Rigorous theory of the radiative and gain characteristics of silicon and germanium lasing media

Group IV Direct Band Gap Photonics: Methods, Challenges, and Opportunities

“…Due to the fact that the Γ-valley energy reduces faster than the one of the L-valley, Ge transforms into a direct band gap semiconductor at ~4.7% uniaxial strain along [100] when the direct transition (black line) decreases below the energy of the indirect recombination (green line). For Ge under biaxial tensile strain or GeSn alloys, the band edges behave similarly with an indirect-to-direct band gap crossover at ~1.6-2.0% strain (El Kurdi et al, 2010;Virgilio et al, 2013;Wen and Bellotti, 2015) or and at a Sn-content of ~9% (Low et al, 2012;Gupta et al, 2013b;Wirths et al, 2015) for a fully relaxed layer.…”

“…Hence, loss processes related to these transitions will become critical. Furthermore, the gain as predicted by a Green's functional approach (Wen and Bellotti, 2015) tend to be smaller than the commonly used joint density of state formalism as applied for …”

“…The latter is not restricted to the Brillouin-zone center but describes the full energy dispersion. In other works, the empirical pseudopotential method (Dutt et al, 2013;Wen and Bellotti, 2015), density functional theory (Tahini et al, 2012), and the tight-binding model (Dutt et al, 2012) are employed. The agreement between the models is generally found to be satisfactory.…”

“…As mentioned above, due to the lack of experimental data for direct gap Ge or GeSn, the energy dependence of the loss as measured by pump-probe experiments for Ge (Carroll et al, 2012) has been used for Figure 4. Its proper inclusion possibly adds significant contribution to the loss (Wen and Bellotti, 2015).…”

“…This can be understood from the Drude dependence of the free carrier absorption modified by dipole allowed intervalence band transitions. Because the emission wavelength increases when approaching the direct band gap configuration, and the initially degenerate heavy and light hole bands split due to strain, the parasitic absorption will strongly increase in direct band gap systems and may, thus, obstruct the efficiency of lasing (Wen and Bellotti, 2015). Applying the above introduced optical characterization methods should allow to investigate these effects in detail, which, together with evolving theoretical results, will enable to complete our understanding.…”