Trions in two-dimensional monolayers within the hyperspherical harmonics method: Application to transition metal dichalcogenides

We develop the theoretical formalism and study the formation of valley trions in two-dimensional monolayers within the framework of a nonrelativistic potential model using the method of hyperspherical harmonics (HH) in four-dimensional space. We present the solution of the three-body Schr\"odinger equation with the Rytova-Keldysh (RK) potential by expanding the wave function of a trion in terms of the HH. We consider a long-range approximation when the RK potential is approximated by the Coulomb potential and a short-range limit when this potential is approximated by the logarithmic potential. In a diagonal approximation, the coupled system of differential equations for the hyperradial functions is decoupled in both limits. Our approach yields the analytical solution for binding energy and wave function of trions in the diagonal approximation for these two limiting cases: the Coulomb and logarithmic potentials. We obtain exact analytical expressions for eigenvalues and eigenfunctions for negatively and positively charged trions. The corresponding eigenvalues can be considered as the lower and upper limits for the trions binding energies. We apply the proposed theoretical approach to describe trions in transition metal dichalcogenides (TMDC) and address the energy difference between the binding energies of ${X}^{\ensuremath{-}}$ and ${X}^{+}$ in TMDC. Results of numerical calculations for the ground state energies with the RK potential are in good agreement with similar calculations and in reasonable agreement with experimental measurements of trion binding energies.