  A neutron scattering study of heavily hole-overdoped superconducting
KFe$_2$As$_2$ revealed a well-defined low-energy incommensurate spin
fluctuation at [$\pi(1\pm2\delta$),0] with $\delta$ = 0.16. The incommensurate
structure differs from the previously observed commensurate peaks in
electron-doped $A$Fe$_2$As$_2$ ($A$ = Ba, Ca, or Sr) at low energies. The
direction of the peak splitting is perpendicular to that observed in Fe(Te,Se)
or in Ba(Fe,Co)$_2$As$_2$ at high energies. A band structure calculation
suggests interband scattering between bands around the $\Gamma$ and X points as
an origin of this incommensurate peak. The perpendicular direction of the peak
splitting can be understood within the framework of multiorbital band
structure. The results suggest that spin fluctuation is more robust in
hole-doped than in electron-doped samples, which can be responsible for the
appearance of superconductivity in the heavily hole-doped samples.
