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

Abstract

Most commonly known hard transition-metal nitrides crystallize in rocksalt structure (B1). The discovery of ultraincompressible pyrite-type ${\mathrm{PtN}}_{2}$ 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 ${\mathrm{PtN}}_{2}$ and other transition-metal pernitrides ($M{\mathrm{N}}_{2}$) with density functional theory. Apart from ${\mathrm{PtN}}_{2}$, the three hardest phases are found among them in the $3d$ transition-metal period. They are ${\mathrm{MnN}}_{2}$, ${\mathrm{CoN}}_{2}$, and ${\mathrm{NiN}}_{2}$, with computed Vickers hardness (${H}_{\mathrm{V}}$) values of 19.9 GPa, 16.5 GPa, and 15.7 GPa, respectively. Harder than all of these is ${\mathrm{PtN}}_{2}$, with a ${H}_{\mathrm{V}}$ 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 $E$, an indicator of metallicity. (c) Often cited monotonic variation of ${H}_{\mathrm{V}}$ 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 $M\ensuremath{-}M$ 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.