Superconductivity in the dilute single band limit in reduced Strontium Titanate

We report on superconductivity in single crystals of SrTiO$_{3-\delta}$ with carrier densities $\textit{n} < 1.4 \times10^{18}cm^{-3}$, where only a single band is occupied. For all samples in this regime, the resistive transition occurs at $T_{c} \approx 65 \pm 25 \ mK$. We observe a zero resistance state for $\textit{n}$ as low as $1.03 \times10^{17}cm^{-3}$, and a partial resistive transition for $\textit{n} = 3.85 \times10^{16}cm^{-3}$. We observe low critical current densities, relatively high and isotropic upper critical fields, and an absence of diamagnetic screening in these samples. Our findings suggest an inhomogeneous superconducting state, embedded within a homogeneous high-mobility 3-dimensional electron gas. $T_{c}$ does not vary appreciably when $\textit{n}$ changes by more than an order of magnitude, inconsistent with conventional superconductivity.