# On Thermodynamic Aspects of Oxide Crystal Growth
## 1 Introduction  
### Oxide Bulk Crystal Growth  
The first oxide material from which crystals were grown on an industrial scale is \(\alpha\)-quartz (SiO\({}_{2}\)). Quartz crystals are still of high technical relevance; their growth technique, as described in 1953 by [PERSON] [1], relies on hydrothermal solutions. However, the equilibria between solvents and solutes will not be in the focus of this article. Rather, equilibria between oxides in the solid (crystalline) and molten state will be described, which are important for the growth of crystals from the melt, e.g., using the Czochralski, Bridgman, or float-zone methods [2; 3]. Here, the Czochralski method is demonstrated first for CaWO\({}_{4}\), which is now often used for different materials such as LiNbO\({}_{3}\), YAG (Y\({}_{3}\)Al\({}_{5}\)O\({}_{12}\)) and other garnets, and partially for \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) (usually named sapphire, but \"corundum\" would be more accurate) [4]. These materials can be grown in different atmospheres, ranging from air to \"pure\" argon or nitrogen with ca. 99.9...99.999% purity. Independently of these different conditions, the constituent metal cations prevail almost completely in one oxidation state. Here, these are Ca\({}^{2+}\), W\({}^{6+}\), Li\({}^{+}\), Nb\({}^{5+}\), Y\({}^{3+}\), or Al\({}^{3+}\), respectively. Some other substances, especially many transparent conducting oxides (TCO's), behave differently: crystals of \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) (melting point \(T_{\mathrm{f}}\approx 1800\)\({}^{\circ}\)C, [5]) and ZnO (\(T_{\mathrm{f}}=1975\)\({}^{\circ}\)C, [6]) can be grown only if a sufficiently high oxygen partial pressure \(p_{\mathrm{O}_{2}}\) prevails in the atmosphere. With current FactSage data [7] one calculates that at their melting points, Ga\({}_{2}\)O\({}_{3}\) (sol/liq) is stable for \(p_{\mathrm{O}_{2}}\gtrsim 10^{-6}\) bar, and ZnO (sol/liq) for \(p_{\mathrm{O}_{2}}\gtrsim 5\times 10^{-4}\) bar. If \(p_{\mathrm{O}_{2}}\) falls below these limits, Ga\({}^{3+}\) or Zn\({}^{2+}\) is reduced to the corresponding metal, which occurs here in a gaseous state. On the other side, sufficiently high \(p_{\mathrm{O}_{2}}\) will always stabilize Ga\({}^{3+}\) or Zn\({}^{2+}\) and, hence, Ga\({}_{2}\)O\({}_{3}\) or ZnO because no higher oxidation state exists. The opposite holds for the melt growth of Ti\({}^{3+}\) doped Al\({}_{2}\)O\({}_{3}\) crystals (\"Ti:sapphire\") because Ti\({}^{3+}\) is the lowest possible valence state of titanium under all conditions, where Al\({}_{2}\)O\({}_{3}\) can begrown from the melt. Hence, \(p_{\text{O}_{2}}\) should be as low as possible under realistic experimental conditions, e.g., by introducing graphite parts in the growth setup [8; 9]. The situation is more complicated if an intermediate oxidation state is desired. This is the case, for example, for w The formation of Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) from powders of CaCO\({}_{3}\) and different iron oxides (FeO, Fe\({}_{3}\)O\({}_{4}\), Fe\({}_{2}\)O\({}_{3}\)) was investigated by [PERSON] et al. [28]. [PERSON] et al. [29] succeeded to grow Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) single crystals of good quality from the melt by the optical float zone (OFZ) technique, which is a strong indication for congruent melting. For this growth method, a polycrystalline stoichiometric feed rod is moved through a hot zone, which is heated by the focused radiation of lamps; [PERSON] et al. used two 500 W halogen lamps for heating. (More details for the OFZ method can be found elsewhere [3]). Increased oxygen mobility in the crystals above ca. 450 \({}^{\circ}\)C was observed by thermogravimetric (TG) measurements of \({}^{18}\)O enriched material in \({}^{16}\)O\({}_{2}\) atmosphere. The observed mass loss was finished at ca. 600 \({}^{\circ}\)C and was attributed to the \({}^{18}\)O/\({}^{16}\)O exchange.