Quaternary borocarbides: A testbed for DFT for superconductors

Using ab initio density functional theory for superconductors (SCDFT), we systematically study the quaternary borocarbides $R{M}_{2}{\mathrm{B}}_{2}\mathrm{C}$. Treating the retarded (frequency-dependent) interaction $W(\ensuremath{\omega})$ within the random-phase approximation, we find good agreement with experiments for the calculated superconducting critical temperature ${T}_{c}$ in the nonmagnetic Ni- and Pd-based compounds. Replacing the density functional theory (DFT) bands with a one-shot $GW$-derived quasiparticle band structure in the SCDFT calculations further improves the agreement with experiment for several of the tested systems. We argue that the problem of accurately placing the $f$ bands within DFT, and possibly the lack of an explicit magnetic pair-breaking mechanism, explains the difficulties of SCDFT in reproducing ${T}_{c}$ in members of the magnetic $R{\mathrm{Ni}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ series ($R$ rare-earth elements with partially filled $4f$ states). While the calculated ${T}_{c}$ is overestimated, SCDFT qualitatively captures the experimentally observed trend along the rare-earth series, which indicates that the electron-phonon couplings and dynamically screened interactions have a significant effect on ${T}_{c}$.