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We study the effects of dilute La and Rh doping on the electronic structureof the relativistic Mott insulator Sr$_2$IrO$_4$ using fully relativistic andmagnetically non-collinear density functional theory with the inclusion of anon-site Hubbard $U$. To model doping effects, we have adopted the supercellapproach that allows for a realistic treatment of structural relaxations andelectronic effects beyond a purely rigid band approach. By means of the bandunfolding technique we have computed the spectral function and constructed theeffective band structure and Fermi surface (FS), which are readily comparablewith available experimental data. Our calculations clearly indicate that La andRh doping can be interpreted as effective electron and (fractional) holedoping, respectively. In Sr$_{2-x}$La$_x$IrO$_4$ the IMT is accompanied by amoderate renormalization of the electronic correlation substantiated by areduction 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 therelativistic Mott gap leads to the emergence of connected elliptical electronpockets at ($\pi$/2,$\pi$/2) and less intense features at $X$ in the FS. Thesubstitution of Ir with the nominally isovalent Rh is accompanied by asubstantial hole transfer from the Rh site to the nearest neighbor Ir sites.This shifts down the chemical potential, creates almost circular disconnectedhole pockets in the FS and establishes the emergence of a two-dimensionalmetallic state formed by conducting Rh-planes intercalated by insulatingIr-planes. Finally, our data indicate that hole doping causes a flipping of thein-plane net ferromagnetic moment in the Rh plane and induces a magnetictransition from the AF-I to the AF-II ordering.

Electron and hole doping in the relativistic Mott insulator Sr$_2$IrO$_4$: a first-principles study using band unfolding technique