Supplementary Online Material

(1) details of the critical point fitting method used to fit the differentiated SE data; (2) discussion of the expected critical thickness of the Si$_x$Ge$_{1-x-y}$Sn$_y$ epitaxial layers on Ge and GaAs substrates; (3) ellipsometry data for the Set A samples, optical constants for the non-Si$_x$Ge$_{1-x-y}$Sn$_y$ layers, comparison of the parametric model and point-by-point fits, depolarization spectra, and the fit parameters obtained for all samples; (4) details of the endpoint energies and bowing parameters for theoretical transition energy estimates (5) XRD reciprocal space maps for all Set A samples.