Impact of order and disorder on phase formation in (In${}_x$Ga${}_{1-x}$)${}_2$O${}_3$ investigated by transmission electron microscopy

We investigate the phase formation in (In${}_x$Ga${}_{1-x}$)${}_2$O${}_3$ solid solutions experimentally by means of transmission electron microscopy (TEM) and with computational support using cluster expansion. In the case of epitaxial growth on crystalline substrates, we find strong ordering on the cation sublattices of (In${}_x$Ga${}_{1-x}$)${}_2$O${}_3$, energetically driven by the tendency of In and Ga to each assume their preferred coordination environment. Based on this experimental finding, we modify the model of the ideal mixture by considering the configurational entropy on the respective cation sublattices with different coordination separately in order to calculate it realistically. The resulting phase diagram is characterized by narrow thermodynamically stable ranges for each phase, while wide composition ranges of metastable compounds are predicted, which can be achieved at temperatures typical for epitaxy: the monoclinic phase is metastable in the composition range x<0.5, the hexagonal phase for 0.550.91. If amorphous (In${}_x$Ga${}_{1-x}$)${}_2$O${}_3$ is crystallized in-situ in the TEM, the spinel phase, which is described as a disordered variant of the monoclinic phase, is formed in the composition range up to x<0.22, while above this composition, the bixbyite phase is stable. This shift in stability is explained by the maximum amount of configurational entropy present during crystallization, which strongly influences the formation enthalpy in structures with diverse coordination environments of the cations. The last part of the work deals with the influence of the lattice order on the material contrast when imaging by HAADF (High Angle Annular Dark Field) STEM. It is shown that the excitation of the 2s-Bloch wave state leads to long-period contrast oscillations, which complicate the quantitative determination of the composition by Z-contrast but allows to quantify the order parameter for a given composition.