Trends in elasticity and electronic structure of transition-metal nitrides and carbides from first principles

The elastic properties of selected transition-metal (TM) nitrides and carbides in ${B}_{1}$ structure are studied using the ab initio density-functional perturbation theory. We find that (1) the inequality $B>{G}^{\ensuremath{'}}>G>0$ holds for all these materials, where $B=({C}_{11}+2{C}_{12})∕3$, ${G}^{\ensuremath{'}}=({C}_{11}\ensuremath{-}{C}_{12})∕2$, and $G={C}_{44}$ with ${C}_{ij}$ the elastic constants, and (2) $G$ has large values when the number of electrons per unit cell ${Z}_{V}=8$ or 9. The fitted curve of $G$ vs. ${Z}_{V}$ predicts that rocksalt MoN is unstable, and TM carbonitrides (e.g., $\mathrm{Zr}{\mathrm{C}}_{x}{\mathrm{N}}_{1\ensuremath{-}x}$) and di-TM carbides (e.g., ${\mathrm{Hf}}_{x}{\mathrm{Ta}}_{1\ensuremath{-}x}\mathrm{C}$) have maximum $G$ at ${Z}_{V}\ensuremath{\approx}8.3$.