Disorder-induced damping of spin excitations in Cr-doped BaFe$_2$As$_2$

In doped Hund's metals, such as the iron-based superconductors, effects like charge doping and chemical pressure are often considered the dominant factors. Partial chemical substitution, however, inevitably introduces disorder. Here, we investigate spin excitations in Ba(Fe$_{1-x}$Cr$_x$)$_2$As$_{2}$ (CrBFA) by high-resolution resonant inelastic x-ray scattering (RIXS) for samples with $x = 0, 0.035,$ and $0.085$. In CrBFA, Cr acts as a hole dopant, but also introduces localized spins that compete with Fe-derived magnetic excitations. We found that the Fe-derived magnetic excitations are softened and damped, becoming overdamped for $x = 0.085$. At this doping level, complementary angle-resolved photoemission spectroscopy measurements (ARPES) show increased electronic localization and a suppression of the nematic $d_{xz}/d_{yz}$ band splitting present in the parent compound. We thus propose a localized spin model that explicitly incorporates substitutional disorder and Cr local moments, successfully reproducing our key observations. Our findings reveal a case where disorder dominates over charge doping in the case of a Hund's metal.