Finding the stable structures of W$_x$N$_1$$_-$$_x$ with an $ab$$-$$initio$ high-throughput approach

Using density functional theory calculations, many researchers have predicted that various tungsten-nitride compounds WN$_x$ ($x$ > 1) will be "ultra-compressible" or "superhard", $i.e.$ as hard as or harder than diamond. These compounds are predicted to have large bulk and shear moduli (> 200 GPa) and to be elastically and vibrationally stable. Compounds with such desirable properties must be energetically stable against decomposition into other compounds. This stability can only be found after the determination of the convex hull for W$_x$N$_1$$_-$$_x$ lines which connect the lowest enthalpy structures as a function of composition. The phase diagram of the W-N structure is uncertain, both experimentally and computationally. Complex van der Waals forces play a significant role in determining the structure of solid N$_2$. Here we use high-throughput calculations to map out the convex hull and other low energy structures for the W-N system. We find that the ground state of the system is the NbO structure, and that the WN$_2$ structures found by Wang $et$ $al$. are also stable when the van der Waals forces are neglected. Other proposed structures are above the convex hull of the W-N system. We show how the choice of density functional influences the shape of the curve and the structures that form the hull. In nitrogen-rich compounds, the choice of functional can dramatically change the structural parameters and mechanical behavior. Using any of the functionals, the bulk and shear moduli of the NbO phase are comparable to the WN$_x$ compounds that have been claimed to be ultra-compressible for superhard.