Radiative Carrier Lifetime in Ge$_{1-x}$Sn$_x$ Mid-Infrared Emitters

Ge$_{1-x}$Sn$_x$ semiconductors hold the premise for large-scale, monolithic mid-infrared photonics and optoelectronics. However, despite the successful demonstration of several Ge$_{1-x}$Sn$_x$-based photodetectors and emitters, key fundamental properties of this material system are yet to be fully explored and understood. In particular, little is known about the role of the material properties in controlling the recombination mechanisms and their consequences on the carrier lifetime. Evaluating the latter is in fact fraught with large uncertainties that are exacerbated by the difficulty to investigate narrow bandgap semiconductors. To alleviate these limitations, herein we demonstrate that the radiative carrier lifetime can be obtained from straightforward excitation power- and temperature- dependent photoluminescence measurements. To this end, a theoretical framework is introduced to simulate the measured spectra by combining the band structure calculations from the k.p theory and the envelope function approximation (EFA) to estimate the absorption and spontaneous emission. Based on this model, the temperature-dependent emission from Ge$_{0.83}$Sn$_{0.17}$ samples at a biaxial compressive strain of $-1.3\%$ was investigated. The simulated spectra reproduce accurately the measured data thereby enabling the evaluation of the steady-state radiative carrier lifetimes, which are found in the 3-22 ns range for temperatures between 10 and 300 K at an excitation power of 0.9 kW/cm$^2$. For a lower power of 0.07 kW/cm$^2$, the obtained lifetime has a value of 1.9 ns at 4 K. The demonstrated approach yielding the radiative lifetime from simple emission spectra will provide valuable inputs to improve the design and modeling of Ge$_{1-x}$Sn$_x$-based devices.