and methods. It is commonly accepted that TM mononitrides often crystallize in a cubic rock-salt structure (B1-type). However, the B1-type MoN is thermodynamically unstable and does not exist in the equilibrium phase diagram of the Mo-N binary systems at ambient condition [14]. There are currently at least four known crystal structures of MoN: three different forms of the hexagonal structures (δ 1 -MoN, δ 2 -MoN, and δ 3 -MoN) [15] and one cubic structure (γ -MoN) [16], as shown in Fig. 1.
The δ 1 -MoN belongs to the space group P6m2 configuration [15] with lattice parameters of a = 2.8512 Å and c = 2.7823 Å. It has been assigned to a tungsten-carbidelike structure, which Mo atoms occupy the 1a Wyckoff sites (0.000, 0.000, 0.000) and N atoms on 6c (0.6667, 0.3333, 0.5000). The δ 2 -MoN crystallizes in a NiAs-type structure with lattice constants of a = 5.729 Å, c = 5.604 Å, and a space group of P6 3 /mmc symmetry [15]. It shows a six-fold N coordination for each Mo atom. The hexagonal layers of Mo atoms are stacked along the c axis, and N atoms are orderly arrayed in the ab plane and also stacked along the c axis. This structure contains two Mo atoms occupying the sites at 2a (0.000, 0.000, 0.000) and two N atoms at 2c (0.3333, 0.6667, 0.2500). The δ 3 -MoN adopts the hexagonal structure of P6 3 mc space group symmetry with lattice constants of a = b = 5.754 Å, c = 5.674 Å, which consists of triangular Mo clusters and ordered arrays of N atoms [15]. The unit cell is composed of 16 atoms, with eight Mo atoms take the two inequivalent Wyckoff 2a (0.000, 0.000, 0.2555) and 6c (0.4876, 0.5124, 0.2482) positions, and eight N atoms occupy two inequivalent sites in the Wyckoff 6c (0.1667, 0.8333, 0.9934) and 2b (0.3333, 0.6667, 0.5203) positions, respectively. The γ -MoN, the cubic phase, crystallizes rather differently from those of hexagonal phases, where both Mo and N atoms occupy sites of square-planar coordination with a simple four-connected three-dimensional (3D) net. It exhibits the NbO-type structure, which can be described as an ordered defected rock-salt structure with one-fourth of the Mo and N atoms missing from the corners and center of the cube. The γ -MoN contains three Mo and three N atoms per unit cell with a lattice constant of a = b = c = 4.118 Å. Here the Mo atoms occupy the Wyckoff positions 3d (0.500, 0.000, 0.000) and N atoms occupy the 3c (0.000, 0.500, 0.500) positions.
Recently, Wang et al. [10] synthesized hexagonal δ 3 -MoN and cubic γ -MoN through an ion-exchange reaction at moderate pressures up to 5 GPa. They further reported that δ 3 -MoN and γ -MoN exhibit excellent hardness of about 30 and 23 GPa at the loads of 0.49 N, and superconducting transition temperature Tc of 13.8 and 5.5 K, respectively. These MoN phases are so far the two hardest known superconducting metal nitrides. The underlying mechanisms for the observed superior hardness remain unexplored at a fundamental level. In the present work, we report a systematic study of the pure and indentation shear strength of all the synthesized MoN phases. The calculated results show that the indentation strength of hexagonal δ 3 -MoN is expected to lie between 23.1 and 45.7 GPa, and that for the cubic γ -MoN is 23.0 GPa. These results are in good agreement with the experimentally reported Vickers hardness of about 30 GPa for δ 3 -MoN and 23 GPa for γ -MoN [10]. Our findings establish detailed atomistic mechanisms for bond stiffening or softening and bond breaking modes in various MoN phases, which provide a comprehensive description for some anomalous stress responses and unexpected indentation strength variations that stem from the differences in the crystal structures of MoN phases that have the same chemical stoichiometry. These results offer crucial insights for understand fundamental structure-property relations in TM nitrides and may also provide useful guidance for studying other materials, especially the related borides and carbides.

