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Electronic and optical properties of Si$_{x}$Ge$_{1-x-y}$Sn$_{y}$ alloys lattice-matched to Ge

We present a combined experimental and theoretical analysis of the evolution of the near-band gap electronic and optical properties of Si$_{x}$Ge$_{1-x-y}$Sn$_{y}$ alloys lattice-matched to Ge and GaAs substrates. We perform photoreflectance (PR) and photoluminescence (PL) measurements on Si$_{x}$Ge$_{1-x-y}$Sn$_{y}$ epitaxial layers grown via chemical vapour deposition, for Si (Sn) compositions up to $x =$ 9.6% ($y =$ 2.5%). Our measurements indicate the presence of an indirect fundamental band gap, with PL observed $\approx$ 200-250 meV lower in energy than the direct $E_0$ transition identified by PR measurements. The measured PL is Ge-like, suggesting that the alloy conduction band (CB) edge is primarily derived from the Ge L-point CB minimum. Interpretation of the PR and PL measurements is supported by atomistic electronic structure calculations. Effective alloy band structures calculated via density functional theory confirm the presence of an indirect fundamental band gap, and reveal the origin of the observed inhomogeneous broadening of the measured optical spectra as being alloy-induced band hybridisation occurring close in energy to the CB edge. To analyze the evolution of the band gap, semi-empirical tight-binding (TB) calculations are employed to enable calculations for large supercell sizes. TB calculations reveal that the alloy CB edge is hybridized in nature, consisting at low Si and Sn compositions of an admixture of Ge L-, $Γ$- and X-point CB edge states, and confirm that the alloy CB edge retains primarily Ge L-point CB edge character. Our experimental measurements and theoretical calculations confirm a direct transition energy close to 1 eV in magnitude for Si and Sn compositions $x =$ 6.8 - 9.6% and $y =$ 1.6 - 2.2%.