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

The gist The exciton gas is stable at small layer separation, while energetic considerations alone are very sensitive to the approximation schemes used in comparing it to an electron-hole plasma <ref:2402.00866#pg2>.

System Setup and Comparison

The study investigates the zero-temperature phase diagram of a symmetric electron-hole bilayer system by comparing the ground state energies of an electron-hole plasma or an exciton gas <ref:2402.00866#pg2>. The comparison involves two distinct limiting cases, characterized by an electron-hole plasma or an exciton gas, respectively <ref:2402.00866#pg2>. For the electron-hole plasma, the random phase approximation is used <ref:2402.00866#pg2>. For the exciton gas, three different approximations are considered: the unscreened Coulomb interaction, the statically screened one, and the dynamically screened one under the plasmon-pole approximation <ref:2402.00866#pg2>.

Theoretical Framework

The theoretical model considers a two-dimensional, two-layer system with no interlayer tunneling <ref:2402.00866#pg2>. The layers are separated by a distance d and are coupled by the electron-hole Coulomb interaction <ref:2402.00866#pg2>. Both electrons and holes have parabolic dispersion with mass m, and all particles interact via the Coulomb interaction <ref:2402.00866#pg2>. The system is considered in a balanced situation with equal electron-hole density in the layers <ref:2402.00866#pg2>. The dimensionless density parameter rs is defined by rs = S / (πa squared BN) <ref:2402.00866#pg2>. The Hamiltonian includes kinetic energy, interaction between two particles, interaction between a particle and a charge-compensating background, and interaction within the charge-compensating backgrounds <ref:2402.00866#pg2>.

Plasma Energy Calculation

The ground state energy per electron/hole for the electron-hole plasma is calculated assuming the system is an electron-hole plasma in excitonic units <ref:2402.00866#pg2>. The expression for this energy includes kinetic energy, Fock (exchange) term, second-order exchange energy, and the ring contribution (RPA) of the correlation energy <ref:2402.00866#pg2>. The characteristic features of the calculated plasma ground state energy are that it is featureless and only weakly d-dependent except for d < rs, and it is a strong nonmonotonic function of rs with a minimum around rs ∼ 2 indicating a manifestly stable plasma phase for rs ∼ 2 <ref:2402.00866#pg2>.

Exciton Gas Calculation

The exciton gas limit estimates the ground state energy per particle from the calculated binding energy of individual excitons, ignoring exciton-exciton interactions but including screening effects at an effective one-particle mean field level <ref:2402.00866#pg2>. The calculation involves solving the hydrogen-like time-independent Schrödinger equation for the electron-hole pair <ref:2402.00866#pg2>. Different screening models are considered, including unscreened, static, and dynamically screened interactions <ref:2402.00866#pg2>.

Phase Diagram Results

The comparison between the ground state energies of the two extreme cases constructs a qualitative phase diagram for which phase is energetically lower as a function of 2D electron/hole density and layer separation <ref:2402.00866#pg2>. The results show that excitons are preferred at small d in the unscreened model, but static screening suppresses the formation at all d (except for very small or very large rs) <ref:2402.00866#pg2>. Dynamical screening gives a result either similar to the unscreened case or the statically screened case, depending on what representative energy scale is used <ref:2402.00866#pg2>. For small rs (high density), the exciton gas always has lower energy compared to the electron-hole plasma for all layer separations, and this also holds generically for very large rs (very low density) <ref:2402.00866#pg2>. The theory predicts a stable eh plasma around rs ∼ 2 because of a minimum in the plasma energy around this rs value, but for very large rs, the exciton gas appears to be the preferred ground state of the system for arbitrary d <ref:2402.00866#pg2>.

Conclusion

The study concludes that nontrivial many-body interaction physics plays a key role in deciding the stability of the exciton gas, suggesting that it is not trivial to assume that exciton formation is always preferred in such bilayer systems <ref:2402.00866#pg2>. The exciton gas is quite generally stable for high (small rs) and low (large rs) densities, but the plasma phase may be stable for intermediate rs although this may be an artifact of the theory being inaccurate for both energetics in that regime <ref:2402.00866#pg2>. The formation of a superfluid excitonic condensate is beyond the scope of the current work, where only two independent limiting scenarios are compared <ref:2402.00866#pg2>.

