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Using Resonant Inelastic X-ray Scattering, we measure the paramagnondispersion and damping of undoped, antiferromagnetic Ca$_2$CuO$_2$Cl$_2$ aswell as doped, superconducting Na$_{x}$Ca$_{2-x}$CuO$_2$Cl$_2$. Our estimationof the spin-exchange parameter and width of the paramagnon peak at the zoneboundary $X=(0.5,0)$ confirms that no simple relation can be drawn betweenthese parameters and the critical temperature $T_\mathrm{c}$. Consistently withother cuprate compounds, we show that upon doping there is a slight softeningat $(0.25,0)$, but not at the zone boundary $X$. In combination with thesemeasurements we perform calculations of the dynamical spin structure factor ofthe one-band Hubbard model using cluster dynamical mean-field theory. Thecalculations are in excellent agreement with the experiment in the undopedcase, both in terms of energy position and width. While the increase in widthis also captured upon doping, the dynamical spin structure factor shows asizable softening at $X$, which provides insightful information on thelength-scale of the spin fluctuations in doped cuprates.

Paramagnon dispersion and damping in doped Na$_{x}$Ca$_{2-x}$CuO$_2$Cl$_2$