First-Principal Investigation of Lattice Constants of Si<sub>1-<i>x</i></sub>Ge<i><sub>x</sub></i>, Si<sub>1-<i>x</i></sub>Sn<i><sub>x</sub></i> and Ge<sub>1-<i>x</i></sub>Sn<i><sub>x</sub></i>

Silicon-based materials are significant candidates for electronic and optoelectronic applications because of their high electron and hole mobility. Si1-xGex, Si1-xSnx and Ge1-xSnx are currently hot materials in the field of fabricanting silicon-based light-emitting sources. At present, GeSn has been experimentally proved to have a direct band gap structure and achieve photoluminescence. But the more practical electroluminescence has not been realized. There are two reasons of these: one is the cost of experiment is high, which makes it impossible to conduct a comprehensive and in-depth study on these materials; Additionally, the variational laws of the lattice constants have not been reported due to the lack of theoretical and experimental data. In this paper, the lattice constants and bowing factor of Si1-xGex, Si1-xSnx and Ge1-xSnx have been studied by the first-principles method based on density functional theory (DFT) combined with the Special Quasirandom Structures (SQS) and hybrid function of Heyd-Scuseria-Ernzerhof (HSE) functional correction. Comparing the calculated data with the reported theoretical and experimental data, the results show our method is more accurate. In addition, the lattice constant fitting formulas of Si1-xGex, Si1-xSnx and Ge1-xSnx are given, it shows Si1-xSnx can reduce the lattice mismatch when Si1-xSnx as the buffer between Si and GeSn alloy.