Quantum Oscillations and Superconductivity in Subband Quantized SrTiO$_3$ Bilayer Delta-Doped Structures

SrTiO$_3$ delta-doped structures show two-dimensional (2D) Shubnikov de-Haas oscillations (SdH) and 2D superconductivity (SC) [1]. Lightly doped systems, with clear SdH signals are ideal to study the link between 2D single electron states and SC [2]. The subbands (SB) should strongly influence SC: their splitting is larger than the superconducting gap. However, the similar spatial extent of the SB in single delta-layers prohibits the detection of SB modulated SC. Growing two delta-layers (DL) in parallel with varying interlayer (IL) thickness $d$, we can spatially separate the SB and identify their contributions to SC and SdH. For small $d$, all SB spread over the DL and the IL. For larger $d$ only lower SB are confined around the DL. From the angular-dependence of the main SdH frequency we find a 2D to three-dimensional crossover for $\sim 60