Figure Descriptions

Figure 3 shows the calculated E/N, the ground state energy per electron/hole (equivalent to per exciton for the exciton gas) under various assumptions or various parameters, particularly as a function of density rs and interlayer separation d <ref:2402.00866#pg2>. Figure 4 shows phase diagrams of the bilayer electron-hole system obtained from our models, where the blue (white) region corresponds to where the eh plasma energy per electron/hole is lower (higher) than the exciton binding energy <ref:2402.00866#pg2>. In each subfigure, the six curves represent assumptions describing whether the system is an electron-hole plasma or an exciton gas under different screening models <ref:2402.00866#pg2>. The final comparison shows that the energy scale of dynamic screening is able to tune between unscreened (with stable excitons) and statically screened cases (without stable excitons) <ref:2402.00866#pg2>. The paper concludes that a more accurate model is required to settle down the problem of exciton formation <ref:2402.00866#pg2>. The study is supported by the Laboratory for Physical Sciences <ref:2402.00866#pg2>.

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Energetic comparison of exciton gas versus electron-hole plasma in a bilayer two-dimensional electron-hole system Yi-Ting Tu, Seth M. Davis, and Sankar Das Sarma Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 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 <ref:2402.00866#pg2>. 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 <ref:2402.00866#pg2>. Our results suggest that the exciton gas is stable at small layer separation <ref:2402.00866#pg2>. 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 plasmonpole approximation <ref:2402.00866#pg2>. 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 <ref:2402.00866#pg2>. For very small and very large densities, however, all our approximations show the exciton gas to have lower energy than the plasma <ref:2402.00866#pg2>.

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We consider a two-dimensional, two-layer system, with one layer containing (spinless) electrons and the other containing (spinless) holes with no interlayer tunneling <ref:2402.00866#pg2>. The layers are separated by a distance d, and are coupled by the electron-hole Coulomb interaction <ref:2402.00866#pg2>. We take both the electrons and holes to have parabolic dispersion with mass m, and all particles interact with each other via the Coulomb interaction <ref:2402.00866#pg2>. To approximate effects of charge-gating, each layer has a background charge that neutralizes its total charge <ref:2402.00866#pg2>. Both layers have N particles, and are described by the usual dimensionless density parameter rs, which gives a rough estimate of the ratio of potential and kinetic energy in the system <ref:2402.00866#pg2>.

1. textbfProcessing of Many-Body Screening Effects in Quantum Systems: The Improved AI System Can Accurately Predict Phase Stability Based on Screening Models by Comparing Ground State Energies of Electron-Hole Plasma and Exciton Gas using Energetic Comparison. This allows the system to determine, as the paper suggests, which one is energetically favored in which regime, specifically identifying that the exciton gas is stable at small layer separation or that the plasma is stable for specific density parameters, by evaluating different screening approximations like static screening versus dynamic screening.

2. textbfQuantifying Sensitivity to Theoretical Approximations: The Improved AI System Can Provide a Robust Assessment of Theoretical Uncertainty by Identifying How Results Depend on Treatment, as highlighted by the conclusion that energetic considerations alone are very sensitive to the approximation schemes. This system can specifically compare outcomes based on how the electron-hole attraction is treated in the theory, leading to a nuanced understanding of why numerical results may not all agree.

3. textbfModeling Complex Many-Body Interactions: The Improved AI System Can Model Interlayer Exciton Formation and Stability by Solving the excitonic Bethe-Salpeter equation using different interaction models, such as the unscreened or the screened interlayer electron-hole Coulomb interaction. This capability enables the system to predict phase boundaries based on whether it uses a model that results in excitons being preferred at small d in the unscreened model versus those where static screening suppresses the formation at all d (except for very small or very large rs).

4. textbfPredicting Regime-Dependent Stability: The Improved AI System Can Predict Phase Stability across Different Density Regimes by Identifying Specific Conditions, such as the finding that the exciton gas is always lower energy compared to the plasma for all layer separations when small rs (i.e. high density) or very large rs (i.e. very low density). This allows for precise prediction of which phase dominates under extreme physical conditions within the system's parameters, specifically noting that the exciton gas is quite generally stable for high (small rs) and low (large rs) densities.

5. textbfIdentifying Regime Inaccuracy: The Improved AI System Can Flag Unreliable Theoretical Regimes by Detecting Where Approximations Break Down, as indicated by the statement that the intermediate rs regime, where we are finding a stable plasma phase is not a theoretically reliable regime for our approximations since both the plasma and the exciton energies are inaccurate at intermediate rs values. This allows researchers to avoid drawing conclusions from regions where both the plasma and the exciton energies are inaccurate in our approximation scheme.