Electronic structure of Rh and Ru doped Sr$_2$IrO$_4$

Sr$_2$IrO$_4$ is a spin-orbit interaction(SOI) assisted insulator. It has been proposed that the weaker SOI in the $4d$-substituted Sr$_2$Ir$_{\mathrm{1-x}}$(Ru, Rh)$_{\mathrm{x}}$O$_4$ closes the insulating gap, rendering it a paramagnetic metal. Rh(${4d}^5$) is isoelectronic to Ir(${5d}^5$) whereas Ru(${4d}^4$) has one less electron in the $4d$-band. The AFM-I/PM-M transition takes place at lower $x$ for Ru than Rh, presumably due to the effect of hole doping. X-ray absorption near edge structure (XANES) and x-ray magnetic circular dichroism (XMCD) measurements at the Ir $ L_{2,3}$ edges show that $\langle\mathbf{L}.\mathbf{S} \rangle $ is non-zero and independent of $x$. This is indicative of a strong local $5d$ spin orbit interaction that is rather insensitive to the $4d$ doping. In contrast, measurements at the $ L_{2,3}$ edges of Ru and Rh show $\langle\mathbf{L}.\mathbf{S} \rangle \approx 0 $ for all $x$. The results point to the importance of local $4d/5d - 2p$ hybridization as opposed to $4d-5d$ band formation in the Rh and Ru doped Sr$_2$IrO$_4$.