Band Structure of Strained $\mathrm{Ge}_{1-x}~\mathrm{Sn}_{x}$ Alloy: A Full-Zone 30-Band ${k}\cdot{p}$ Model

We extend the previous 30-band k^.p model effectively employed for relaxed Ge_1-xSn_x alloy to the case of strained Ge_1-xSn_x alloy. The strain-relevant parameters for the 30-band k^.p model are obtained by using linear interpolation between the values of single crystal of Ge and Sn that are from literatures and optimizations. We specially investigate the dependence of band-gap at L-valley and Γ-valley with different Sn composition under uniaxial and biaxial strain along [100], [110] and [111] directions. The good agreement between our theoretical predictions and experimental data validates the effectiveness of our model. Our 30-band k^.p model and relevant input parameters successfully applied to relaxed and strained Ge_1-xSn_x alloy offers a powerful tool for the optimization of sophisticated devices made from such alloy.

I. Introduction

The Si-based optical platform has attracted considerable interests over the last decade, and its landscape is expanding rapidly with its powerful solutions such as mid-infrared lasers [1], [2], infrared LEDs [3] and photodetectors [4]. There is little doubt that Si photonics is becoming a mature technology as evidenced by its integration in large scale with complementary metal-oxide-semiconductor (CMOS) technology. With all the progress being made, this technology is currently being challenged, however, by the poor efficiency of light emission because of the fundamental material limitation - indirect bandgaps in Si, Ge and SiGe alloy that are employed as building materials for Si-based photonics. One solution that has been investigated extensively over the last decade is to alter their band structure to achieve direct band-gap through material engineering. Realizing the difference between the direct and the indirect bandgap in Si is 2.28eV while that for Ge is only 0.14eV [5], [6], much effort has been directed towards achieving direct bandgap by exploring strain conditions and/or material compositions in Ge or Ge-rich alloy with the goal to lower its -valley below its -valley.