We investigate the superconducting (SC) properties of experimentally realized $\gamma$-BiPd by solving the anisotropic Migdal-Eliashberg equations in conjunction with 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 characterized 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 discernible. While our results show an anisotropic nature of the k-dependent SC gap ${\mathrm{\Delta }}_{\mathbf{k}}$ and EPC strength ${\lambda }_{\mathbf{k}}$ across the FS, calculated superconducting quasiparticle density of states $N_S$ spectra exhibit a U-shaped gap and ${\mathrm{\Delta }}_{\mathbf{k}}$ distribution forms a single peak, being consistent with the spin-singlet $s$-wave superconductivity observed in this material. The calculated $T_c$ is $\sim 2.0\mathrm{K}$, 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 $0.141{\mathrm{mJ}/\mathrm{K}}^2{\mathrm{cm}}^3$ is similar to the experimental ${\gamma }_n$ value $\left(0.119{\mathrm{mJ}/\mathrm{K}}^2{\mathrm{cm}}^3\right)$ of the isoelectronic and isostructural $\mathrm{Bi}({\mathrm{Pd}}_{0.5}{\mathrm{Pt}}_{0.5)}$ alloy.

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• Electron-phonon coupling
• Multiband superconductivity
• Pairing mechanisms
• Spin-orbit coupling
• Superconducting gap
• Superconductivity
• Superconductors
• Ab initio calculations
• Eliashberg theory