# Mathematical modelling of the effect of surface roughness on magnetic field profiles in type II superconductors

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## Abstract

One of the defining characteristics of a superconductor is the Meissner effect, in which an external magnetic field is expelled from the bulk of a sample when cooled below the critical temperature. Although there has been considerable theoretical work on the Ginzburg–Landau theory of superconductors, the effects of interest in this paper can be modelled with the simpler London equation. This equation predicts an exponential decay of the local magnetic field magnitude as a function of the distance into the superconductor from a flat surface in the London limit where \(\kappa =\lambda /\xi \), defined as the ratio between the penetration depth and coherence length, is much greater than 1. However, recent measurements of the field profile in high \(\kappa \) superconductors show that the observed decay is non-exponential near the surface. In particular, the measured field profiles indicate that the decay rate in the field magnitude is smaller than expected from a simple London model on a short length scale \(d\) near the surface. In this paper, we examine the effects of surface roughness on magnetic field penetration into a high \(\kappa \) superconductor. We model the roughness as a sinusoidal perturbation from a flat interface and investigate the effect using both an asymptotic method, based upon a small-amplitude perturbation, and a numerical method, using a finite difference discretization with a coordinate mapping from an underlying rectangular domain. A novel discretization is used in the case of 3D calculations and a fast, preconditioned GMRES solver is developed. A careful comparison of asymptotic and numerical methods validates both approaches for small perturbations, but the numerical approach allows for the investigation of rougher surfaces. Our results show that surface roughness reduces the decay rate in the average magnetic field near the surface relative to a London model. However, the reduction is more gradual than the simple dead layer model currently being used to fit experimental data. In addition, we discover some interesting new phenomena in the 3D case.

### Keywords

Asymptotics Finite differences London equation Superconductor Surface roughness### References

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