  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.
