A Universal Scaling Law for $T_c$ in Unconventional Superconductors

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.