Electronic and bonding analysis of hardness in pyrite-type transition-metal pernitrides

Most commonly known hard transition-metal nitrides crystallize in rocksalt structure (B1). The discovery of ultraincompressible pyrite-type PtN₂ 10 years ago has raised a question about the cause of its exceptional mechanical properties. We answer this question by a systematic computational analysis of the pyrite-type PtN₂ and other transition-metal pernitrides (MN₂) with density functional theory. Apart from PtN₂, the three hardest phases are found among them in the $3d$ transition-metal period. They are MnN₂, CoN₂, and NiN₂, with computed Vickers hardness (HV) values of 19.9 GPa, 16.5 GPa, and 15.7 GPa, respectively. Harder than all of these is PtN₂, with a HV of 23.5 GPa. We found the following trends and correlations that explain the origin of hardness in these pernitrides. (a) Charge transfer from M to N controls the length of the N-N bond, resulting in a correlation with bulk modulus, dominantly by providing Coulomb repulsion between the pairing N atoms. (b) Elastic constant C₄₄, an indicator of mechanical stability and hardness is correlated with total density of states at E, an indicator of metallicity. (c) Often cited monotonic variation of HV and Pugh's ratio with valence electron concentration found in rocksalt-type early transition-metal nitrides is not evident in this structure. (d) The change in M-M bond strength under a shearing strain indicated by crystal orbital Hamilton population is predictive of hardness. This is a direct connection between a specific bond and shear related mechanical properties. This panoptic view involving ionicity, metallicity, and covalency is essential to obtain a clear microscopic understanding of hardness.