Paramagnon dispersion and damping in doped Na_xCa_2-xCuO₂Cl₂

Using resonant inelastic x-ray scattering, we measure the paramagnon dispersion and damping of undoped, antiferromagnetic Formula Presented as well as doped, superconducting Formula Presented. Our estimation of the spin-exchange parameter and width of the paramagnon peak at the zone boundary Formula Presented confirms that no simple relation can be drawn between these parameters and the critical temperature Formula Presented. Consistently with other cuprate compounds, we show that upon doping there is a slight softening at (0.25,0) but not at the zone boundary Formula Presented. In combination with these measurements we perform calculations of the dynamical spin structure factor of the one-band Hubbard model using cluster dynamical mean-field theory. The calculations are in excellent agreement with the experiment in the undoped case, both in terms of energy position and width. While the increase in width is also captured upon doping, the dynamical spin structure factor shows a sizable softening at Formula Presented, which provides insightful information on the length-scale of the spin fluctuations in doped cuprates.