The Persistence of High-Frequency Spin Fluctuations in Overdoped Superconducting La$_{2-x}$Sr$_{x}$CuO$_{4}$ ($x$=0.22)

The wavevector and energy dependence of the spin fluctuations in overdoped superconducting La$_{1.78}$Sr$_{0.22}$CuO$_4$ ($T_c$=26~K) were investigated up to high energy (160~meV) by inelastic neutron scattering. Overdoping was found to suppress the strong magnetic response $\chi^{\prime\prime}(\mathbf{q},\omega)$ present in optimally doped La$_{2-x}$Sr$_{x}$CuO$_4$ which is peaked near 50~meV. The magnetic excitations were less dispersive than in the optimal compound and incommensurate at all energies investigated. Overdoping did not appear to weaken the high-frequency $E\ge 100$~meV response, suggesting that significant antiferromagnetic exchange couplings persist well into the overdoped part of the cuprate phase diagram.