We study the correlated electronic structure of single-layer iridates based on structurally-undistorted Ba$_2$IrO$_4$. Starting from the first-principles band structure, the interplay between local Coulomb interactions and spin-orbit coupling is investigated by means of rotational-invariant slave-boson mean-field theory. The evolution from a three-band description towards an anisotropic one-band ($J=1/2$) picture is traced. Single-site and cluster self-energies are used to shed light on competing Slater- and Mott-dominated correlation regimes. We reveal a clear asymmetry between electron and hole doping, notably in the nodal/anti-nodal Fermi-surface dichotomy at strong coupling. Electron-doped iridates appear comparable to hole-doped cuprates due to the different sign of the next-nearest-neighbor hopping $t$.
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