Ginzburg Landau theory of superconductivity at fractal dimensions

TL;DR

This paper evaluates the Ginzburg-Landau theory of superconductivity in fractal dimensions near 2+2ε, showing that fluctuation effects and dimensionality significantly influence high-Tc superconductor properties, and provides a method to estimate effective sample dimensionality.

Contribution

It introduces a calculation of post-Gaussian corrections in fractal dimensions for the Ginzburg-Landau theory, linking theoretical predictions with experimental data on high-Tc superconductors.

Findings

Post-Gaussian correction terms are larger in fractal dimensions than in 3D.

The effective fractal dimension ε=0.21 best fits experimental data for Tl-2223.

Fluctuation contributions and dimensionality are crucial for understanding high-Tc superconductivity.

Abstract

The post Gaussian effective potential in $D=2+2\veps$ dimensions is evaluated for the Ginzburg-Landau theory of superconductivity. Two and three loop integrals for the post Gaussian correction terms in $D=2+2\veps$ dimensions are calculated and $\veps$-expansion for these integrals are constructed. In $D=2+2\veps$ fractal dimensions Ginzburg Landau parameter turned out to be sensitive to $\veps$ and the contribution of the post Gaussian term is larger than that for D=3. Adjusting $\veps$ to the recent experimental data on $\kappa (T)$ for high -$T_c$ cuprate superconductor $Tl_2Ca_2Ba_2Cu_3O_{10} (T\cl-2223)$, we found that $\veps=0.21$ is the best choice for this material. The result clearly shows that, in order to understand high - $T_c$ superconductivity, it is necessary to include the fluctuation contribution as well as the contribution from the dimensionality of the sample. The method gives a theoretical tool to estimate the effective dimensionality of the samples.