Composite excitonic states in doped semiconductors

We present a theoretical model of composite excitonic states in doped semiconductors. Many-body interactions between a photoexcited electron-hole pair and the electron gas are integrated into a computationally tractable few-body problem, solved by the variational method. We focus on electron-doped monolayer (ML)-$\mathrm{Mo}{\mathrm{Se}}_2$ and ML-$\mathrm{W}{\mathrm{Se}}_2$ due to the contrasting character of their conduction bands. In both cases, the core of the composite is a tightly bound trion (two electrons and a valence-band hole) surrounded by a region depleted of electrons. The composite in ML-$\mathrm{W}{\mathrm{Se}}_2$ further includes a satellite electron with different quantum numbers. The theory is general and can be applied to semiconductors with various energy-band properties, allowing one to calculate their excitonic states and to quantify the interaction with the Fermi sea.

Figure 1

Composite excitonic states in electron-doped monolayer (ML)-$\mathrm{Mo}{\mathrm{Se}}_2$ and ML-$\mathrm{W}{\mathrm{Se}}_2$ following circularly polarized photoexcitation in the $-K$ valley. The left diagrams in (a) and (b) show the corresponding $k$-space configurations. The right diagrams correspond to real-space configurations, showing a core trion surrounded by conduction band (CB) hole(s). The composite in ML-$\mathrm{W}{\mathrm{Se}}_2$ is further accompanied by a satellite electron. Each electron in these composites comes with distinct spin and valley quantum numbers.

Figure 2

(a) Tetron energy vs Fermi energy ($E_{\text{F}}\approx 5$ meV amounts to electron density $n={10}^{12}{\mathrm{cm}}^{-2}$). The dotted line is the contribution from the core trion [i.e., without bandgap renormalization (BGR) and the conduction band (CB) hole]. (b) $k$-space density distributions of particles in the tetron when $E_{\text{F}}=19$ meV. (c) Left: Binding energy of the satellite electron to the hexciton. Right: Overlap between valence band (VB) hole and satellite electron. (d) Interparticle distances in the hexciton, showing average distances within the core trion ($r_{he}$ and $r_{ee}$), between the VB hole and top-valley satellite electron ($r_{ht}$), and between the VB and CB holes ($r_{h\overline{e}}$).