  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-, $\Gamma$- 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%.
