Spin Fluctuations in the Rare-Earth Doped Bilayer Nickelates

Spin fluctuations have been generally believed as the pairing glue of high-$T_c$ superconductivity. Recent inelastic neutron scattering (INS) studies have revealed a weak flat spin-fluctuation signal around 45 meV in the bilayer nickelate La$_3$Ni$_2$O$_{7-\delta}$, suggesting strong interlayer and weak intralayer magnetic couplings ($SJ_{\perp}\approx$ 60 meV, $SJ_{\parallel}\leq$ 3.5 meV) in contrast to cuprate and pnictide superconductors. Here, we report further INS studies on the Pr and Nd doped La$_3$Ni$_2$O$_{7-\delta}$ powder samples at ambient pressure. Besides the crystalline electric field excitations at low energies, we have found that the 45 meV flat mode splits into two modes in doped compounds, along with another weak mode at about 60 meV, where the spin fluctuations in La$_2$NdNi$_2$O$_{7-\delta}$ are stronger than La$_3$Ni$_2$O$_{7-\delta}$ and La$_2$PrNi$_2$O$_{7-\delta}$. Our results are consistent with an enhanced interlayer coupling $SJ_{\perp}$ within the stripe-type Heisenberg model framework, where the estimated $SJ_{\perp}$ value is in the range of about 69 to 73 meV for the rare-earth doped bilayer nickelates.