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

The elastic properties of the $B_1$-structured transition-metal nitrides and\ntheir carbide counterparts are studied using the {\\it ab initio\\} density\nfunctional perturbation theory. The linear response results of elastic\nconstants are in excellent agreement with those obtained from numerical\nderivative methods, and are also consistent with measured data. We find the\nfollowing trends: (1) Bulk moduli $B$ and tetragonal shear moduli\n$G^{\\prime}=(C_{11}-C_{12})/2$, increase and lattice constants $a_{0}$ decrease\nrightward or downward on the Periodic Table for the metal component or if C is\nreplaced by N; (2) The inequality $B > G^{\\prime} > G > 0$ holds for\n$G=C_{44}$; (3) $G$ depends strongly on the number of valence electrons per\nunit cell ($Z_{V}$). From the fitted curve of $G$ as a function of $Z_{V}$, we\ncan predict that MoN is unstable in $B_{1}$ structure, and transition-metal\ncarbonitrides ($e.g.$ ZrC$_{x}$N$_{1-x}$) and di-transition-metal carbides\n($e.g.$ Hf$_{x}$Ta$_{1-x}$C) have maximum $G$ at $Z_{V} \\approx 8.3$.\n