Ultimate limits of biaxial tensile strain and n-type doping for realizing an efficient low-threshold Ge laser

We theoretically investigate the methodology involved in the minimization of the threshold of a Ge-on-Si laser and maximization of the slope efficiency in the presence of both biaxial tensile strain and n-type doping. Our findings suggest that there exist ultimate limits beyond which no further benefit can be realized through increased tensile strain or n-type doping. In this study, we quantify these limits, showing that the optimal design for minimizing threshold involves approximately 3.7% biaxial tensile strain and 2 × 1018 cm−3 n-type doping, whereas the optimal design for maximum slope efficiency involves approximately 2.3% biaxial tensile strain with 1 × 1019 cm−3 n-type doping. Increasing the strain and doping beyond these limits will degrade the threshold and slope efficiency, respectively.