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 (H-V) values of 19.9 GPa, 16.5 GPa, and 15.7 GPa, respectively. Harder than all of these is PtN2, with a H-V 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 C-44, an indicator of mechanical stability and hardness is correlated with total density of states at E-F, an indicator of metallicity. (c) Often cited monotonic variation of H-V 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.