Controllable trions in buckled two-dimensional materials

We predict the formation of intravalley and intervalley controllable trions in buckled two-dimensional (2D) materials such as silicene, germanene, and stanene monolayers in an external electric field. A study is performed within the framework of a nonrelativistic potential model using the method of hyperspherical harmonics (HH). We solve the three-body Schrödinger equation with the Rytova-Keldysh potential by expanding the wave functions of a trion in terms of the HH. A resultant system of coupled differential equations is solved numerically. Controllable ground-state energies of intravalley and intervalley trions by the external electric field are presented. The dependencies of the binding energy (BE) of trions in silicene, germanene, and stanene as a function of the electric field are qualitatively similar. BEs of trions formed by $A$ and $B$ excitons have a non-negligible difference that increases slightly as the electric field increases. It is demonstrated that trion BEs can be controlled by the external electric field, and the dielectric environment has a significant effect on the trion BE. The capability to control the BE and compactness of trions in buckled 2D materials by an external electric field suggests a possible trions crystallization in Xenes.