Trion ground state energy: simple results

TL;DR

This paper presents a precise calculation of the trion binding energy in 3D semiconductors using an exact three-body method, providing a simple interpolating formula valid across all mass ratios.

Contribution

It introduces an exact method for the three-body problem applied to trions and derives a universal interpolating formula for the binding energy across all mass ratios.

Findings

Perfect agreement with existing variational calculations.

Derived a simple interpolating formula for the trion binding energy.

Validated the formula across the full mass ratio range.

Abstract

We investigate the trion binding energy in a three-dimensional semiconductor, with bare Coulomb interaction between charges, and effective mass approximation for the electron and hole dispersion relations. This is done by making use of a previously proposed exact method for the three-body problem. The calculations cover the complete range of electron-to-hole mass ratio. We find a perfect agreement with existing variational calculations. Investigating the small and large mass ratio regimes, we build a three parameters interpolating formula for the trion binding energy $E_b(r)$ in terms of the exciton binding energy, where $r$ is the electron to exciton mass ratio. This formula $E_b(r)=0.71347-0.11527 \,r -0.18580\, \sqrt{1-r}\,$, in atomic units, is in full agreement, within our precision, with our numerical results over the complete range of mass ratio.