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We present an ab initio analysis with density functional theory forsuperconductors (SCDFT) to understand the superconducting mechanism of dopedlayered nitrides \beta-Li$_x$MNCl (M=Ti, Zr, and Hf). The current version ofSCDFT is based on the Migdal-Eliashberg theory and has been shown to reproduceaccurately experimental superconducting-transition temperatures Tc of a widerange of phonon-mediated superconductors. In the present case, however, ourcalculated Tc$\leq$4.3 K (M=Zr) and $\leq$10.5 K (M=Hf) are found to be lessthan a half of the experimental Tc. In addition, Tc obtained in the presentcalculation increases with the doping concentration x, opposite to thatobserved in the experiment. Our results indicate that we need to consider someelements missing in the present SCDFT based on the Migdal-Eliashberg theory.

High-temperature Superconductivity in Layered Nitrides \beta-Li$_x$MNCl (M = Ti, Zr, Hf): Insights from Density-functional Theory for Superconductors