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In doped SrTiO$_{3}$ superconductivity persists down to an exceptionally lowconcentration of mobile electrons. This restricts the relevant energy windowand possible pairing scenarios. We present a study of quantum oscillations andsuperconducting transition temperature, $T_{c}$ as the carrier density is tunedfrom $10^{17}$ to $10^{20}$ $cm^{-3}$ and identify two critical doping levelscorresponding to the filling thresholds of the upper bands. At the firstcritical doping, which separates the single-band and the two-bandsuperconducting regimes in oxygen-deficient samples, the steady increase ofT$_{c}$ with carrier concentration suddenly stops. Near this doping level, theenergy dispersion in the lowest band displays a downward deviation fromparabolic behavior. The results impose new constraints for microscopic pairingscenarios.

Critical doping for the onset of a two-band superconducting ground state in SrTiO$_{3-\delta}$