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

Abstract

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.

Figures (3)

• Figure 1: a) Unit cell of the rutile structure. The blue atoms correspond to cation sites occupied by Ge or Sn atoms, while red sites correspond to oxygen atoms. b) Calculated PBE (blue circles), r$^2$SCAN (green squares), HSE (orange pentagons) lattice constants $a$ (top) and $c$ (bottom), for $\text{Sn}_{1-x}\text{Ge}_x\text{O}_2$ alloys at x = 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1. The values for the alloys are calculated by averaging three SQS supercells, each containing 72 atoms (a $2\times2\times3$ supercell of the primitive unit cell). The error bars represent the standard deviation. c) Calculated HSE bandgaps (orange pentagons) for $\text{Sn}_{1-x}\text{Ge}_x\text{O}_2$ alloys using 72 atom SQSs. The bowing fit is determined using a second-order composition-dependent model. The experimental data for alloy thin films are given by takane_band-gap_2022 (upward triangles), nagashima_deep_2022 (downward triangles), Abed2025 (diamonds), and Kluth2024_GeSnO2_Bowing (rhombuses), while the data for the bulk crystals is given by Feneberg2014_SnO2_Dielectric (rightward triangles) and StapelbroekEvans1978_GeO2_UVabsorption (leftward triangles).
• Figure 2: a) DFT formation energies of the SQS structures evaluated with PBE (blue circles) and r$^{2}$SCAN (green squares) functionals. The dashed lines represent the enthalpy of the disordered phase estimated with the subregular solution model. b) Phase diagrams of the r-$\text{Sn}_{1-x}\text{Ge}_x\text{O}_2$ computed using PBE functional (left) and r²SCAN functional (right). The black solid line indicates the miscibility gap, the blue line the incoherent spinodal, and the orange line the coherent spinodal along the $[001]$ direction. The miscibility gap obtained through free energy integration using the CE is indicated by black dots in the PBE phase diagram. c) Convex hull computed with 80 symmetrically distinct orderings on $\text{r-Sn}_{1-x}\text{Ge}_x\text{O}_2$ predicted by DFT (blue circles) and CE (orange crosses), using the PBE functional. The blue dashed lines are the disordered enthalpy estimated with the subregular solution model fit to the DFT formation energies of the SQS structures. The orange dashed line is the enthalpy of a perfectly disordered solid solution estimated with the CE model.
• Figure 3: Plot of the Ge-Sn SRO as a function of the composition for the first (blue), second (orange), and third (green) nearest-neighbor shells, computed at $2400$ K (solid line) and at $3000$ K (dashed line).