Disorder-Induced Damping of Spin Excitations in Cr-Doped BaFe$_2$As$_2$

Partial chemical substitution inevitably introduces disorder. In doped Hund's metals, such as the iron-based superconductors, effects like charge doping and chemical pressure are often considered dominant. 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 primarily by damping, becoming overdamped for $x = 0.085$. At this doping level, complementary angle-resolved photoemission spectroscopy measurements (ARPES) show the absence of electronic structure reconstruction effects such as the nematic band splitting. We propose a localized spin model that explicitly incorporates substitutional disorder and Cr local moments, successfully reproducing our key observations. These results reveal a case where disorder dominates over charge doping in the case of a correlated Hund's metal.