Possible Nodeless Superconducting Gaps in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ and YBa${}_2$Cu${}_3$O${}_{7-x}$ Revealed by Cross-Sectional Scanning Tunneling Spectroscopy

Pairing in the cuprate high-temperature superconductors and its origin remain among the most enduring mysteries in condensed matter physics. With cross-sectional scanning tunneling microscopy/ spectroscopy, we clearly reveal the spatial-dependence or inhomogeneity of the superconducting gap structure of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ (Bi2212) and YBa${}_2$Cu${}_3$O${}_{7-x}$ (YBCO) along their $c$-axes on a scale shorter than the interlayer spacing. By tunneling into the (100) plane of a Bi2212 single crystal and a YBCO film, we observe both U-shaped tunneling spectra with extended flat zero-conductance bottoms, and V-shaped gap structures, in different regions of each sample. On the YBCO film, tunneling into a (110) surface only reveals a U-shaped gap without any zero-bias peak. Our analysis suggests that the U-shaped gap is likely a nodeless superconducting gap. The V-shaped gap has a very small amplitude, and is likely proximity-induced by regions having the larger U-shaped gap.