We studied the superconducting properties of the one-unit cell FeSe grown on high-doped ${\mathrm{SrTiO}}_{3}$(001) (STO) substrate by in situ transport and angle-resolved photoemission spectroscopy (ARPES) measurements. By comparing with previous data taken for the low-doped STO substrate [Phys. Rev. Lett. 124, 227002 (2020)], it was revealed that the onset of superconductivity occurs at $\ensuremath{\sim}40$ K irrespective of the STO doping level. Furthermore, from the temperature dependence of the critical current, we were able to deduce the superconducting gap that is relevant in the transport phenomena, which gradually increases by cooling the sample down from 40 K. In contrast, the coherent peak emerges at 60 K that likely corresponds to the Cooper pair formation, while the leading edge shift near the Fermi level is clearly observed at 40 K in the ARPES spectra. We speculate that the STO surface presumably acts as an effective ``disorder'' in this system and the Cooper pairs are preformed well above the onset of the resistance drop.
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