  We study the effects of dilute La and Rh doping on the electronic structure
of the relativistic Mott insulator Sr$_2$IrO$_4$ using fully relativistic and
magnetically non-collinear density functional theory with the inclusion of an
on-site Hubbard $U$. To model doping effects, we have adopted the supercell
approach that allows for a realistic treatment of structural relaxations and
electronic effects beyond a purely rigid band approach. By means of the band
unfolding technique we have computed the spectral function and constructed the
effective band structure and Fermi surface (FS), which are readily comparable
with available experimental data. Our calculations clearly indicate that La and
Rh doping can be interpreted as effective electron and (fractional) hole
doping, respectively. In Sr$_{2-x}$La$_x$IrO$_4$ the IMT is accompanied by a
moderate renormalization of the electronic correlation substantiated by a
reduction of the effective on-site Coulomb repulsion $U-J$ from 1.6 eV ($x=0$)
to 1.4 eV (metallic regime $x=12.5\%$). The progressive closing of the
relativistic Mott gap leads to the emergence of connected elliptical electron
pockets at ($\pi$/2,$\pi$/2) and less intense features at $X$ in the FS. The
substitution of Ir with the nominally isovalent Rh is accompanied by a
substantial hole transfer from the Rh site to the nearest neighbor Ir sites.
This shifts down the chemical potential, creates almost circular disconnected
hole pockets in the FS and establishes the emergence of a two-dimensional
metallic state formed by conducting Rh-planes intercalated by insulating
Ir-planes. Finally, our data indicate that hole doping causes a flipping of the
in-plane net ferromagnetic moment in the Rh plane and induces a magnetic
transition from the AF-I to the AF-II ordering.
