<i>Ab initio</i> study of orbital-selective superconductivity in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> -BiPd

We investigate the superconducting (SC) properties of experimentally realized $\ensuremath{\gamma}$-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 $\ensuremath{\gamma}$-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 $\ensuremath{\gamma}$-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 ${\mathrm{\ensuremath{\Delta}}}_{\mathbf{k}}$ and EPC strength ${\ensuremath{\lambda}}_{\mathbf{k}}$ across the FS, calculated superconducting quasiparticle density of states ${N}_{S}$ spectra exhibit a U-shaped gap and ${\mathrm{\ensuremath{\Delta}}}_{\mathbf{k}}$ distribution forms a single peak, being consistent with the spin-singlet $s$-wave superconductivity observed in this material. The calculated ${T}_{c}$ is $\ensuremath{\sim}\phantom{\rule{0.16em}{0ex}}2.0\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, agreeing in order of magnitude with the experimental value of 3.3 K in $\ensuremath{\gamma}$-BiPd thin films. The predicted EPC-enhanced Sommerfeld coefficient ${\ensuremath{\gamma}}_{n}$ of $0.141\phantom{\rule{0.16em}{0ex}}{\mathrm{mJ}/\mathrm{K}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{3}$ is similar to the experimental ${\ensuremath{\gamma}}_{n}$ value $(0.119\phantom{\rule{0.16em}{0ex}}{\mathrm{mJ}/\mathrm{K}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{3})$ of the isoelectronic and isostructural $\mathrm{Bi}({\mathrm{Pd}}_{0.5}\phantom{\rule{0.16em}{0ex}}{\mathrm{Pt}}_{0.5)}$ alloy.