Universal $T$-linear resistivity and Planckian limit in overdoped cuprates

The perfectly linear temperature dependence of the electrical resistivity observed as $T \rightarrow$ 0 in a variety of metals close to a quantum critical point is a major puzzle of condensed matter physics . Here we show that $T$-linear resistivity as $T \rightarrow$ 0 is a generic property of cuprates, associated with a universal scattering rate. We measured the low-temperature resistivity of the bi-layer cuprate Bi2212 and found that it exhibits a $T$-linear dependence with the same slope as in the single-layer cuprates Bi2201, Nd-LSCO and LSCO, despite their very different Fermi surfaces and structural, superconducting and magnetic properties. We then show that the $T$-linear coefficient (per CuO$_2$ plane), $A_1$, is given by the universal relation $A_1 T_F = h / 2e^2$, where $e$ is the electron charge, $h$ is the Planck constant and $T_F$ is the Fermi temperature. This relation, obtained by assuming that the scattering rate 1 / $\tau$ of charge carriers reaches the Planckian limit whereby $\hbar / \tau = k_B T$, works not only for hole-doped cuprates but also for electron-doped cuprates despite the different nature of their quantum critical point and strength of their electron correlations.