General Scaling Behavior of Superconductors

The physics of high-T_c superconductors, which has been a major topic in condensed matter physics for more than thirty years, reveals some features of conventional superconductors. We analyze the scaling of the condensation energy \({{E}_{\Delta }}\) divided by \(\gamma \), \({{E}_{\Delta }}{\text{/}}\gamma \simeq N(0)\Delta _{1}^{2}{\text{/}}\gamma \), that equally applicable to both conventional and unconventional high-T_c superconductors. Here, \(N(0)\) is the density of states, \({{\Delta }_{1}}\) is the maximum value of the superconducting gap, and \(\gamma \) is the Sommerfeld coefficient. Basing on this observation, we analyze experimental facts that reveal the general scaling properties of both high-T_c and ordinary superconductors, and theoretically explain that the Homes’ law \({{\rho }_{{s0}}} \propto {{T}_{{\text{c}}}}\sigma ({{T}_{{\text{c}}}})\) is applicable to the both types of superconductors. Here, \(\sigma \) is the conductivity, \(T\) is the temperature, T_c is the superconducting transition temperature, \({{\lambda }_{D}}\) is the zero-T penetration depth, and \({{\rho }_{{s0}}}\) is the superconducting electron density. For the first time, we also explain the reason of violation of the Homes’ law. Our theoretical results agree well with experimental facts.