Phonon-assisted exciton/trion conversion efficiency in transition metal dichalcogenides

Abstract

Photoluminescence spectra show that monolayer transition-metal dichalcogenides (ML-TMDs) possess charged exciton binding energies, conspicuously similar to the energy of optical phonons. This enigmatic coincidence has offered opportunities to investigate many-body interactions between trion ${X}_{\ensuremath{-}}$, exciton $X$, and phonon and led to efficient excitonic anti-Stokes processes with the potential for laser refrigeration and energy harvesting. In this study, we show that in ${\mathrm{WSe}}_{2}$ materials, the trion binding energy matches two phonon modes, the out-of-plane ${A}_{1}^{\ensuremath{'}}$ and the in-plane ${E}^{\ensuremath{'}}$ modes. In this respect, using the Fermi golden rule together with the effective mass approximation, we investigate the rate of the population transfers between $X$ and ${X}_{\ensuremath{-}}$, mediated by a single phonon. We demonstrate that, while the absolute importance of the two phonon modes on the up-conversion process strongly depends on the experimental conditions such as the temperature and the dielectric environment (substrate), both modes lead to an up-conversion process on time scales in the range of few picoseconds to subnanoseconds, consistent with recent experimental findings. The conjugate process is also investigated in our study, as a function of temperature $T$ and electron density ${N}_{e}$. We prove that the exciton to trion down-conversion process is very unlikely at low electron density ${N}_{e}l{10}^{10}\phantom{\rule{4.pt}{0ex}}{\text{cm}}^{\ensuremath{-}2}$ and high temperature $Tg50\phantom{\rule{4.pt}{0ex}}\text{K}$ while it increases dramatically to reach few picoseconds time scale at low temperature and for electron density ${N}_{e}g{10}^{10}\phantom{\rule{4.pt}{0ex}}{\text{cm}}^{\ensuremath{-}2}$. Finally, our results show that the conversion processes occur more rapidly in exemplary monolayer molybdenum-based dichalcogenides (${\mathrm{MoSe}}_{2}$ and ${\mathrm{MoTe}}_{2}$) than tungsten dichalcogenides.