Direct Relation between the Low-Energy Spin Excitations and Superconductivity of Overdoped High-TcSuperconductors

The dynamic spin susceptibility, ${\chi }^{\prime \prime }(\omega )$, has been measured over the energy range of $2\le \omega \le 10\mathrm{meV}$ for overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. Incommensurate (IC) spin excitations are observed at 8 K for all superconducting samples for $0.25\le x\le 0.28$ with ${\chi }^{\prime \prime }$ peaking at $\sim 6\mathrm{meV}$. 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 ${\chi }^{\prime \prime }$ decreases linearly with $T_c$(onset) in the overdoped region, implying a direct cooperative relation between the spin fluctuations and the superconductivity.