Interface superconductivity in bi-layers of insulating and overdoped metallic La$_{2-x}$Sr$_{x}$CuO$_{4 + \delta}$

We report on properties of thin superconducting (SC) sheets obtained in La$_{2-x}$Sr$_{x}$CuO$_{4 + \delta}$ bi-layers of overdoped, non-superconducting $(x = 0.45)$ and insulating $(x = 0)$ films grown by molecular beam epitaxy. Superconductivity is confined to a thickness of $\approx$ 2 nm from the interface. The observed transition temperatures have values $T_{c} \approx 15$~K, 30 K and 50 K depending on the layering sequence and oxidation state of the insulating material. Transport measurements are used to determine the screening properties of the quasi-2D SC sheets and the data in bi-layers are compared to results from single-phase films and bulk crystals.