II. COMPUTATIONAL METHODS AND DETAILS
We performed first-principles stress-strain calculations to obtain the ideal pure shear and indentation strength following a computational approach that has been recently developed and applied to many materials [17][18][19][20][21][22][23][24][25]. The Vienna ab initio simulation package (VASP) code [26] was employed, and the total energy calculations and structural relaxations were carried out using the density functional theory (DFT) within the generalized gradient approximation (GGA) [27]. The electron-ion interaction is described by means of projector augmented wave (PAW) method [28] with the 4p 6 4d 5 5s 1 and 2s 2 2p 3 electrons treated as valence for Mo and N atoms, respectively. A cutoff energy of 600 eV is used for the planewave expansion, together with an adequately fine Monkhorst-Pack k-point sampling [29] in the Brillouin zone. The resulting enthalpy calculations are converged with 1 meV/atom. The quasistatic ideal strength and relaxed loading path in various crystallographic directions are determined using a previously developed method [17][18][19][20][21][22][23][24][25]. The lattice vectors are incrementally deformed in the direction of the applied strain. At each step the atomic basis vectors and all the atoms inside the unit cell are simultaneously relaxed until all residual components of the Hellmann-Feynman stress tensor orthogonal to the applied strain are less than 0.1 GPa [30]. The shape of the unit cell is determined by the full atomic relaxation without any imposed boundary conditions. To explore the mechanical and dynamic stabilities of the four low-energy phases of MoN, we calculated the elastic constants and phonon dispersion relations under ambient condition. The phonon calculations were performed using a supercell approach as implemented in the PHONOPY code [31]. The obtained elastic constants are listed in Table SI [32]. All structures satisfy the Born-Huang stability criteria [33], indicating their mechanical stability. The calculated elastic constants are somewhat sensitive to the exchange-correlation functionals used in the calculations [32]. The calculated phonon dispersions show that there are no imaginary phonon modes in the entire Brillouin zone in all the cases, thus confirming the dynamic stability of these structures.

III. RESULTS AND DISCUSSIONS
The Vickers hardness measurements show that the synthesized single crystal δ 3 -MoN exhibits a Vickers hardness of ∼30 GPa [10]. To assess the experimental results, we calculated stress-strain curves of δ 3 -MoN under different loading conditions. Under compressive strains, it is clear that the weakest peak stresses is 76.9 GPa in the <110 > direction. Under tensile strains, the calculated peak stresses are 55.1, 34.3, and 30.5 GPa in the <001 >, <110 >, and <110 > directions, respectively, which indicate that <110 > is the weakest tensile direction, and thus the (110)  To further elucidate the mechanisms underlying the abnormal stress responses and strain stiffening effects that are rarely seen among TM light-element compounds, we plot the structural snapshots of δ 3 -MoN under pure shear and (Vickers) indentation shear deformation right before and after the bond breaking strains. At the equilibrium strain ε = 0, the Mo atoms are each coordinated with six N atoms constituting a strongly three-dimensional (3D) trigonal prisms, and part of the Mo atoms, those at the center, form two equilateral triangle with the Mo-Mo bond lengths of 2.663 Å (see Fig. 1), which are shorter than the Mo-Mo bonds in Mo metal (2.800 Å) [34]. Structural snapshots show that, as the pure shear strain increases to 0.260, the Mo-N covalent bonds in the trigonal prisms do not change very much. In contrast, one of the Mo-Mo metallic bonds in the equilateral triangle stretches from 2.663 Å in the intact structure to 2.930 Å, which is larger than 2.800 Å in Mo metal and obviously breaks, and then one of the abscised Mo atoms forms a bent Mo-Mo metallic bond with another adjacent Mo atom, as shown in Fig. 2(b). After passing the peak stress, the Mo-Mo bond flips, and one of the Mo-N covalent bonds in the trigonal prism breaks, leading to the sudden decrease of the shear strength to 1.4 GPa. Under indentation shear deformation, the normal compressive pressure beneath the indenter predominantly compresses the neighboring Mo atoms without appreciably stretching the Mo-N covalent bonds. The Mo-Mo bond lengths decrease from 2.663 Å at ε = 0 to 2.371 Å at ε = 0.190. The rigid linear Mo-Mo metallic bonds, combined with the Mo-N covalent bonds in the trigonal prisms, create a strongly 3D network that is the main load-bearing component and resists large shear deformation under the Vickers indentation loads, which generates a significant enhancement of the indentation strength by more than 54%. As the strain increases to 0.195, the 3D network breaks, as shown in Fig. 2(c), which suddenly releases the indentation stress to 3.7 GPa. Similar phenomena are also seen in δ 3 -MoN along the (110)[001] direction under indentation shear deformation. However, the enhancement of the indentation strengths is just 6.5% largely because only part of the linear Mo-Mo bonds are involved in the deformation process, while other bonds are S-shape bent (see Fig. 3).
