A Universal Scaling Law for $T_c$ in Unconventional Superconductors

TL;DR

This paper establishes a universal scaling law relating the maximum superconducting transition temperature to microscopic parameters across diverse unconventional superconductors, suggesting a shared underlying mechanism.

Contribution

It introduces a universal quantitative relationship connecting $T_c$, Coulomb interaction, and Cooper pair size, validated across 173 compounds and 13 superconductor families.

Findings

The scaling law holds over four orders of magnitude in $T_c$.

Different superconductor families satisfy the universal relationship.

The law provides a benchmark for exploring high-temperature superconductivity.

Abstract

Understanding the pairing mechanism of unconventional superconductors remains a core challenge in condensed matter physics, particularly the ongoing debate over whether the related effects caused by electron-electron interactions unify various unconventional superconductors (UcSs). To address this challenge, it is necessary to establish a universal quantitative relationship for the superconducting transition temperature ($T_c$), which can be directly obtained from experiments and correlated with microscopic parameters of different material systems. In this work, we establish a relation: $N_{\text{CP}}\cdot k_{B}T_{c}^\star = \alpha\cdot U $, where $\alpha = 1/(16\pi)$ is a universal constant, $k_B$ is the Boltzmann constant, $T_{c}^\star$ is the maximal $T_{c}$, $U$ is the on-site Coulomb interaction, and $N_{\text{CP}}$($\propto(\xi_0/a)^D$) quantifies the spatial extent of Cooper pairs ($\xi_0$) relative to lattice parameter ($a$) in $D$ dimensions. The validity of this scaling relationship is empirically demonstrated, across a four order-of-magnitude $T_c^\star$ range (0.08--133 K), by database from 173 different compounds spanning 13 different UcS families in over 500 experiments. The fact that the unified relationship is satisfied by different materials of different UcS families reveals that they may share superconducting mechanisms. In addition, the scaling relationship indicates the existence of a maximum $T_{c}^\star$ determined by the minimum $N_{\text{CP}}$, providing a benchmark for theoretical and experimental exploration of high-temperature superconductivity.