  Iridates are of considerable current interest because of the strong
spin-orbit coupling that leads to a variety of new phenomena. Using
density-functional studies, we predict the formation of a spin-orbital
entangled two-dimensional electron gas (2DEG) in the $\delta$-doped iridate
La$_\delta$Sr$_2$IrO$_4$, where a single SrO layer is replaced by a LaO layer.
The extra La electron resides close to the $\delta$-doped layer, partially
occupying the $J_{\rm eff}= 1/2 $ upper Hubbard band and thereby making the
interface metallic. The magnetic structure of the bulk is destroyed near the
interface, with the Ir$_0$ layer closest to the interface becoming
non-magnetic, while the next layer (Ir$_1$) continues to maintain the AFM
structure of the bulk, but with a reduced magnetic moment. The Fermi surface
consists of a hole pocket and an electron pocket, located in two different Ir
layers (Ir$_0$ and Ir$_1$), with both carriers derived from the $J_{\rm eff}=
1/2 $ upper Hubbard band. The presence of both electrons and holes at the
$\delta$-doped interface suggests unusual transport properties, leading to
possible device applications.
