Nodal and nodeless gap in proximity induced superconductivity: application to monolayer CuO₂ on BSCCO substrate

We present a detailed analysis on the hopping between monolayer CuO$_2$ and bulk CuO$_2$ plane in the Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ substrate. With a two-band model, we demonstrate that the nodeless gap can only exist when the hole concentration in monolayer CuO$_2$ plane is very large or very small. We argued that the possible phase separation may play important role in the recent experimental observation of nodeless gap.