Strong-Coupling Phases of Trions and Excitons in Electron-Hole Bilayers at Commensurate Densities

We introduce density imbalanced electron-hole bilayers at a commensurate $2:1$ density ratio as a platform for realizing novel phases of electrons, excitons, and trions. Through the independently tunable carrier densities and interlayer spacing, competition between kinetic energy, intralayer repulsion, and interlayer attraction yields a rich phase diagram. By a combination of theoretical analysis and numerical calculation, we find a variety of strong-coupling phases in different parameter regions, including quantum crystals of electrons, excitons, and trions. We also propose an “electron-exciton supersolid” phase that features electron crystallization and exciton superfluidity simultaneously. The material realization and experimental signature of these phases are discussed in the context of semiconductor transition metal dichalcogenide bilayers.

Figure 1

The classical phase diagram of the electron-hole bilayer at $n_e/n_h=2$ for $a/d>1$. The top depicts the checkerboard composite crystal (left) and the honeycomb composite crystal (right). The bottom shows the energy and energy difference of the two composite crystals, with a dashed line indicating the phase transition.

Figure 2

Soft and hard optical phonon frequencies at the center of the Brillouin zone as function of $a/d$ for the classical composite crystals. The low-frequency branches correspond to the displacement of the dipole’s center of mass relative to the electron, while the high-frequency branches are associated with oscillations in the dipole’s internal structure.

Figure 3

The electron (hole) density is shown in the left (right) panel. The coloring scheme is consistent for both densities and is indicated by the color bar in units of $1/a_{B,h}^2$ for $a_{B,h}=(4\pi \epsilon {\hslash }^2/e^2{m}_h)$. The calculation parameters are $m_e=0.8m_0$, $m_h=0.4m_0$, $d=4.83a_{B,h}$, hole $r_s=8$, $n_h=36$, and equal electron spin populations.

Figure 4

Schematic phase diagram for the electron-hole bilayer at commensurate densities $n_e/n_h=2$, with analytically known boundaries marked in red and limiting phases labeled.