Many-body theory of radiative lifetimes of exciton-trion superposition states in doped two-dimensional materials

Optical absorption and emission spectra of doped two-dimensional (2D) materials exhibit sharp peaks that are often mistakenly identified with pure excitons and pure trions (or charged excitons), but both peaks have been recently attributed to superpositions of two-body exciton and four-body trion states and correspond to the approximate energy eigenstates in doped 2D materials. In this paper, we present the radiative lifetimes of these exciton-trion superposition energy eigenstates using a many-body formalism that is appropriate given the many-body nature of the strongly coupled exciton and trion states in doped 2D materials. Whereas the exciton component of these superposition eigenstates are optically coupled to the material ground state and can emit a photon and decay into the material ground state provided the momentum of the eigenstate is within the light cone, the trion component is optically coupled only to the excited states of the material and can emit a photon even when the momentum of the eigenstate is outside the light cone. In an electron-doped 2D material, when a four-body trion state with momentum outside the light cone recombines radiatively, and a photon is emitted with a momentum inside the light cone, the excess momentum is taken by an electron-hole pair left behind in the conduction band. The radiative lifetimes of the exciton-trion superposition states with momenta inside the light cone are found to be in the few hundred femtoseconds to a few picoseconds range and are strong functions of the doping density. The radiative lifetimes of exciton-trion superposition states with momenta outside the light cone are in the few hundred picoseconds to a few nanoseconds range and are again strongly dependent on the doping density. The doping density dependence of the radiative lifetimes of the two peaks in the optical emission spectra follows the doping density dependence of the spectral weights of the same two peaks observed in the optical absorption spectra, as both have their origins in the Coulomb coupling between the excitons and trions in doped 2D materials.

Figure 1

The nature of couplings involving two-body exciton and four-body trion states are depicted for an electron-doped material. The four-body trion states are coupled to the two-body exciton states via electron-electron and electron-hole Coulomb interactions. Only the exciton states are coupled to the material ground state via optical coupling. The trion states are optically coupled to excited states of the material consisting of a CB electron-hole pair. The trion states include both bound and unbound trion states.

Figure 2

A 2D TMD monolayer in the $z=0$ plane is shown. The two light polarizations are also illustrated.

Figure 3

Two different kinds of photon emission processes are depicted. (a) Photon emission resulting in a decay of the energy eigenstate into the material ground state. The transition rate is determined by $|{\alpha }_n{|}^2$, the weight of the exciton component of the energy eigenstate in Eq. (3). This transition is possible only if the momentum $\vec{Q}$ of the energy eigenstate is within the light cone. (b) Photon emission resulting in a decay of the energy eigenstate into an excited state of the material that has a CB electron-hole pair. The CB electron-hole pair is left behind after photon emission from the trion components of the energy eigenstate. The transition rate is determined by $|{\beta }_{m,s^{\prime }}{|}^2$ in Eq. (3). This transition is possible even if the momentum $\vec{Q}$ of the energy eigenstate is outside the light cone.

Figure 4

Calculated real part of the optical conductivity, ${\sigma }_{0,s}(\vec{Q}=0,\omega )$, for in-plane light polarization is plotted for different electron densities for electron-doped monolayer 2D $\mathrm{Mo}{\mathrm{Se}}_2$. Only the lowest energy exciton state is considered in the calculations. The spectra are all normalized to the peak optical conductivity value at zero electron density. $T=5\mathrm{K}$. The frequency axis is offset by the exciton eigenenergy $E_0^{\text{ex}}(\vec{Q}=0,s)$ of the two-body Schrödinger equation. Two prominent peaks are seen in the spectra. Each peak corresponds to an energy eigenstate state that is a superposition of exciton and trion states, as shown in Eq. (3). Figure is reproduced from the paper by Rana et al. [12].

Figure 5

The zero-momentum radiative lifetimes,${\tau }_{n=0,s}^{\text{lo}}(\vec{Q}=0)$ and ${\tau }_{n-0,s}^{\text{hi}}(\vec{Q}=0)$, of the lower and higher energy eigenstates, respectively, of the coupled exciton-trion system (and corresponding to the lower and higher energy peaks in the optical absorption spectra in Fig. 4) are plotted as a function of the electron densities for an electron-doped monolayer 2D $\mathrm{Mo}{\mathrm{Se}}_2$ suspended in air. $T=5\mathrm{K}$. The inset shows the same data on a linear scale.

Figure 6

The radiative lifetimes,${\tau }_{n=0,s}^{\text{lo}}\left(\vec{Q}\right)$ and ${\tau }_{n-0,s}^{\text{hi}}\left(\vec{Q}\right)$, of the lower and higher energy eigenstates, respectively, of the coupled exciton-trion system (and corresponding to the lower and higher energy peaks in the optical absorption spectra in Fig. 4) are plotted as a function of the in-plane momentum $Q$ for different electron densities (${10}^{12}{\mathrm{cm}}^{-2}$ and $6\times {10}^{12}{\mathrm{cm}}^{-2}$) for an electron-doped monolayer 2D $\mathrm{Mo}{\mathrm{Se}}_2$ suspended in air. $T=5\mathrm{K}$.

Figure 7

The radiative lifetimes,${\tau }_{n=0,s}^{\text{lo}}\left(\vec{Q}\right)$ and ${\tau }_{n-0,s}^{\text{hi}}\left(\vec{Q}\right)$, of the lower and higher energy eigenstates, respectively, of the coupled exciton-trion system (and corresponding to the lower and higher energy peaks in the optical absorption spectra in Fig. 4) are plotted as a function of the in-plane momentum $Q$ for different electron densities (${10}^{12}{\mathrm{cm}}^{-2}$ and $6\times {10}^{12}{\mathrm{cm}}^{-2}$) for an electron-doped monolayer 2D $\mathrm{Mo}{\mathrm{Se}}_2$ suspended in air. The lifetimes shown correspond to the process depicted in Fig. 3 for radiative decay into excited states of the material. $T=5\mathrm{K}$. The lifetimes shown are three to four orders of magnitude longer than the lifetimes shown earlier in Fig. 6 for the process depicted in Fig. 3 for radiative decay into the material ground state.

Figure 8

The radiative lifetimes,${\tau }_{n=0,s}^{\text{lo}}\left(\vec{Q}\right)$ and ${\tau }_{n-0,s}^{\text{hi}}\left(\vec{Q}\right)$, of the lower and higher energy eigenstates, respectively, of the coupled exciton-trion system (and corresponding to the lower and higher energy peaks in the optical absorption spectra in Fig. 4) are plotted as a function of the electron densities for an electron-doped monolayer 2D $\mathrm{Mo}{\mathrm{Se}}_2$ suspended in air. $T=5\mathrm{K}$. The momentum value is chosen to be just outside the light cone $Q\sim {10}^7$ 1/m. The lifetimes shown correspond to the process depicted in Fig. 3 for radiative decay into the excited states of the material.

Figure 9

Certain processes that have been proposed in the literature for photon emission involving excitons and trions in electron-doped materials are depicted. (a) Photon emission process involving a three-body trion state in which the CB electron recombines with the VB hole leaving behind another CB electron which is deposited outside the Fermi sea [1, 20, 30]. (b) Photon emission process involving an exciton in which an uncorrelated CB electron from the Fermi sea recombines with the VB hole, leaving behind an electron-hole pair in the CB [31]. (c) Photon emission process involving a trion in which an uncorrelated CB electron from the Fermi sea recombines with the VB hole, leaving behind two electron-hole pairs in the CB [31].