Many-body theory of trions in two-dimensional nanostructures

A many-body theory of trions is presented for strongly correlated systems with an analytical expression of trion binding energy being obtained. When there are extra electrons at present, an optical excitation with lower energy may occur besides the exciton peak (\(X\)), which is usually attributed to the creation of a negatively charged exciton (\(X^-\)), commonly known as a trion. The energy difference between the \(X\) and \(X^-\) peaks was commonly regarded for the trion binding energy \( \Delta _{X^-} \), which is later however proposed to be \( \Delta _{X^-} + \Delta E \) with an energy part \( \Delta E \) not accurately known for decades. In this work it is deduced that \( \Delta E = U_{ee} - \Delta _{qp}(N\text{+1 }) \) for a confined N-electron system where \( U_{ee} \) is the interaction energy of two electrons and \( \Delta _{qp}(N\text{+1 }) \) is the quasiparticle gap of the system with an extra charge. By using a configuration interaction approach, the newly developed theory is applied to study the correlated trion states in phosphorene nanostructures. The energy part \( \Delta E \) is shown to be crucial to obtain the trion binding energies that have the correct dielectric dependence. In the case of \( \text{ SiO}_2 \) substrate, our result finds that the binding energy of a negative trion in a rectangular phosphorene nanoflake with 98 atoms is around 63 meV, which agrees well with the recent experimental value of 70 meV.