Insensitivity of Tc to the residual resistivity in high-Tc cuprates and the tale of two domes

One of the few undisputed facts about hole-doped high- T c cuprates is that their superconducting gap Δ has d -wave symmetry. According to ‘dirty’ d -wave BCS theory, even structural (non-magnetic) disorder can suppress Δ, the transition temperature T c and the superfluid density ρ s . The degree to which the latter is affected by disorder depends on the nature of the scattering. By contrast, T c is only sensitive to the total elastic scattering rate (as estimated from the residual resistivity ρ 0 ) and should follow the Abrikosov-Gor’kov pair-breaking formula. Here, we report a remarkable robustness of T c in a set of Bi2201 single crystals to large variations in ρ 0 . We also survey an extended body of data, both recent and historical, on the LSCO family which challenge key predictions from dirty d -wave theory. We discuss the possible causes of these discrepancies, and argue that either we do not understand the nature of disorder in cuprates, or that the dirty d -wave scenario is not an appropriate framework. Finally, we present an alternative (non-BCS) scenario that may account for the fact that the superconducting dome in Tl2201 extends beyond that seen in Bi2201 and LSCO and suggest ways to test the validity of such a scenario.