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Si$_x$Ge$_{1-x-y}$Sn$_y$ ternary alloys are a candidate material system foruse in solar cells and other optoelectronic devices. We report on the directtransition energies and structural properties of Ge-richSi$_x$Ge$_{1-x-y}$Sn$_y$ alloys with six different compositions up to 10 % Siand 3 % Sn, lattice-matched to Ge or GaAs substrates. The direct transitionsoccurring between 0.9 and 5.0 eV were investigated using spectroscopicellipsometry (SE), and the resulting data was used to obtain the dielectricfunctions of the Si$_x$Ge$_{1-x-y}$Sn$_y$n layer by fitting a multi-layermodel. Values for the $E_0$, $E_1$, $\Delta_1$, $E_0'$ and $E_2$ transitionenergies were then found by differentiating these dielectric functions toextract the locations of critical points. Structurally, the composition of thesamples was measured using energy-dispersive X-ray measurements (EDX). Thelattice constants predicted from these compositions are in good agreement withreciprocal space maps obtained through X-ray diffraction (XRD). The resultsconfirm that a 1 eV direct absorption edge can be achieved using relatively lowSi and Sn fractions ($<$ 10 % and $<$ 3 % respectively), while thehigher-energy critical points show smaller shifts relative to Ge and matchresults previously observed or predicted in the literature.

Compositional dependence of direct transition energies in Si$_x$Ge$_{1-x-y}$Sn$_y$ alloys lattice-matched to Ge/GaAs