Superconductivity near lattice instability: the case of NbC_(1-x)N_x and NbN

Using density functional theory within the local density approximation we report a study of the electron-phonon coupling in NbC$_{1-x}$N$_x$ and NbN crystals in the rocksalt structure. The Fermi surface of these systems exhibits important nesting. The associated Kohn anomaly greatly increases the electron-phonon coupling and induces a structural instability when the electronic density of states reaches a critical value. Our results reproduce the observed rise in $T_{c}$ from 11.2 K to 17.3 K as the nitrogen doping is increased in NbC$_{1-x}$N$_x$. To further understand the contribution of the structural instability to the rise of the superconducting temperature, we develop a model for the Eliashberg spectral function in which the effect of the unstable phonons is set apart. We show that this model together with the McMillan formula can reproduce the increase of $T_{c}$ near the structural phase transition.