Data for Manuscript: High-pressure floating zone crystal growth of Sr$_2$IrO$_4$

Here we demonstrate the floating zone crystal growth of the $J_\mathrm{eff}=1/2$ Mott insulator Sr$_2$IrO$_4$. Historically, the growth of iridates from a ternary melt has been precluded by the extreme vapor pressure of the metal oxide species and the difficulty of maintaining the correct oxidation state of Ir at high temperatures. Here, we show that the application of a high-pressure oxygen growth environment stabilizes the Sr$_2$IrO$_4$ phase, leading to the first demonstration of cm$^{3}$-scale crystals. In contrast to the conventional SrCl$_2$ flux growth method, where poor control over disorder leads to strong sample dependence, the high-pressure floating zone growth enables active control over the homogeneity of the melt. Crystals grown via this technique possess qualitatively similar properties to those grown via flux, with a relatively sharp onset of antiferromagnetic order observed in temperature-dependent magnetization. Further, we demonstrate that by tuning the mixing rate of the melt, we are able to grow natively hole-doped Sr$_2$Ir$_{1-y}$O$_4$, which exhibits a strongly modified magnetic and electronic response.