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Compositional dependence of the direct and indirect band gaps in $\text{Ge}_{1-y}\text{Sn}_{y}$ alloys from room temperature photoluminescence

The compositional dependence of the lowest direct and indirect band gaps in $\text{Ge}_{1-y}\text{Sn}_{y}$ has been determined from room-temperature photoluminescence measurements. This technique is particularly attractive for a comparison of the two transitions because distinct features in the spectra can be associated with the direct and indirect gaps. However, detailed modeling of these room temperature spectra is required to extract the band gap values with the high accuracy required to determine the Sn concentration $y_{c}$ at which the alloy becomes a direct gap semiconductor. For the direct gap, this is accomplished using a microscopic model that allows the determination of direct gap energies with meV accuracy. For the indirect gap, it is shown that current theoretical models are inadequate to describe the emission properties of systems with close indirect and direct transitions. Accordingly, an ad hoc procedure is used to extract the indirect gap energies from the data. For $y$ < 0.1 the resulting direct gap compositional dependence is given by $ΔE_{0} = -(3.57 \pm 0.06)y$ (in eV). For the indirect gap, the corresponding expression is $ΔE_{\text{ind}} = -(1.64\pm 0.10)y$ (in eV). If a quadratic function of composition is used to express the two transition energies over the entire compositional range $0 \leq y \leq 1$, the quadratic (bowing) coefficients are found to be $b_{0} = 2.46 \pm 0.06$ eV (for $E_{0}$) and $b_{\text{ind}} = 0.99 \pm 0.11$ eV (for $E_{\text{ind}}$). These results imply a crossover concentration $y_{c} = 0.073_{-0.006}^{+0.007}$, much lower than early theoretical predictions based on the virtual crystal approximation, but in better agreement with predictions based on large atomic supercells.