  The nature of superconductivity in the dilute semiconductor SrTiO$_3$ has
remained an open question for more than 50 years. The extremely low carrier
densities ($10^{18}$ - $10^{20}$ cm$^{-3}$) at which superconductivity occurs
suggests an unconventional origin of superconductivity outside of the adiabatic
limit on which the Bardeen-Cooper-Schrieffer (BCS) and Migdal-Eliashberg (ME)
theories are based. We take advantage of a newly developed method for
engineering band alignments at oxide interfaces and access the electronic
structure of Nb-doped SrTiO$_3$ using high resolution tunneling spectroscopy.
We observe strong coupling to the highest energy longitudinal optic (LO) phonon
branch and estimate the doping evolution of the dimensionless electron-phonon
interaction strength ($\lambda$). Upon cooling below the superconducting
transition temperature ($T_{\mathrm{c}}$), we observe a single superconducting
gap corresponding to the weak-coupling limit of BCS theory, indicating an order
of magnitude smaller coupling ($\lambda_{\textrm{BCS}} \approx 0.1$). These
results suggest that despite the strong normal state interaction with
electrons, the highest LO phonon does not provide a dominant contribution to
pairing. They further demonstrate that SrTiO$_3$ is an ideal system to probe
superconductivity over a wide range of carrier density, adiabatic parameter,
and electron-phonon coupling strength.
