Formation of spin-orbital entangled 2D electron gas in layer delta-doped bilayer iridate La$_{\delta}$Sr$_3$Ir$_2$O$_7$

5$d$ transition metal oxides host a variety of exotic phases due to the comparable strength of Coulomb repulsion and spin-orbit coupling. Herein, by pursuing density-functional studies on a delta-doped quasi-two-dimensional iridate Sr$_3$Ir$_2$O$_7$, where a single SrO layer is replaced by LaO layer, we predict the formation of a spin-orbital entangled two-dimensional electron gas (2DEG) which is sharply confined on two IrO$_2$ layers close to the LaO layer. In this bilayer crystal structure, an existing potential well is further augmented with the inclusion of positively charged LaO layer which results in confining the extra valence electron made available by the La$^{3+}$ ion. The confined electron is bound along crystal $a$ direction and is highly mobile in the $bc$ plane. From the band structure point of view, now the existing half-filled $J_{eff}$ = 1/2 states are further electron doped to destroy the antiferromagnetic Mott insulating state of IrO$_2$ layers near to the delta-doped layer. This leads to partially occupied Ir upper-Hubbard subbands which host the spin-orbital entangled 2DEG. The IrO$_2$ layers far away from the interface remain insulating and preserve the collinear G-type magnetic ordering of pristine Sr$_3$Ir$_2$O$_7$. The conductivity tensors calculated using semi-classical Boltzmann theory at room temperature reveal that the 2DEG exhibits large electrical conductivity of the order of 10$^{19}$.