Phonon-mediated superconductivity in ternary silicides $X_4$CoSi ($X=$ Nb, Ta)

The superconducting properties of two recently synthesized ternary silicides with unit formula $X_4$CoSi ($X=$ Nb, Ta) are investigated through \textit{ab initio} calculations combined with Eliashberg theory. Interestingly, their crystal structure comprises interlocking honeycomb networks of Nb/Ta atoms. Nb$_4$CoSi is found to harbor better conditions for phonon-mediated superconductivity, as it possesses a higher density of states at the Fermi level, fostering stronger electron-phonon coupling. The superconducting critical temperatures ($T_c$) follow the same trend, with Nb$_4$CoSi having a twice higher value than Ta$_4$CoSi. Furthermore, the calculated $T_c$ values (6.0 K vs. 3.2 K) agree excellently with the experimentally obtained ones, establishing superconductivity in this new materials class as mediated by the electron-phonon coupling. Furthermore, our calculations show that the superconducting properties of these compounds do not simply correlate with the parameters of their honeycomb networks, contrary to proposals raised in the literature. Rather, their complete fermiology and phonon spectrum should be taken into account in order to explain their respective superconducting properties.