It is worth noting that the strengths of δ 3 -MoN exhibit a large degree of anisotropy along different deformation directions; for example, there is a significant (more than 31.  Our study shows that all the MoN phases in hexagonal structures, i.e., δ 1 -MoN, δ 2 -MoN, and δ 3 -MoN, exhibit significant strain stiffening that produces considerably enhanced indentation strengths, especially in the (110)[110] direction where the indentation strength exceeds 40 GPa, which is the threshold value commonly set for superhard materials. This remarkable hardness enhancement is attributed to the presence of the rigid linear Mo-Mo metallic bonds, which combine with the Mo-N covalent bonds in the trigonal prisms in the hexagonal structure to create a strongly 3D network that effectively resists large shear deformation under Vickers indentation strains. In contrast, the cubic γ -MoN shows a strain softening under all the indentation shear deformations with a reduction in shear strength at large strains. In particular, the ideal indentation strengths of MoN in the (110)[001] direction is reduced by more than 56%. This contrasting behavior is attributed to that the cubic γ -MoN has large vacancies at the corners and in the center of the cubic cell, which reduces the ability of γ -MoN to resist large shear deformation under Vickers indentation strains. Meanwhile, we also find strain softening phenomena in the hexagonal structures of MoN under various shear directions. Such strength reductions stem from the anisotropic nature of the linear Mo-Mo metallic bonds that are unable to resist large strains along certain weak directions despite their ability to do much better in the strong directions. Furthermore, it is also interesting to note that the ideal indentation strengths of δ 2 -MoN and δ 3 -MoN are visibly larger than that of δ 1 -MoN. Specifically, the strengths are 34.9 GPa at ε = 0.125 for δ 2  The calculated peak values by LDA functional are, about 5.0 GPa, higher than those of GGA and GGA + U functionals, however, the trend of stress-strain curves are the same. The detailed results are shown in the Supplemental Material [32].

IV. CONCLUSION
We performed systematic first-principles calculations to examine the stress responses, especially under pure and indentation shear strains that are most relevant to the experimental hardness measurements, for four reportedly synthesized crystal phases of MoN along various deformation modes. The calculated results show that crystal structural symmetry and the associated bonding configurations play an important role in determining key mechanical properties of crystalline solids, including those with the same chemical stoichiometry as in the present case. As a result, the three hexagonal structures of MoN share some common trends and similar stressstrain relations. These include the strong anisotropy in their peak stresses along various crystallographic directions and abnormal strain-stiffening effect in the (110) cleavage plane under Vickers indentation deformations. The strain stiffening behavior is attributed to the rigid linear Mo-Mo metallic bonds between the adjacent MoN 6 octahedra, which is combined with the Mo-N covalent bonds to create a strongly 3D bonding network that is capable of resisting large shear deformations under Vickers indentation strains. In contrast, the cubic structure of γ -MoN shows nearly isotropic stress response and a drastically contrasting strain softening phenomenon along all crystallographic directions. In this later case, the normal pressure beneath the indenter drives the face-centered N atoms to sink into the vacancies of the γ -MoN crystal lattice, which further weakens the Mo-N bonds. This effect leads to a considerably reduced strength of cubic γ -MoN, producing an ideal indentation strength of 23 GPa, which is in good agreement with the measured Vickers hardness, while the measured value of 30 GPa for hexagonal δ 3 -MoN is considerably (30%) higher. These insights expand the fundamental understanding of atomistic mechanisms underlying the experimentally measured hardness, which is directly related to the indentation strength [17][18][19][20][21][22][23][24][25], and the associated structural deformation of various MoN phases. The present findings also provide useful guidelines for further exploring novel TM nitrides, borides, and carbides, especially distinguishing and identifying among many distinct crystal structures with rich bonding configurations those that are most likely to possess superior mechanical properties under diverse loading conditions.