  The cuprate high-temperature superconductors exhibit many unexplained
electronic phases, but it was often thought that the superconductivity at
sufficiently high doping is governed by conventional mean-field
Bardeen-Cooper-Schrieffer (BCS) theory[1]. However, recent measurements show
that the number of paired electrons (the superfluid density) vanishes when the
transition temperature Tc goes to zero[2], in contradiction to expectation from
BCS theory. The origin of this anomalous vanishing is unknown. Our scanning
tunneling spectroscopy measurements in the overdoped regime of the
(Pb,Bi)2Sr2CuO6+{\delta} high-temperature superconductor show that it is due to
the emergence of puddled superconductivity, featuring nanoscale superconducting
islands in a metallic matrix[3,4]. Our measurements further reveal that this
puddling is driven by gap filling, while the gap itself persists beyond the
breakdown of superconductivity. The important implication is that it is not a
diminishing pairing interaction that causes the breakdown of superconductivity.
Unexpectedly, the measured gap-to-filling correlation also reveals that
pair-breaking by disorder does not play a dominant role and that the mechanism
of superconductivity in overdoped cuprate superconductors is qualitatively
different from conventional mean-field theory.
