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We have measured the doping and temperature dependence of lattice vibrationsand magnetic excitations in the prototypical doped spin-orbit Mott insulatorSr$_{2-x}$La$_x$IrO$_4$ (x=0, 0.015, and 0.10). Our findings show that thepseudospin-lattice coupling -- responsible for the renormalization of severallow energy phonon modes -- is preserved even when long-range magnetic order issuppressed by doping. In our most highly doped sample, the single magnon(Gamma-point) excitation disappears while the two-magnon mode softens andbecomes heavily damped. Doping induced electron-phonon coupling is alsoobserved in a higher energy phonon mode. We observe two differentelectron-phonon interaction channels, which provide evidence of the coexistenceof fluctuating magnetic moments and mobile carriers in doped iridates.

Raman scattering study of vibrational and magnetic excitations in Sr$_{2-x}$La$_x$IrO$_4$