We investigate the mechanical, electronic, and optical properties of monolayer GeX (X= S, Se, and Te) with $\gamma$ structure based on density-functional theory calculations. We find that the mechanical anisotropy of $\gamma$-GeS is higher than those of $\gamma$-GeSe and $\gamma$-GeTe, because of its strongest ionic bond. In the unstrained case, $\gamma$-GeX is an indirect-gap semiconductor with the Mexican-hat (MH) dispersion in the valence band. By applying tensile and biaxial strains, both energy band gap and valley positions are controlled. On the other hand, by applying an external electric field, the energy band gap is decreased to zero because of the downward interlayer band. We also find that the in-plane optical absorptions of monolayer $\gamma$-GeX in the visible-light region are comparable to that of the monolayer transition metal dichalcogenides such as MoS${}_2$. Due to the unique structure in the $z$ direction, the monolayer $\gamma$-GeX also shows a high value of the out-of-plane optical absorptions. The strain engineering significantly modifies the optical absorption in the visible light, while the effect of the external electric field on the optical properties is weak. Our results will be helpful to design the electro-optical devices based on monolayer materials with MH band.

我们研究了单层 GeX（X= S、Se 和 Te）的机械、电子和光学特性 $\gamma$基于密度泛函理论计算的结构。我们发现机械各向异性$\gamma$-GeS 高于 $\gamma$-GeSe 和 $\gamma$-GeTe，因为它的离子键最强。在不紧张的情况下，$\gamma$-GeX 是一种间接带隙半导体，在价带具有墨西哥帽 (MH) 色散。通过施加拉伸和双轴应变，能带隙和谷位置都得到控制。另一方面，通过施加外部电场，由于层间带向下，能带隙减小到零。我们还发现单层的面内光学吸收$\gamma$-GeX 在可见光区域与单层过渡金属二硫化物如 MoS2 相当${}_2$. 由于其独特的结构$z$ 方向，单层 $\gamma$-GeX 还显示出面外光吸收的高值。应变工程显着改变了可见光中的光学吸收，而外部电场对光学性质的影响较弱。我们的研究结果将有助于设计基于MH波段单层材料的电光器件。