  The interpretation of how superconductivity disappears in cuprates at large
hole doping has been controversial. To address this issue, we present an
experimental study of single-crystal and thin film samples of
La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) with $x\ge0.25$. In particular, measurements of
bulk susceptibility on LSCO crystals with $x=0.25$ indicate an onset of
diamagnetism at $T_{c1}=38.5$ K, with a sharp transition to a phase with full
bulk shielding at $T_{c2}=18$ K, independent of field direction. Strikingly,
the in-plane resistivity only goes to zero at $T_{c2}$. Inelastic neutron
scattering on $x=0.25$ crystals confirms the presence of low-energy
incommensurate magnetic excitations with reduced strength compared to lower
doping levels. The ratio of the spin gap to $T_{c2}$ is anomalously large. Our
results are consistent with a theoretical prediction for strongly overdoped
cuprates by Spivak, Oreto, and Kivelson, in which superconductivity initially
develops within disconnected self-organized grains characterized by a reduced
hole concentration, with bulk superconductivity occurring only after
superconductivity is induced by proximity effect in the surrounding medium of
higher hole concentration. Beyond the superconducting-to-metal transition,
local differential conductance measurements on an LSCO thin film suggest that
regions with pairing correlations survive, but are too dilute to support
superconducting order. Future experiments will be needed to test the degree to
which these results apply to overdoped cuprates in general.
