Direct relation between the low-energy spin excitations and superconductivity of overdoped high-$T_c$ superconductors

The dynamic spin susceptibility, ${\chi }^{\mathrm{\prime }\mathrm{\prime }}(\omega )$, has been measured over the energy range of $2\le \omega \le 10$ meV for overdoped La${}_{2-x}$Sr${}_x$CuO${}_4$. Incommensurate (IC) spin excitations are observed at 8 K for all superconducting samples for $0.25\le x\le 0.28$ with ${\chi }^{\mathrm{\prime }\mathrm{\prime }}$ peaking at $\sim 6$ meV. The IC peaks at 6 meV become smaller in intensity with increasing $x$ and, finally, become unobservable for a sample with $x=0.30$ which has no bulk superconductivity. The maximum ${\chi }^{\mathrm{\prime }\mathrm{\prime }}$ decreases linearly with $T_c$(onset) in the overdoped region, implying a direct cooperative relation between the spin fluctuations and the superconductivity.