First-principles study of effective cluster interactions and enthalpies of formation of substoichiometricVC1−x

We have performed extensive first-principles electronic structure calculations on a series of ordered ${\mathrm{VC}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ compounds using the full-potential linear-muffin-tin orbitals method. The results are fitted to equations of state and volume-dependent effective cluster interactions (ECI's) are determined. A set of eight composition-independent ECI's including pair interactions up to the third-nearest neighbors as well as some multiplet clusters are found to reproduce the total electronic energies of all ordered ${\mathrm{VC}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ compounds within 1 mRy/atom. These ECI's are used to derive the enthalpy of formation at 0 K, ${\mathrm{\ensuremath{\Delta}}}^{0}$${\mathit{H}}^{\mathrm{dis}}$(x), for ${\mathrm{VC}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ with a completely disordered vacancy configuration. ${\mathrm{\ensuremath{\Delta}}}^{0}$${\mathit{H}}^{\mathrm{dis}}$(x) is higher than ${\mathrm{\ensuremath{\Delta}}}^{0}$H[VC] for all vacancy concentrations x\ensuremath{\ne}0. The energy gain due to ordering is approximately 10 mRy/V atom. This energy is enough to yield enthalpies of formation of some ordered structures (e.g., ${\mathrm{V}}_{8}$${\mathrm{C}}_{7}$ and ${\mathrm{V}}_{8}$${\mathrm{C}}_{7}$ and ${\mathrm{V}}_{6}$${\mathrm{C}}_{5}$) that are below the enthalpy of formation for VC. We conclude that the ground state of ${\mathrm{VC}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ at zero temperature occurs for x\ensuremath{\ne}0. Lattice relaxations around vacant sites are not accounted for by our ECI's. These will further decrease the total energy of ${\mathrm{VC}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ and thus stabilize the vacancies. We present a simple band-filling model which qualitatively explains the limited stability range of vacancy formation.