  We have performed neutron inelastic scattering and resonant inelastic X-ray
scattering (RIXS) at the Cu-$L_3$ edge to study high-energy magnetic
excitations at energy transfers of more than 100 meV for overdoped
La$_{2-x}$Sr$_{x}$CuO$_{4}$ with $x=0.25$ ($T_c=15$ K) and $x=0.30$
(non-superconducting) using identical single crystal samples for the two
techniques. From constant-energy slices of neutron scattering cross-sections,
we have identified magnetic excitations up to ~250 meV for $x=0.25$. Although
the width in the momentum direction is large, the peak positions along the (pi,
pi) direction agree with the dispersion relation of the spin-wave in the
non-doped La$_{2}$CuO$_{4}$ (LCO), which is consistent with the previous RIXS
results of cuprate superconductors. Using RIXS at the Cu-$L_3$ edge, we have
measured the dispersion relations of the so-called paramagnon mode along both
(pi, pi) and (pi, 0) directions. Although in both directions the neutron and
RIXS data connect with each other and the paramagnon along (pi, 0) agrees well
with the LCO spin-wave dispersion, the paramagnon in the (pi, pi) direction
probed by RIXS appears to be less dispersive and the excitation energy is lower
than the spin-wave of LCO near (pi/2, pi/2). Thus, our results indicate
consistency between neutron inelastic scattering and RIXS, and elucidate the
entire magnetic excitation in the (pi, pi) direction by the complementary use
of two probes. The polarization dependence of the RIXS profiles indicates that
appreciable charge excitations exist in the same energy range of magnetic
excitations, reflecting the itinerant character of the overdoped sample. A
possible anisotropy in the charge excitation intensity might explain the
apparent differences in the paramagnon dispersion in the (pi, pi) direction as
detected by the X-ray scattering.
