Electron-Hole Liquid in 2D Semiconductors

• Electron-hole liquid is a quantum phase characterized by strong electron-hole correlations, incompressibility, and abrupt photoluminescence shifts.
• Research in monolayer TMDCs reveals that reduced dielectric screening and enhanced Coulomb interactions enable EHL formation at or above room temperature.
• Experimental studies show clear phase diagrams with threshold excitation, validating theoretical models and enabling novel optoelectronic device applications.

An electron-hole liquid (EHL) is a dense, strongly correlated, macroscopically coherent quantum phase of electrons and holes stabilized by many-body exchange-correlation effects. It forms via a first-order phase transition when the density of photoexcited carriers in a semiconductor exceeds a critical value and the temperature is below a material-specific critical temperature. EHLs exhibit incompressibility, a distinct thermodynamic signature, and are characterized by a threshold in photoluminescence, broad linewidths due to Fermi degeneracy, and phase coexistence with a dilute exciton or plasma phase. While traditional observations required cryogenic temperatures, quasi-two-dimensional (quasi-2D) systems such as monolayer transition metal dichalcogenides (TMDCs) enable EHL formation at or above room temperature due to reduced screening and enhanced Coulomb interactions.

1. Microscopic Theory and Central Hamiltonian

The EHL is described microscopically by a many-body Hamiltonian for conduction-band electrons and valence-band holes in a 2D (or quasi-2D) geometry. The canonical second-quantized Hamiltonian for a photoexcited monolayer TMDC is

$$H = \sum_{i=e,h} \sum_{k} \epsilon_{k,i} c^{\dagger}_{k,i} c_{k,i} + \tfrac{1}{2} \sum_{q} V(q) \sum_{i,j=e,h} \sum_{k,k'} c^{\dagger}_{k+q,i} c^{\dagger}_{k'-q,j} c_{k',j} c_{k,i}.$$

Here $\epsilon_{k,i} = \hbar^2 k^2 / (2m_i)$ is the kinetic energy of carriers of mass $m_i$, and $V(q)$ is the interaction potential. In monolayer TMDCs, $V(q)$ is the strictly 2D Keldysh potential,

$$V(q) = \frac{4\pi e^2}{(\varepsilon_1 + \varepsilon_2) q [1 + r_0 q]},$$

with $\varepsilon_{1,2}$ being dielectric constants of the environment and $r_0$ the in-plane screening length (e.g., $r_0 \approx 43.9$ Å for MoS$_2$). The ground and thermally excited states are obtained by evaluating the grand or Helmholtz free energy, incorporating kinetic, exchange, and random-phase approximation (RPA) correlation contributions,

$F(n,T) = F_0(n,T) + F_X(n,T) + F_C(n,T).$

The phase diagram is constructed by analyzing the chemical potential $\mu(n,T) = \partial F / \partial n$ and identifying points and regions of negative compressibility, which signal the onset of the EHL phase (Rustagi et al., 2017).

2. Phase Diagram and Thermodynamics

The condensation of the EHL is a first-order gas-liquid-like phase transition. Thermodynamic stability requires $\partial\mu/\partial n > 0$. The coexistence (binodal) and instability (spinodal) lines are defined via:

• Binodal: $\mu(n_1) = \mu(n_2)$ and $P(n_1) = P(n_2)$, with $P=-F + \mu n$.
• Spinodal: $\partial\mu/\partial n = 0$.
• Critical point: conditions $\partial\mu/\partial n = 0$, $\partial^2\mu/\partial n^2 = 0$.

The phase transition in a suspended monolayer MoS$_2$ occurs at $n_c \approx 3.8 \times 10^{11}$ cm$^{-2}$ and $T_c \approx 515$ K. The Mott density, where the exciton binding vanishes, is $n_M \approx 2.8 \times 10^{12}$ cm$^{-2}$. The critical temperature generally follows $k_B T_c \sim 0.1\, E_\text{ex}$, with $E_\text{ex}$ being the 2D exciton binding energy. The resulting phase diagram exhibits regions of dilute excitonic gas, phase separation (coexistence), and dense EHL at high density and low temperature (Rustagi et al., 2017, Yu et al., 2017, Pekh et al., 2021).

3. Distinctive Features: Compressibility, Spectroscopy, and Collective Behavior

The EHL is an incompressible Fermi liquid of electrons and holes, distinct from an exciton gas (hydrogenic, compressible, with sharp PL lines). Key features include:

• Incompressibility: The EHL maintains fixed density above threshold excitation, reflected in PL spectra that show a constant peak position and linewidth over a range of pumping intensities.
• Photoluminescence: The EHL exhibits a sudden red-shift and narrowing of the emission line when the density crosses the coexistence threshold ($\sim 10^{11}$–$10^{12}$ cm$^{-2}$). A density-independent “liquid” peak appears, and exciton/trion lines are quenched above the Mott density.
• First-order features: Abrupt jumps in PL intensity (by factors of 4–5) mark the threshold for condensation. The threshold power for EHL formation matches the theoretical density ($n_c$), and a luminescent ring forms due to high mobility and surface tension.
• Compressibility: Experimentally, PL linewidth and peak remain unchanged above threshold, evidencing $\partial\mu/\partial n \approx 0$ (flat compressibility) (Yu et al., 2017).

