A Universal Scaling Law for $T_c$ in Unconventional Superconductors

TL;DR

The paper addresses whether unconventional superconductors across diverse families share a universal energy scale for pairing. It introduces a universal law $N_CP \cdot k_B T_c^* = \alpha \cdot U$ with $N_CP = \gamma \xi_0^D / \Omega_SC$ and $\alpha = 1/(16\pi)$, validated against a database of 173 compounds from 13 UcS families. The data show a near-linear relation in $log$-$log$ space, with slope close to 1 and high $R^2$, implying a common Coulomb-driven pairing mechanism and an upper bound for $T_c^*$ set by the minimal $N_CP$. The framework subsumes Uemura's and Homes' laws and offers practical predictive power for new materials, as well as guidance for lattice engineering to maximize $T_c$.

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 = α\cdot U $, where $α= 1/(16π)$ 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 $N_{\text{CP}}$($\propto(ξ_0/a)^D$) quantifies the spatial extent of Cooper pairs ($ξ_0$) 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.

Figures (3)

• Figure 1: Universal scaling of $T_{c}^\star$ in UcSs. The normalized maximal critical temperature $T_{c}^\star/U$ shows a universal linear dependence on $1/N_{\text{CP}}$ across 160+ UcSs from 13 families involved in over 500 experiments. The solid red circle marks the limiting case at $N_{\text{CP}}=2$, giving $k_{\text{B}}T_{c}^\star/U\approx0.01$. The pink dashed line corresponds to Eq. (\ref{['Eq_Ncp']}).
• Figure 2: Distinctive scaling regimes of UcSs versus BCS superconductors. (a) Superconductors that diverge from the universal scaling relation in Eq. (\ref{['Eq_Ncp']}) are distinctly color-coded. The cyan dashed boundary separates BCS superconductors (green circles) from UcSs. The intermediate parameter space (between cyan demarcation and purple universal line in Fig. \ref{['fig:1']}) hosts transitional UcS candidates. (b) The deviation scaling relationship of A$_{3}$C$_{60}$ under different degrees of alkali metal intercalation is displayed..
• Figure 3: This figure presents the statistical distributions of key parameters for the fitting, including slope ($c_1$), intercept ($c_2$), and coefficient of determination ($R^2$), they are derived from 100,000 Monte Carlo sampling trials across three error propagation scenarios. (a), (d), (g) correspond to the slope $c_1$ under uniform sampling, normal sampling ($\sigma$=1), and normal sampling ($\sigma$=1.96), respectively; (b), (e), (h) show the intercept $c_2$ for the same three scenarios; (c), (f), (i) display the $R^2$ statistic.