A superconductor, when in the Meissner state, expels magnetic fields from its interior. Very near its surface, there is an exponential decay in field strength that is predicted by the London equation, a special limit of the Ginzburg-Landau equations, provided the surface is flat. In the superconductivity literature, the assumption of a flat interface was taken for granted, but due to experimental measurements of a non-exponential decay in field strength near the surface of a superconductor, experimentalists asked the question of whether small-amplitude perturbations could have an effect on the field profile. With colleagues, I studied these effects through idealized mathematical models of sinusoidal surfaces with the use of asymptotics and numerical methods; then, we further expanded the work by using experimental data measuring the roughness of the surfaces to more accurately describe the magnetic field perturbations.

applied magnetic field and superconductor surface
The superconductor has a surface that is not flat. An external magnetic field is applied within a vacuum. We are interested in the magnetic field within the superconductor.
AFM image of superconductor surface
Measurements of superconductor surface roughness with Atomic Force Microscopy.
magnetic field magnitude vs depth into superconductor
The mathematical model yields a delayed decay rate of the magnetic field but does not produce a true dead layer, i.e. a depth where no decay is present.
distribution of magnetic field magnitudes at various depths
Due to the rough geometry of real superconductors, our model predicts that the magnetic field magnitude follows a distribution of values as the penetration depth increases, rather than a single value at each depth.