  Atomically precise epitaxial structures are unique systems for tunneling
spectroscopy that minimize extrinsic effects of disorder. We present a
systematic tunneling spectroscopy study, over a broad doping, temperature, and
bias range, in epitaxial c-axis
La$_{2-x}$Sr$_{x}$CuO$_{4}$/La$_{2}$CuO$_{4}$/La$_{2-x}$Sr$_{x}$CuO$_{4}$
heterostructures. The behavior of these superconductor/insulator/superconductor
(SIS) devices is unusual. Down to 20 mK there is complete suppression of c-axis
Josephson critical current with a barrier of only 2 nm of La$_{2}$CuO$_{4}$,
and the zero-bias conductance remains at 20-30% of the normal-state
conductance, implying a substantial population of in-gap states. Tunneling
spectra show greatly suppressed coherence peaks. As the temperature is raised,
the superconducting gap fills in rather than closing at $T_{c}$. For all doping
levels, the spectra show an inelastic tunneling feature at $\sim$ 80 meV,
suppressed as $T$ exceeds $T_{c}$. These nominally simple epitaxial cuprate
junctions deviate markedly from expectations based on the standard
Bardeen-Cooper-Schrieffer (BCS) theory.
