Fundamental constraints on the strength of transition-metal borides: The case of CrB4

Abstract

Recent theoretical calculations predict an ideal shear strength over 50 GPa for CrB${}_{4}$, placing it well above ultrahard ReB${}_{2}$ in terms of strength and thus suggesting possible superhardness of CrB${}_{4}$. This result, however, is contradicted by the latest experimental measurements that produced a relatively low Vickers hardness around 23 GPa, which is about the same as the hardness value of ReB${}_{2}$. To solve this intriguing problem, we have performed a systematic first-principles study that unveils two fundamental constraints that limit the strength of CrB${}_{4}$: (i) a quantum-mechanical effect involving a transition between two-center and three-center bonding among the boron atoms that reduces the rigidity and directionality of the boron bonding and (ii) a mechanistic effect caused by the pressure beneath the indenter that drives a lateral bond and volume expansion that further stretches and weakens the boron bonds in addition to the shear deformation in the CrB${}_{4}$ structure under Vickers indentation hardness tests. These effects lead to considerably reduced strength of CrB${}_{4}$, producing an ideal (i.e., an upper bound) indentation strength of 27 GPa that is consistent with the experimental results. These constraints also explain previous results on the pure shear and indentation strength for ReB${}_{2}$, WB${}_{3}$, and MoB${}_{3}$, limiting their ideal (Vickers) indentation strength below 30 GPa irrespective of the composition and structural details. The present results suggest that transition-metal boron compounds are unlikely to become superhard as previously predicted.