  We theoretically investigate how the threshold of a Ge-on-Si laser can be
minimized and how the slope efficiency can be maximized in presence of both
biaxial tensile strain and n-type doping. Our finding shows that there exist
ultimate limits beyond which point no further benefit can be realized through
increased tensile strain or n-type doping. Here were quantify these limits,
showing that the optimal design for minimizing threshold involves about 3.7%
biaxial tensile strain and 2x1018 cm-3 n-type doping, whereas the optimal
design for maximum slope efficiency involves about 2.3% biaxial tensile strain
with 1x1019 cm-3 n-type doping. Increasing the strain and/or doping beyond
these limits will degrade the threshold or slope efficiency, respectively.
