Strong-coupling anisotropic superconductivity in hexagonal HfRuAs from anisotropic Migdal-Eliashberg theory

We present a comprehensive theoretical investigation of the superconducting (SC) properties of hexagonal HfRuAs ($h$-HfRuAs) by solving anisotropic Migdal–Eliashberg (ME) equations with the inputs from $\mathit{\text{ab}}\mathit{\text{initio}}$ calculations of electronic structure, phonon dispersion and electron phonon coupling matrix elements. The calculated Eliashberg spectral function reveals strong electron–phonon coupling (EPC) with a constant $\lambda \approx 1.56$, dominated by low-frequency phonon modes associated primarily with Hf and Ru vibrations. The SC state is characterized by a single anisotropic gap with overall $s$-wave symmetry, as evidenced by the fully gapped quasiparticle density of states. The momentum-resolved EPC and SC gap exhibit pronounced anisotropy across different Fermi surface sheets, with the largest variations occurring on the hole-like bands. The SC gap is centered around $\mathrm{\Delta }\approx 2.9$ meV with a spread of $\sim 0.8$ meV, indicating significant multiband anisotropy. The resulting gap ratio $2\mathrm{\Delta }(0)\mathrm{/}{k}_B{T}_c\approx 4.2$ exceeds the BCS weak-coupling limit, establishing $h$-HfRuAs as a strong-coupling superconductor. The calculated transition temperature, $T_c$, agrees in the order of magnitude with experiments. Overall, our results identify $h$-HfRuAs as a phonon-mediated, strongly coupled anisotropic superconductor and provide detailed insights into the role of momentum-dependent electron–phonon interactions in determining its SC properties.