An $ab\;initio$ answer to long-debated questions about superconducting Nb$_3$Sn

We present the first fully $ab\;initio$ microscopic description of cubic and tetragonal Nb$_3$Sn. We compute the anharmonic free energy surface, phonon spectra, and solve the full-bandwidth anisotropic Migdal-Eliashberg equations for the superconducting gap of the two phases. Our results show that anharmonic effects are crucial to stabilize both the cubic and tetragonal structures, yielding phonon spectra in excellent agreement with neutron scattering data. We find that the martensitic transition is weakly first-order and that the superconducting gap is strongly anisotropic yet fully-open, with contributions from both longitudinal and transverse Nb $d$-orbitals, revealing an unexpected three-dimensional pairing mechanism. We also find that the experimentally observed reduction of the upper critical field $H_{c2}$ across the transition is explained by a combination of overall weaker electron-phonon coupling and a redistribution of Fermi velocities, which shifts parts of the Fermi surface to longer coherence lengths and limits $H_{c2}$. Based on these insights, we propose that Sn-site doping could enhance transverse-state coupling and gap isotropy, potentially improving both $T_c$ and $H_{c2}$, while Nb-site doping reinforce $H_{c2}$ at the cost of lowering $T_c$.