Recently, spin excitations in doped cuprates have been measured using resonant inelastic x-ray scattering. The paramagnon dispersions show the large hardening effect in the electron-doped systems and seemingly doping independence in the hole-doped systems, with the energy scales comparable to that of the antiferromagnetic (AFM) magnons. This anomalous hardening effect and the lack of softening were partially explained by using the strong-coupling t−J model but with a three-site term [Nat. Commun. 5, 3314 (2014)], although the hardening effect is already present even without the latter. By considering the t−t′−t"−J model and using the slave-boson mean-field theory, we obtain, via the spin-spin susceptibility, the spin excitations in qualitative agreement with the experiments. The doping-dependent bandwidth due to the strong correlation physics is the origin of the hardening effect. We also show that dispersions in the AFM regime, different from those in the paramagnetic (PM) regime, hardly vary with dopant density. These excitations are mainly collective in nature instead of particle-hole-like. We further discuss the interplay and different contributions of these two kinds of excitations in the PM phase and show that the dominance of the collective excitation increases with decreasing dopant concentrations
