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 $B>G^{\prime }>G>0$ holds for all these materials, where $B=(C_{11}+2C_{12})∕3$, $G^{\prime }=(C_{11}-C_{12})∕2$, 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., $\mathrm{Zr}{\mathrm{C}}_x{\mathrm{N}}_{1-x}$) and di-TM carbides (e.g., ${\mathrm{Hf}}_x{\mathrm{Ta}}_{1-x}\mathrm{C}$) have maximum $G$ at $Z_V\approx 8.3$.

Figure 1

The theoretical (a) lattice constants $a_0$, (b) bulk moduli $B$, and (c) zone-center band gap $\Delta E_{\Gamma }$ for selected transition-metal nitrides and carbides with $B_1$ structure. Filled symbols denote nitrides, and open symbols refer to carbides.

Figure 2

(Color online) The electronic band structures of selected transition-metal nitrides and carbides with $B_1$ structure. The red (dark gray) curves are band structures of NbN and NbC under a finite shear strain $({ϵ}_{12}={ϵ}_{21}=0.1)$.

Figure 3

The theoretical (a) tetragonal shear moduli $G^{\prime }=(C_{11}-C_{12})∕2$, and (b) shear moduli $G=C_{44}$ with respect to number of valence electrons per unit cell $\left(Z_V\right)$. The solid curve in (b) is obtained by quadratic fitting to average shear moduli at each $Z_V$.