Spin excitations in hole-overdoped iron-based superconductors Ba$_{\mathrm{1-x}}$K$_{\mathrm{x}}$Fe$_{\mathrm{2}}$As$_{\mathrm{2}}$*

Understanding the overall features of magnetic excitation is essential for clarifying the mechanism of Cooper pair formation in iron-based superconductors. Here we report a study on the spin fluctuations of the hole-overdoped iron-based superconductors Ba$_{\mathrm{1-x}}$K$_{\mathrm{x}}$Fe$_{\mathrm{2}}$As$_{\mathrm{2}}$ (x $=$ 0.5 and 1.0; $T_{\mathrm{c}} \quad =$ 36 K and 3.4 K, respectively) over the entire Brillouin zone using inelastic neutron scattering. We find that their spin spectra consist of spin wave and chimney-like dispersions. The chimney-like dispersion can be attributed to the itinerant character of magnetism. The band width of the spin wave-like dispersion is almost constant from the non-doped to optimum-doped region, which is followed by a large reduction in the overdoped region. This suggests that the superconductivity is suppressed by the reduction of magnetic exchange couplings, indicating a strong relationship between magnetism and superconductivity in iron-based superconductors.