We calculate the band structures of \(\hbox {Ge}_{1-x}\hbox {Sn}_{x}\) nanowires under the [110] direction stress via the effective-mass \({\textbf {k}} \cdot {\textbf {p}}\) theory. Interestingly, it is found that octuple equivalent indirect L-valleys can be split into \(L_1\) -valleys and \(L_2\) -valleys with quadruple degeneracy by the imposed stress, which results in the inflection point of electron filling ratio at the direct \(\Gamma \) -valley whether Sn content is low or high. Moreover, the optical gain along the z direction will not only appear an inflection point, but also can be enhanced several times with the increase of the stress, which will be much greater than that along the x direction. For \(\hbox {Ge}_{0.92}\hbox {Sn}_{0.08}\) nanowires, the result indicates that the z direction peak gain will invariably exceed the FCA loss as increasing the stress to 4 GPa when the total electron concentration reaches the order of \(10^{19}\) \(\hbox {cm}^{-3}\) , which is obviously better than the case of \(\hbox {Ge}_{0.95}\hbox {Sn}_{0.05}\) nanowires due to the larger electron filling ratio at the \(\Gamma \) -valley. While for \(\hbox {Ge}_{1-x}\hbox {Sn}_{x}\) nanowires with high Sn contents, the FCA loss is almost negligible and the considerable positive net peak gain can be acquired under the stress even if the total electron concentration is as low as the order of \(10^{18} \, \hbox {cm}^{-3}\) . Therefore, in the field of silicon photonics, our researches manifest that [110] direction strained \(\hbox {Ge}_{1-x}\hbox {Sn}_{x}\) nanowires as lasers are advantageous over unstrained \(\hbox {Ge}_{1-x}\hbox {Sn}_{x}\) nanowires, but inferior to [100] direction strained \(\hbox {Ge}_{1-x}\hbox {Sn}_{x}\) nanowires because of the splitting of the L-valleys.
