  We report on the emergence of two disconnected superconducting domes in
alkali-metal potassium (K)-doped FeSe ultra-thin films grown on graphitized
SiC(0001). The superconductivity exhibits hypersensitivity to K dosage in the
lower-Tc dome, whereas in the heavily electron-doped higher-Tc dome it becomes
spatially homogeneous and robust against disorder, supportive of a conventional
Cooper-pairing mechanism. Furthermore, the heavily K-doped multilayer FeSe
films all reveal a large superconducting gap of ~ 14 meV, irrespective of film
thickness, verifying the higher-Tc superconductivity only in the topmost FeSe
layer. The unusual finding of a double-dome superconducting phase has stepped
towards the mechanistic understanding of superconductivity in FeSe-derived
superconductors.
