Disappearance of antiferromagnetic spin excitations in over-doped La$_{2-x}$Sr$_{x}$CuO$_{4}$

Magnetic excitations in the energy range up to 100 meV are studied for over-doped La$_{2-x}$Sr$_{x}$CuO$_{4}$ with $x=0.25$ and 0.30, using time-of-flight neutron spectroscopy. Comparison of spectra integrated over the width of an antiferromagnetic Brillouin zone demonstrates that the magnetic scattering at intermediate energies, $20 \lesssim \omega \lesssim 100$ meV, progressively decreases with over-doping. This strongly suggests that the magnetism is not related to Fermi surface nesting, but rather is associated with a decreasing volume fraction of (probably fluctuating) antiferromagnetic bubbles.