Universal scaling of the critical temperature and the strange-metal scattering rate in unconventional superconductors

Dramatic evolution of properties with minute change in the doping level is a hallmark of the complex chemistry which governs cuprate superconductivity as manifested in the celebrated superconducting domes as well as quantum criticality taking place at precise compositions. The strange metal state, where the resistivity varies linearly with temperature, has emerged as a central feature in the normal state of cuprate superconductors. The ubiquity of this behavior signals an intimate link between the scattering mechanism and superconductivity. However, a clear quantitative picture of the correlation has been lacking. Here, we report observation of quantitative scaling laws between the superconducting transition temperature $T_{\rm c}$ and the scattering rate associated with the strange metal state in electron-doped cuprate $\rm La_{2-x}Ce_xCuO_4$ (LCCO) as a precise function of the doping level. High-resolution characterization of epitaxial composition-spread films, which encompass the entire overdoped range of LCCO has allowed us to systematically map its structural and transport properties with unprecedented accuracy and increment of $\Delta x = 0.0015$. We have uncovered the relations $T_{\rm c}\sim(x_{\rm c}-x)^{0.5}\sim(A_1^\square)^{0.5}$, where $x_c$ is the critical doping where superconductivity disappears on the overdoped side and $A_1^\square$ is the scattering rate of perfect $T$-linear resistivity per CuO$_2$ plane. We argue that the striking similarity of the $T_{\rm c}$ vs $A_1^\square$ relation among cuprates, iron-based and organic superconductors is an indication of a common mechanism of the strange metal behavior and unconventional superconductivity in these systems.