Energetic comparison of exciton gas versus electron-hole plasma in a bilayer two-dimensional electron-hole system

We study the zero-temperature phase diagram of a symmetric electron-hole bilayer system by comparing the ground-state energies of two distinct limiting cases, characterized by an electron-hole plasma or an exciton gas, respectively. For the electron-hole plasma, the random-phase approximation is used; for the exciton gas, we consider three different approximations: the unscreened Coulomb interaction, the statically screened one, and the dynamically screened one under the plasmon-pole approximation. Our results suggest that the exciton gas is stable at small layer separation. However, static screening in general suppresses the formation of excitons, and dynamic screening gives different results depending on the representative energy scale we used in the plasmon-pole approximation. We conclude that energetic considerations alone are very sensitive to the approximation schemes, and the phase diagram of the system may depend crucially on exactly how the electron-hole attraction is treated in the theory. For very small and very large densities, however, all our approximations show the exciton gas to have lower energy than the plasma.

Figure 1

The Feynman diagrams used in the calculation of the eh plasma energy. Red solid line: electron; green solid line: hole; red wavy line: Coulomb interaction between holes; green wavy line: Coulomb interaction between electrons; black wavy line: interlayer Coulomb interaction. (a) The Fock energy. (b) The second-order exchange correlation energy $E_2^{\text{(b)}}$. (c) The RPA correlation energy $E_{\text{corr}}$, which includes all ring diagrams of second or higher orders.

Figure 2

The diagrams for the Bethe-Salpeter equation describing the excitons. (a) Unscreened electron-hole Coulomb coupling; (b) screened Coulomb coupling; (c) the screening approximation in RPA. Here the solid lines with arrows are the electron and hole propagators in each layer (in the single exciton limit), and the wavy/double wavy lines are the unscreened and screened interlayer electron-hole coupling between the two layers with the polarization bubble being the screening function approximated in either the zero-frequency limit (“static screening”) or a fixed finite-frequency limit (“plasmon pole screening”) as described in the main text.

Figure 3

Ground-state energy per electron/hole of bilayer electron-hole system as a function of $d$ and of $r_s$ under different assumptions. In each subfigure, the six curves represent the assumptions that the system is electron-hole plasma (gray), exciton gas with unscreened Coulomb interaction (blue), statically screened Coulomb interaction (red), and the plasmon-pole approximated dynamically screened Coulomb interaction with external frequency $\omega$ being the unscreened exciton binding energy (light green), the Fermi energy of the eh plasma (green), and the plasmon energy at the Fermi momentum (dark green). For very large $r_s$, the eh plasma curves will become positive again (not shown).

Figure 4

Phase diagrams of the bilayer electron-hole system obtained from our models. The five subfigures describe different screening models of the interlayer Coulomb interaction used in the calculation of excitonic binding energy. The “phase” here is determined by comparing the ground-state energy assuming the two extremes, so the blue (white) region corresponds to where the eh plasma energy per electron/hole is lower (higher) than the exciton binding energy.