Exciton-Trion Polaritons in Doped Two-Dimensional Semiconductors

We present a many-body theory of exciton-trion polaritons (ETPs) in doped two-dimensional semiconductor materials. ETPs are robust coherent hybrid excitations involving excitons, trions, and photons. In ETPs, the 2-body exciton states are coupled to the material ground state via exciton-photon interaction, and the 4-body trion states are coupled to the exciton states via Coulomb interaction. The trion states are not directly optically coupled to the material ground state. The energy-momentum dispersion of ETPs exhibit three bands. We calculate the energy band dispersions and the compositions of ETPs at different doping densities using Green’s functions. The energy splittings between the polariton bands, as well as the spectral weights of the polariton bands, depend on the strength of the Coulomb coupling between the excitons and the trions, which in turn depends sensitively on the doping density. The doping density dependence of the ETP bands and the charged nature of the trion states could enable novel electrical and optical control of ETPs.

Figure 1

(a) The nature of couplings among bound and unbound trion states, exciton states, the material ground state, and photons in exciton-trion polaritons (ETPs) is depicted for an electron-doped 2D material (${\mathrm{MoSe}}_2$) [21, 22]. (b) A 2D material monolayer embedded inside an optical microcavity.

Figure 2

Calculated real part of the optical conductivity, $\sigma (\overrightarrow{Q}=0,\omega )$, for in-plane (TE) light polarization, is plotted for three different electron densities ($n_e={10}^{10},2\times {10}^{12},8\times {10}^{12}{\mathrm{cm}}^{-2}$) for electron-doped monolayer 2D ${\mathrm{MoSe}}_2$. 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 energy $E_{n=0,s}^{\mathrm{ex}}(\overrightarrow{Q}=0)$. The position of the cavity optical mode is also indicated (see Fig. 3). Two prominent peaks are seen in the absorption spectra when the electron density exceeds $\sim {10}^{12}{\mathrm{cm}}^{-2}$. Each peak corresponds to a state that is a superposition of the exciton and trion states [21]. The spectral weight shifts from the higher energy peak to the lower energy peak with the increase in the electron density.

Figure 3

Calculated exciton-trion polariton (ETP) energy dispersions (dashed lines) and the spectral densities of the photon [$S^{\mathrm{ph}}(\overrightarrow{Q},\omega )$], the transverse exciton [$S_{n=0,T}^{\mathrm{ex}}(\overrightarrow{Q},\omega )$], and the transverse bound trion [$S_{n=0,m=0,T}^{\mathrm{tr}}(\overrightarrow{Q},\omega )$], are plotted for three different electron densities ($n_e={10}^{10},2\times {10}^{12},8\times {10}^{12}{\mathrm{cm}}^{-2}$) for an electron-doped monolayer 2D ${\mathrm{MoSe}}_2$ inside an optical cavity [Fig. 1]. In each case, the cavity optical mode is tuned $\sim 20\mathrm{meV}$ below the lower energy peak in the optical absorption spectra (as indicated in Fig. 2). $T=5\mathrm{K}$. The unit in the color bar is ${10}^{-13}\mathrm{s}$.