Investigating the ranges of (meta)stable phase formation in ${\left({\mathrm{In}}_x{\mathrm{Ga}}_{1\text{−}x}\right)}_2{\mathrm{O}}_3$: Impact of the cation coordination

We investigate the phase diagram of the heterostructural solid solution ${\left({\mathrm{In}}_x{\mathrm{Ga}}_{1\text{−}x}\right)}_2{\mathrm{O}}_3$ both computationally, by combining cluster expansion and density functional theory, and experimentally, by means of transmission electron microscopy (TEM) measurements of pulsed laser deposited (PLD) heteroepitaxial thin films. The shapes of the Gibbs free energy curves for the monoclinic, hexagonal, and cubic bixbyite alloy as a function of composition can be explained in terms of the preferred cation coordination environments of indium and gallium. We show by atomically resolved scanning TEM that the strong preference of indium for sixfold coordination results in ordered monoclinic and hexagonal lattices. This ordering impacts the configurational entropy in the solid solution and thereby the ${\left({\mathrm{In}}_x{\mathrm{Ga}}_{1\text{−}x}\right)}_2{\mathrm{O}}_3$ phase diagram. The resulting phase diagram is characterized by very limited solubilities of gallium and indium in the monoclinic, hexagonal, and cubic ground state phases, respectively, but exhibits wide metastable ranges at realistic growth temperatures. On the indium rich side of the phase diagram a wide miscibility gap up to temperatures higher than 1400 K is found, which results in phase separated layers. The experimentally observed indium solubilities in the PLD samples are in the range of $x=0.45$ and $x=0.55$ for monoclinic and hexagonal single-phase films, while for phase separated films we find $x=0.5$ for the monoclinic phase, $x=0.65–0.7$ for the hexagonal phase and $x\ge 0.9$ for the cubic phase. These values are consistent with the computed metastable ranges for each phase.

Figure 1

The lattice symmetries for the three sesquioxides considered in this work. The monoclinic, cubic bixbyite, and hexagonal phases (referred to as β, c, and h phase in here) have mixed four-/sixfold (green/blue), only sixfold, and mixed five- (orange)/sixfold coordinated cation positions with oxygen, respectively. Made with vesta [30].

Figure 2

TEM images of single-phase (a) monoclinic ${\left({\mathrm{In}}_{0.45}{\mathrm{Ga}}_{0.55}\right)}_2{\mathrm{O}}_3$ and (b) hexagonal ${\left({\mathrm{In}}_{0.55}{\mathrm{Ga}}_{0.45}\right)}_2{\mathrm{O}}_3$ thin films on sapphire grown, respectively, at $T_g=953\mathrm{K}$ and $T_g=913\mathrm{K}$. On top are TEM bright field images showing the morphology of the layer. The bottom left on each side shows the diffraction pattern of the ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate (white arrows) and the layer (colored arrows) together. For the monoclinic film, four different in-plane orientations can be recognized. Two of them are also identified in the STEM image, which shows the atomic pattern of a $\left[010\right]$ and a $\left[1\overline{3}2\right]$ grain and their common growth along the $\left(\overline{2}01\right)$ planes. The STEM image of the hexagonal phase is taken in the $\left[11\overline{2}0\right]$ orientation and presents some stacking faults.

Figure 3

(a) TEM bright field image of a phase separated ${\left({\mathrm{In}}_{0.75}{\mathrm{Ga}}_{0.25}\right)}_2{\mathrm{O}}_3$ thin film on sapphire substrate grown at $T_g=953$ K. The local indium content $x$ in the monoclinic, hexagonal, and cubic regions is determined by the STEM-EDXS measurement shown in (b).

Figure 4

Experimental high-magnification STEM-HAADF images (several images summed to enhance contrast) of the β-, h-, and $\mathrm{c}\text{−}{\left({\mathrm{In}}_x{\mathrm{Ga}}_{1\text{−}x}\right)}_2{\mathrm{O}}_3$ alloys with overlay of the stick-and-ball models without oxygen atoms. The lower plots show experimental HAADF intensity line profiles (black, multiple averaged) from the STEM images along the two differently coordinated atom columns in the β and h lattices, compared to the simulated profiles (red) for ordered structures. The intensity difference between the four-/fivefold and sixfold atomic columns, respectively, proves the preferential incorporation of the heavier indium to the sixfold lattice sites.

Figure 5

(a) Mixing enthalpies ($\mathrm{\Delta }H$) at 0 K for the lowest energy configurations of the monoclinic (β), hexagonal (h), and cubic bixbyite (c) lattices calculated by DFT using the PBEsol functional with fit lines to the data. (b) Configurational entropy ($\mathrm{\Delta }{S}_{c\mathrm{onfig}}$) for the three phases (green and blue curves are overlapped), calculated as described in the text. (c) and (d) Free energies ($\mathrm{\Delta }G$) at $T=600\mathrm{K}$ and $T=1000\mathrm{K}$ as a function of indium content for the β, h, and c phase. The global convex hull is indicated by the black crosses and line. The spinodal limits at elevated temperatures are indicated with red crosses.

Figure 6

Mean effective coordination number of indium and gallium and corresponding $\mathrm{\Delta }H$ values for about 100 randomly generated structures for $\mathrm{c}\text{−}\mathrm{InGa}{\mathrm{O}}_3$ (left), $\mathrm{h}\text{−}\mathrm{InGa}{\mathrm{O}}_3$ (middle), and $\beta \text{−}\mathrm{InGa}{\mathrm{O}}_3$ (right), i.e., $x=0.5$.

Figure 7

Computed temperature dependent phase diagram for ${\left({\mathrm{In}}_x{\mathrm{Ga}}_{1\text{−}x}\right)}_2{\mathrm{O}}_3$ including binodal and spinodal lines for the monoclinic (blue), hexagonal (green), and cubic (orange) phases. Thermodynamic stable composition ranges are color-filled, metastable ranges are grey, and for the white area below the spinodal lines phase separation is expected. The black vertical lines indicate the critical compositions where the lowest energy phase changes from (1) monoclinic to hexagonal and from (2) hexagonal to cubic. Experimental data points from PLD layers studied in this work (circles and diamonds) and from other synthesis methods found in literature (triangles) are added as symbols. The white diamonds denote the global compositions of the phase separated films.