Influence of tunneling matrix elements and quantum well states on Josephson effects in Scanning Tunneling Microscopy

Superfluid density is crucial for establishing phase coherence (stiffness) in superconductors, making it essential for understanding high-temperature superconductivity mechanisms, such as those in cuprates, where low carrier density induces strong phase fluctuations. Recently, atomic-resolution Scanning Josephson Tunnelling Microscopy (SJTM) has been used to investigate superfluid density in superconductors, revealing pair density waves (PDW) and inhomogeneous superfluid in unconventional superconductors. However, the impact of tunneling matrix elements on the detection of superfluid density in SJTM experiments remains unexplored, which is fundamental for interpreting experimental results. Conventional superconductors like lead, with their large coherence length and spatially homogeneous superfluid, provide an ideal platform to study these effects. By manipulating Pb atoms on the Pb (111) surface, we demonstrated that selective tunneling into different bands in lead results in up to a 10% variation in detected superfluid density, exceeding most known modulations in superconducting systems, especially comparing with observed PDW. Additionally, we observed that quantum well states contribute to further modulation of superfluid density beyond tunneling matrix elements, the mechanism of which remains unclear. Our findings provide crucial insights for future SJTM result interpretation and highlight the significant role of tunneling matrix elements in these experiments.