Universal Planckian relaxation in the strange metal state of the cuprates

A major puzzle in understanding high-$T_{\mathrm{c}}$ superconductivity is the microscopic origin of the linear-in-temperature ($T$-linear) resistivity in the strange metal state, which persists up to very high temperatures. Implicit to existing theoretical discussions of this universal `{Planckian}’ relaxation rate is the assumption that it must also be independent of doping, $p$. Applied to the cuprates, however, this apparently contradicts the observed strong doping-dependence ($\propto 1\mathrm{/}p$) of the slope of the $T$-linear resistivity over a wide doping range. Here, we show through a combination of measurements, including optical conductivity and entropy, that the plasma frequency squared ${\omega }_p^2$ scales as $p$ over a similar doping range. Together, these dependences provide compelling evidence that the relaxation rate $1\mathrm{/}\tau$ is indeed doping-independent (i.e. universal) throughout the entire strange metal state. Furthermore, we argue that the entire doping dependence of ${\omega }_p^2$ can be understood to arise from an effective mass enhancement of a specific form proposed by Anderson [\emph{Science} $\mathbf{\text{235}}$, 1196 (1987)] in the context of doped Mott insulators. Such a mass enhancement originates from strong short-range repulsive interactions, while the Planckian relaxation rate itself appears to be an emergent phenomenon of independent physical origin.