How surface roughness and defects reduce the vortex-entry field
In plain words
Small bumps and impurities on the metal can let vortices enter early. The question is how much each kind of imperfection lowers the limit.
Precise statement
For a type-II superconductor with surface topography of rms height $h$ and lateral scale $l$, and with local suppression of the order parameter over a region of size comparable to the coherence length $\xi$, compute the reduction of the vortex penetration field relative to $H_{\mathrm{sh}}$ of a flat surface. An answer is a formula or computed map $H_p(h/\xi, l/\xi, \kappa)$ usable to set surface specifications.
What would settle it
Ginzburg-Landau simulations of realistic measured surfaces whose predicted $H_p$ matches cavity quench fields.
Status in the literature
Ginzburg-Landau simulations of idealized defects and roughness exist (2015 to 2020); predictions from measured cavity surfaces that match quench fields are lacking.