The Fractional London Equation and The Fractional Pippard Model For Superconductors

TL;DR

This paper introduces a fractional calculus approach to modify the London and Pippard models of superconductors, incorporating non-local interactions and media effects to better describe new superconducting phenomena.

Contribution

It applies fractional derivatives to the London equation and proposes a fractional Pippard model, offering a novel phenomenological framework for superconductors.

Findings

Magnetic field distribution calculated for mesoscopic superconductors.

Fractional parameter alpha can characterize different superconducting behaviors.

Enhanced model includes non-locality and media effects.

Abstract

With the discovery of new superconductors there was a running to find the justifications for the new properties found in these materials. In order to describe these new effects some theories were adapted and some others have been tried. In this work we present an application of the fractional calculus to study the superconductor in the context of London theory. Here we investigated the linear London equation modified by fractional derivatives for non-differentiable functions, instead of integer ones, in a coarse grained scenario. We apply the fractional approach based in the modified Riemann-Liouville sense to improve the model in order to include possible non-local interactions and the media. It is argued that the e ects of non-locality and long memory, intrinsic to the formalism of the fractional calculus, are relevant to achieving a satisfactory phenomenological description. In order to compare the present results with the usual London theory, we calculated the magnetic field distribution for a mesoscopic superconductor system. Also, a fractional Pippard-like model is proposed to take into account the non-locality beside effects of interactions and the media. We propose that parameter alfa of fractionality can be used to create an alternative way to characterize superconductors.