Some new aspects of fractal superconductivity

Abstract

The purpose of this paper is to develop, in fractal dimension, the semi-quantitative extension of the Ginzburg-Landau theory of superconductivity. Our theoretical analysis is based on the notion of the non-standard Lagrangian approach which, based on recent studies, offer new insights in the theory of nonlinear differential equations and quantum mechanics. Besides, the fractal dimensional approach was based on the concept of the product-like fractal measure introduced by Li and Ostoja-Starzewski in order to study anisotropic media. The extended Ginzburg-Landau equation in fractal dimension was derived and several features have been revealed in particular for fractal dimension α=1/2. In particular, the theory is characterized by discrete effective mass and effective coherent length which is independent of the temperature and it don't diverge at critical temperature. This outcome is motivating since it allows manufacturing temperature-independent superconductors with a high current density without the need of refrigeration or explicit substances. The effective discrete coherent length was found to decrease with the energy levels and such a shrinking is detected in several superconductors including YBCO, Zr-doped (GdY) BCO tapes and SQUIDs. We have also studied the Abrikosov's anisotropic vortex lattice. It was found that the extended fractal theory is deformed due to the effective potential of theory which holds an inverse square part and the associated Schrödinger equation is analogous to the biconfluent Heun's differential equation. We have obtained the solution in terms of the Hermite polynomials and the energy equation was found to be altered considerably. However, it was observed that a transition between a type-II and a type-I superconductors takes place if only the non-standard parameter is ξ0≈0.99orξ0≈0.13where both numerical values result on a Ginzburg-Landau parameter κ≈1/2 although the theory is characterized by a deformed periodic lattice structure due to its anisotropic feature and non-standard Lagrangian structure. Further points are discussed and analyzed.