Superconducting properties in Li$_x$ZrNCl from first principles

ZrNCl is a layered band insulator. Upon Li intercalation it undergoes a metal-superconductor transition at 5$\%$ Li content. The superconducting critical temperature displays a very peculiar behaviour as it is maximal (T$_C$ =15.2K) for $\approx 5\%$ intercalation, at the edge of the insulator-superconductor transition. When the Li concentration is increased the critical temperature decreases monotonically, stabilizing at T$_C$ =11.5 K for x=0.3. We present calculations beyond DFT on Li$_x$ZrNCl as a function of doping. We show that an ultradense sampling of the Brillouin zone (both for phonon and electron momentum) is necessary to describe the phonon spectra and the electron-phonon coupling in this material. We show that marked Kohn-anomalies, overlooked in all previous calculations, occur in the phonon spectrum at different energy scales. Finally, we discuss the role of correlation effects in determining the superconducting properties of Li$_x$ZrNCl.