  Two-dimensional Rashba superconductor
Si(111)-$\sqrt{3}\times\sqrt{3}$-(Tl,Pb) is a candidate platform of mixed
spin-singlet and -triplet superconductivity. A recent scanning tunneling
microscope (STM) experiment revealed a pseudogap at the vortex core, suggesting
the finite triplet component [T. Nakamura $\textit{et al.}$, Phys. Rev. B $\bf{
98}$, 134505 (2018)]. Detailed spectroscopic information of the superconducting
gap and the low-energy band structure is necessary to establish the putative
triplet superconductivity. Here, we performed high-energy-resolution
spectroscopic imaging experiments on Si(111)-$\sqrt{3}\times\sqrt{3}$-(Tl,Pb)
using an ultra-low temperature STM. We found that various spectroscopic
features, including the vortex-core spectrum, are consistent with spin-singlet
$s$-wave superconductivity, having no sign of the triplet component. The
apparent contradiction with the previous STM result suggests that the nature of
superconductivity changes within the same system. From the analysis of the
quasiparticle interference patterns, we found that the Fermi energy is in the
close vicinity of the saddle point near the $\overline{\rm{M}}$ point. We
speculate that the nature of superconductivity varies depending on the
saddle-point energy with respect to the Fermi energy, which is sample-dependent
due to different band filling.
