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The path from a Mott insulating phase to high temperature superconductivityencounters a rich set of unconventional phenomena involving theinsulator-to-metal transition (IMT) such as emergent electronic orders andpseudogaps that ultimately affect the condensation of Cooper pairs. A hugehindrance to understanding the origin of these phenomena in the curates is thedifficulty in accessing doping levels near the parent state. Recently, theJ$_{eff}$=1/2 Mott state of the perovskite strontium iridates has revealedintriguing parallels to the cuprates, with the advantage that it providesunique access to the Mott transition. Here, we exploit this accessibility tostudy the IMT and the possible nearby electronic orders in the electron-dopedbilayer iridate (Sr$_{1-x}$La$_x$)$_3$Ir$_2$O$_7$. Using spectroscopic imagingscanning tunneling microscopy, we image the La dopants in the top as well asthe interlayer SrO planes. Surprisingly, we find a disproportionatedistribution of La in these layers with the interlayer La being primarilyresponsible for the IMT, thereby revealing the distinct site-dependent effectsof dopants on the electronic properties of bilayer systems. Furthermore, wediscover the coexistence of two electronic orders generated by electron doping:a unidirectional electronic order with a concomitant structural distortion; andlocal resonant states forming a checkerboard-like pattern trapped by La. Thisprovides evidence that multiple charge orders may exist simultaneously in Mottsystems, even with only one band crossing the Fermi energy.

Doping induced Mott collapse and the density wave instabilities in (Sr$_{1-x}$La$_x$)$_3$Ir$_2$O$_7$