Understanding and modeling technological conventional superconductors using ab initio quantum-mechanical methods

This thesis provides the first fully \textit{ab initio} description of the two workhorse superconductors in large-scale applications, NbTi and Nb$_3$Sn. Despite their technological relevance, these two materials had never been comprehensively described by first principles methods. In fact, in these cases, the underlying assumption of calculations based on the Migdal-Eliashberg theory -- dynamically stable structures -- breaks down in the standard harmonic approximation. In this thesis, we address this issue by including anharmonic effects in the lattice dynamics through the Stochastic Self-Consistent Harmonic Approximation (SSCHA). This is achieved by implementing a novel approach that integrates the existing code with Machine Learning Interatomic Potentials (MLIP), thereby overcoming the computational workload of SSCHA, which would have made the calculations infeasible. The methodology was first tested on NbTi due to its apparent simplicity. We demonstrate that the dynamical instabilities predicted at the harmonic level are eliminated if lattice dynamics is treated in an anharmonic framework, enabling the first \textit{ab initio} study of the superconducting properties of this material, over 60 years after its discovery. However, including anharmonic effects alone is not sufficient to reconcile predictions with experiments: we show that one must include additional, often-disregarded effects -- such as finite-bandwidth corrections, energy-dependent Coulomb interactions, and lattice disorder -- to obtain an accurate prediction of the superconducting $T_c$. Our idea of thermal averaging over supercells to model disorder is one of the first attempts to include this effect in electron-phonon calculations. After validation on NbTi, the SSCHA-MLIP workflow is applied to Nb$_3$Sn. This material is currently under optimization for adoption in medical devices and large international projects, such as the ITER fusion reactor and upgrades to the Large Hadron Collider at CERN. To this end, we offer a reliable theoretical foundation for understanding and improving its performance. This is achieved by addressing several questions that are crucial to material optimization and have been debated for over 70 years. In particular, we provide a microscopic explanation for the reduction in the upper critical field $\mathrm{B}_{c2}$ observed in samples that undergo the distinctive martensitic transition of A15 superconductors. The explanation stems from a simple, yet original, way of including Fermi surface anisotropy and strong-coupling effects in the evaluation of $\mathrm{B}_{c2}$. Our treatment of the Fermi surface anisotropy provides a clear picture of how different doping strategies affect superconducting performance, suggesting practical strategies for optimization. As in the case of NbTi, an accurate description of Nb$_3$Sn has required overcoming several limitations of standard \textit{ab initio} methods. Effects such as anharmonicity, energy-dependent Coulomb interactions, anisotropy, and disorder appear to be general features of technologically relevant superconductors. This thesis contributes to the community's efforts to extend the capabilities of first principles methods to describe real-world materials and to guide their optimization.