Vacancy-induced mechanical stabilization of cubic tungsten nitride

Abstract

First-principles methods are employed to determine the structural, mechanical, and thermodynamic reasons for the experimentally reported cubic WN phase. The defect-free rocksalt phase is both mechanically and thermodynamically unstable, with a negative single crystal shear modulus ${C}_{44}=\phantom{\rule{0.16em}{0ex}}\ensuremath{-}86\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ and a positive enthalpy of formation per formula unit ${H}_{f}=\phantom{\rule{0.16em}{0ex}}0.623\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ with respect to molecular nitrogen and metallic W. In contrast, WN in the NbO phase is stable, with ${C}_{44}=\phantom{\rule{0.16em}{0ex}}175\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ and ${H}_{f}=\phantom{\rule{0.16em}{0ex}}\ensuremath{-}0.839\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$. A charge distribution analysis reveals that the application of shear strain along [100] in rocksalt WN results in an increased overlap of the ${t}_{2g}$ orbitals which causes electron migration from the expanded to the shortened W-W $\ensuremath{\langle}110\ensuremath{\rangle}$ bond axes, yielding a negative shear modulus due to an energy reduction associated with new bonding states 8.1--8.7 eV below the Fermi level. A corresponding shear strain in WN in the NbO phase results in an energy increase and a positive shear modulus. The mechanical stability transition from the NaCl to the NbO phase is explored using supercell calculations of the NaCl structure containing ${C}_{v}=\phantom{\rule{0.16em}{0ex}}0%\ensuremath{-}25%$ cation and anion vacancies, while keeping the N-to-W ratio constant at unity. The structure is mechanically unstable for ${C}_{v}l\phantom{\rule{0.16em}{0ex}}5%$. At this critical vacancy concentration, the isotropic elastic modulus $E$ of cubic WN is zero, but increases steeply to $E=445\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ for ${C}_{v}=\phantom{\rule{0.16em}{0ex}}10%$, and then less steeply to $E\phantom{\rule{0.16em}{0ex}}=561\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ for ${C}_{v}=\phantom{\rule{0.16em}{0ex}}25%$. Correspondingly, the hardness estimated using Tian's model increases from 0 to 15 to 26 GPa as ${C}_{v}$ increases from 5% to 10% to 25%, indicating that a relatively small vacancy concentration stabilizes the cubic WN phase and that the large variations in reported mechanical properties of WN can be attributed to relatively small changes in ${C}_{v}$.