$\textit{Ab Initio}$ Theory of the Impact from Grain Boundaries and Substitutional Defects on Superconducting Nb$_3$Sn

Grain boundaries play a critical role in applications of superconducting Nb$_3$Sn: in dc applications, grain boundaries preserve the material's intrinsically high critical-current by pinning flux, while in ac applications grain boundaries can provide weak points for flux entry and lead to significant dissipation. We present the first $\textit{ab initio}$ study to investigate the physics of different boundary types in Nb$_3$Sn using density functional theory. We identify an energetically favorable selection of tilt and twist grain boundaries of distinct orientations. We find that clean grain boundaries free of point defects reduce the Fermi-level density of states by a factor of two, an effect that decays back to the bulk electronic structure $\sim1-1.5$ nm from the boundary. We further calculate the binding free-energies of tin substitutional defects to multiple boundaries, finding a strong electronic interaction that extends to a distance comparable to that of the reduction of density of states. Associated with this interaction, we discover a universal trend in defect electronic entropies near a boundary. We probe the effects of defect segregation on grain boundary electronic structure and calculate the impact of substitutional impurities on the Fermi-level density of states in the vicinity of a grain boundary, finding that titanium and tantalum have little impact regardless of placement, whereas tin, copper, and niobium defects each have a significant impact but only on sites away from the boundary core. Finally, we consider how all of these effects impact the local superconducting transition temperature $T_\textrm{c}$ as a function of distance from the boundary plane.