Electronic structure, electron-phonon coupling and superconductivity in noncentrosymmetric ThCoC2 from ab initio calculations

Superconductors without inversion symmetry in their crystal structure are known to exhibit unconventional properties. Recently, based on the measured temperature dependence of the magnetic field penetration depth, superconductivity in noncentrosymmetric $\mathrm{ThCoC_2}$ was proposed to be a nodal $d$-wave and mediated by the spin fluctuations. Moreover, a non-BCS behavior of the temperature dependence of the electronic specific heat and the magnetic upper critical field were reported. In this work, the electronic structure, phonons and electron-phonon coupling are studied in $\mathrm{ThCoC_2}$ on the basis of \textit{ab initio} computations. The effect of the spin-orbit coupling on the electronic structure and electron-phonon interaction is analyzed, and a large splitting of the electronic band structure is found. The calculated electron-phonon coupling constant $\lambda = 0.59$ remains in decent agreement with the experimental estimates, suggesting that the electron-phonon interaction is strong enough to explain superconductivity with $T_c \simeq 2.5$~K. Nevertheless, we show that the conventional isotropic Eliashberg formalism is unable to describe the thermodynamic properties of the superconducting state, as calculated temperature dependence of the electronic specific heat and magnetic penetration depth deviate from experiments, which is likely driven by the strong spin-orbit coupling and inversion symmetry breaking. In addition, to shed more light on the pairing mechanism, we propose to measure the carbon isotope effect, as our calculations based on the electron-phonon coupling predict the observation of the isotope effect with an exponent $\alpha \simeq 0.15$.