Ab initio study of orbital-selective superconductivity in $\gamma$-BiPd

We investigate the superconducting (SC) properties of experimentally realised $\gamma$-BiPd by solving the anisotropic Migdal-Eliashberg equations in conjunction with {\it ab initio} relativistic calculations of the electron and phonon band structures as well as electron-phonon coupling (EPC) matrix elements. Our study reveals that $\gamma$-BiPd possesses a complex Fermi surface (FS), consisting of two electron pockets and one hole pocket, each characterised by distinct atomic orbitals. Our key finding is that the superconductivity in $\gamma$-BiPd is primarily orbital-selective, arising from Bi $p$-orbitals, and distributed anisotropically on the FS, although contribution from Pd $d$-orbitals, particularly on the hole pocket, is also discernable. While our results show an anisotropic nature of the {\bf k}-dependent SC gap $\Delta_{\bf k}$ and EPC strength $\lambda_{\bf k}$ across the FS, calculated superconducting quasiparticle density of states $N_S$ spectra exhibit a U-shaped gap and $\Delta_{\bf k}$ distribution forms a single peak, being consistent with the spin-singlet $s$-wave superconductivity observed in this material. The calculated $T_c$ is $\sim$2.0 K, agreeing in order of magnitude with the experimental value of 3.3 K in $\gamma$-BiPd thin films. The predicted EPC-enhanced Sommerfeld coefficient $\gamma_n$ of $0.141$ mJ/K$^2$cm$^3$ is similar to the experimental $\gamma_n$ value ($0.119$ mJ/K$^2$cm$^3$) of the isoelectronic and isostructural Bi(Pd$_{0.5}$Pt$_{0.5}$) alloy.