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Thermodynamic and electronic properties of rutile Sn$_{1-x}$Ge$_x$O$_2$ alloys from first principles

Rutile Sn$_{1-x}$Ge$_x$O$_{2}$ alloys are promising materials for high-power electronic applications due to their dopability and tunable ultra-wide band gaps. We use first-principles density functional theory and statistical mechanics to investigate the crystallographic, electronic, and thermodynamic properties of rutile $\text{Sn}_{1-x}\text{Ge}_x\text{O}_2$ alloys. We predict that the lattice parameters follow Vegard's law, while band gaps calculated with the hybrid HSE06 functional exhibit strong bowing, consistent with experiment. We also predict that the disordered phase has a large positive mixing enthalpy and a slight tendency for Ge-Sn clustering, indicated by weakly negative short-range order parameters. This large positive mixing enthalpy produces a miscibility gap with a critical temperature above 2300 K, implying that the high Ge and Sn solubilities observed in thin-film synthesis cannot be explained by the incoherent phase diagram alone. We demonstrate that coherency strain during epitaxial growth substantially alters phase stability. Calculations of the coherent spinodal show significant suppression of the miscibility gap, reducing the critical temperature to $\approx 900$ K. These coherent phase boundaries account for the experimentally observed high solubilities at typical growth temperatures. Our results indicate that coherency strain stabilizes these metastable alloys and enables bandgap engineering in this ultrawide-bandgap material system.