Electronic structure of lonsdaleite $\mathrm{Si}_{x}\mathrm{Ge}_{1-x}$ alloys

Conventional diamond-structured silicon (Si) and germanium (Ge) possess indirect fundamental band gaps, limiting their potential for applications in light-emitting devices. However, $\mathrm{Si}_{x}\mathrm{Ge}_{1-x}$ alloys grown in the lonsdaleite (“hexagonal diamond”) phase have recently emerged as a promising direct-gap, Si-compatible material system, with experimental measurements demonstrating strong room temperature photoluminescence. When grown in the lonsdaleite phase, Ge possesses a narrow ( $\sim 0.3 \mathrm{eV}$ ) “pseudo-direct” fundamental band gap. Lonsdaleite Si is indirect-gap ( $\sim 0.8 \mathrm{eV}$ ), creating the possibility to achieve direct-gap lonsdaleite $\mathrm{Si}_{x}\mathrm{Ge}_{1-x}$ alloys across a Ge-rich composition range. We present a first principles analysis of the electronic and optical properties of lonsdaleite $\mathrm{Si}_{x}\mathrm{Ge}_{1-x}$ alloys, elucidate the electronic structure evolution and direct- to indirect-gap transition, and describe the impact of alloy band mixing effects on inter-band optical transition strengths.