Doping dependence of magnetic excitations in Sr$_{2-x}$La$_x$IrO$_4$

We have studied the magnetic excitations of electron-doped Sr$_{2-x}$La$_x$IrO$_4$ ($0 leq x leq 0.10$) using resonant inelastic x-ray scattering (RIXS) at the Ir L$_3$-edge. The long range magnetic order is rapidly lost with increasing $x$, but two-dimensional short-range order (SRO) and dispersive magnon excitations with nearly undiminished spectral weight persist up to $x=0.10$. The magnons in the SRO phase are heavily damped and exhibit anisotropic softening. Their dispersions are well described by a pseudospin-1/2 Heisenberg model with exchange interactions whose spatial range increases with doping. We also find a doping-independent high-energy magnetic continuum, which is not described by this model. The spin-orbit excitons arising from the pseudospin-3/2 manifold of the Ir ions broaden substantially in the SRO phase, but remain largely separated from the low-energy magnons. Pseudospin-1/2 models are therefore a good starting point for the theoretical description of the low-energy magnetic dynamics of doped iridates.