Radiative carrier lifetime in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:msub><mml:mi>Ge</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Sn</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:math> midinfrared emitters

${\mathrm{Ge}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}$ semiconductors have promise for large-scale, monolithic, midinfrared photonics and optoelectronics. However, despite the successful demonstration of several ${\mathrm{Ge}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}$-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 for the carrier lifetime. Evaluating the latter is in fact fraught with large uncertainties that are exacerbated by the difficulty in investigating narrow-band-gap semiconductors. To alleviate these limitations, herein we demonstrate that the behavior of the radiative carrier lifetime can be evaluated 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 to estimate the absorption and spontaneous emission. The model computes explicitly the momentum matrix element to estimate the strength of the optical transitions in single bulk materials, unlike the joint density of states model that assumes a constant matrix element. Based on this model, the temperature-dependent emission from ${\mathrm{Ge}}_{0.83}{\mathrm{Sn}}_{0.17}$ samples at a biaxial compressive strain of $\ensuremath{-}1.3\mathrm{%}$ is 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 range 3--22 ns for temperatures between $10$ and 300 K at an excitation power of $0.9\phantom{\rule{0.2em}{0ex}}\mathrm{kW}/{\mathrm{cm}}^{2}$. For a lower power of $0.07\phantom{\rule{0.2em}{0ex}}\mathrm{kW}/{\mathrm{cm}}^{2}$, the obtained lifetime has a value of $1.9\phantom{\rule{0.2em}{0ex}}\mathrm{ns}$ 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 ${\mathrm{Ge}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}$-based devices.

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