Universal correlations of the superconducting transition temperature, the superfluid density, the linear-in-T coefficient, and the slope of resistivity–temperature dependence with the fractal dimensions of electron structures

Superconductivity transition temperature ( T c ) has been found to strongly correlate with the superfluid density or the Cooper pair density at T = 0 , ρ s 0 , and the linear-in-T coefficient, A 1 , defined from the empirical experimental equation ρ = ρ 0 + A 1 T with ρ as the resistivity, and the slope of resistivity–temperature dependence, d ρ / d T . These correlations in the literature are sometimes material-specific, and cannot explain why they work and under what conditions they would be violated. Here we provide a theoretical treatment of these parameters based on the T c equation derived previously for any superconductors. Our approach correlates T c with ρ s 0 , A 1 , and d ρ / d T , with the fractal dimensions of electron structures. Related empirical laws available in the literature including the universal Uemura’s law, Homes’ law, and the linearity between d ρ / d T and λ m 2 are derived, where λ m is the magnetic penetration depth. Our findings provide theoretical supports for these scaling rules empirically discovered among various superconductors, deepening our understanding of superconductivity mechanisms. • Several famous scaling laws, such as the Uemura’s law, Homes’ law, the linear-in-T coefficient A1, and the linearity between d ρ /dT and λ m 2 , etc. are theoretically derived based on the superconductivity transition temperature Tc equation. • The application scopes of these laws are theoretically obtained and the conditions that they would be violated are presented. • The fractal dimensions of electron structures are introduced into theoretical derivations. Related empirical laws available in the literature are special cases of these generic correlations. • All these scaling correlations observed experimentally are adequately explained with our approach.