Optical critical points of thin-filmGe1−ySnyalloys: A comparativeGe1−

The ${E}_{0}$, ${E}_{0}+{\ensuremath{\Delta}}_{0}$, ${E}_{1}$, ${E}_{1}+{\ensuremath{\Delta}}_{1}$, ${E}_{0}^{\ensuremath{'}}$, and ${E}_{2}$ optical transitions have been measured in ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{Sn}}_{y}$ alloys $(y<0.2)$ using spectroscopic ellipsometry and photoreflectance. The results indicate a strong nonlinearity (bowing) in the compositional dependence of these quantities. Such behavior is not predicted by electronic structure calculations within the virtual crystal approximation. The bowing parameters for ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{Sn}}_{y}$ alloys show an intriguing correlation with the corresponding bowing parameters in the ${\mathrm{Ge}}_{1\ensuremath{-}x}{\mathrm{Si}}_{x}$ system, suggesting a scaling behavior for the electronic properties that is the analog of the scaling behavior found earlier for the vibrational properties. A direct consequence of this scaling behavior is a significant reduction (relative to prior theoretical estimates within the virtual crystal approximation) of the concentration ${y}_{c}$ for a crossover from an indirect- to a direct-gap system.