Direct bandgap of group IV semiconductors by uni-axial stress

We theoretically examine the possibility of converting typical group IV semiconductors Si, SiGe (zinc blende), and Ge into direct bandgap materials by uniaxial stress along the $<$111$>$ and $<$100$>$ directions. For silicon, the required tensile strain is too large to be practical. For SiGe and Ge along $<$111$>$, although band splitting at the L point lowers the conduction band edge at L, a direct bandgap can still be achieved through a supralinear decrease in the energy of the conduction band edge at $\Gamma $. The required longitudinal strains along $<$111$>$ are 8{\%} and 4{\%} for GeSi and Ge, respectively. For strain along $<$100$>$, the position of the conduction band edge at the $\Gamma $ point varies sub-linearly with strain; therefore strain along $<$100$>$ is less efficient: GeSi is unlikely to achieve a direct gap by extension along $<$100$>$ and Ge requires a 6{\%} longitudinal strain. The full dependence of the indirect/direct transition on arbitrary combinations of uniaxial/hydrostatic tensile strain is given for both GeSi and Ge.