Collective properties such as surface tension, droplet formation, and hydrodynamics (e.g., the expansion of a ring-shaped luminous region) directly reflect the quantum liquid nature of the EHL (Yu et al., 2017).

4. Material Systems and Scaling Laws

System	Exciton $E_b$ (eV)	$T_c$ (K)	$n_c$ ($10^{11}$ cm$^{-2}$)
Monolayer MoS$_2$ (susp.)	0.5	515	3.8
Monolayer MoSi$_2$N$_4$	0.432	415	1.0
MoSi$_2$As$_4$	0.466	302	0.7
Bulk (Ge, Si)	0.01–0.02	$\leq$ 50	$>10^3$ (3D $\to$ per $a_B^3$)

For quasi-2D TMDC monolayers:

• High $E_b$ and $T_c$: Weak dielectric screening and quantum confinement enhance $E_b$, pushing $T_c$ to room temperature and above (Rustagi et al., 2017, Yadav et al., 2023).
• Equilibrium EHL density scales as $n_0 \sim \nu^{3/2}A^{3/2}/(3\pi a_B^2)$, where $\nu$ is the valley degeneracy, $A$ is a correlation constant, and $a_B$ is the 2D Bohr radius.
• Comparison to 3D systems: Three-dimensional semiconductors like Ge, Si, or GaP have $T_c \lesssim 50$ K and much lower $|E_0|$, owing to strong 3D screening (Pekh et al., 2020, Pekh et al., 2021, Yadav et al., 2023).

Monolayer MoS$_2$ yields $E_\text{ex} \approx 0.5$ eV, $a_B \approx 0.65$ nm, and $n_c \sim 10^{13}$ cm$^{-2}$, all supporting EHLs stable well above room temperature.

5. Experimental Realizations and Signatures

EHLs have been demonstrated at room temperature in monolayer MoS$_2$ via suspension over vacuum or encapsulation in low-$\varepsilon$ dielectrics (Yu et al., 2017, Rustagi et al., 2017). Key experimental parameters:

• Excitation: CW or pulsed lasers exceeding $I \sim 0.8$ kW/cm$^2$ to drive the density past $n_c$.
• Detection: PL and differential absorption spectroscopy monitor the collapse of exciton resonances, appearance of sub-gap absorption, and threshold behavior in emission.
• Carrier density estimation: Consistency between density deduced from lifetime, bandgap renormalization (empirically $\Delta E_\text{BG}(n)\propto (n a_B^2)^{1/3} E_b$), and free-energy minimization (yielding $n \approx 3$–$4 \times 10^{13}$ cm$^{-2}$) confirms EHL formation (Yu et al., 2017).
• Lifetime and mobility: The EHL exhibits carrier lifetimes $\tau \gg 1$ ns and high mobility, distinct from the substrate-supported (ns) lifetime.

Similar EHL signatures have been observed in MoTe$_2$ photocells, again at 297 K, using time-resolved dynamic photoresponse microscopy, where ring-like spatial patterns and abrupt power-law breakdown in photocurrent response signify phase transition (Arp et al., 2017).

6. Control Parameters and Tunability

EHL properties can be tuned by:

• Valley polarization: Circularly polarized excitation redistributes carrier populations among valleys, shifting the equilibrium EHL density, PL energy, and linewidth (density changes up to 30–40%) (Ray et al., 2022).
• Substrate and environment: The effective dielectric constant $(\varepsilon_1 + \varepsilon_2)/2$ modulates screening, thereby shifting $E_b$ and $n_c$.
• Doping and multilayer engineering: Increasing the number of conduction-band valleys (e.g., via bilayer or heterostructures) further raises $T_c$ and $|E_0|$ (Ratnikov, 2022).
• Strain: Mechanical strain can favor indirect gaps confining holes, extending EHL lifetimes and enabling tunable emission (Ray et al., 2022).

7. Broader Implications and Applications

Room-temperature EHLs in 2D semiconductors and van der Waals heterostructures open pathways to:

• Ultrahigh-power broadband photonics and light sources: The incompressibility and fast recombination facilitate efficient lasers and LEDs.
• Quantum fluids and macroscopic coherence: EHLs provide a platform for studying Fermi-liquid hydrodynamics, droplet physics, and surface tension-driven patterns.
• Collective many-body states: Possibility of quartet and other exotic condensates (biexciton-like Cooper quartets) within the EHL phase (Guo et al., 2022).
• Device technologies: Integrated optoelectronic circuits leveraging high carrier densities, room-temperature operation, and device tunability via valley or doping engineering.

A plausible implication is that EHL realization in monolayer or heterostructure TMDC devices will usher in new functionalities in quantum nanophotonics, as well as enable controlled studies of quantum phase transitions—including the interplay with Mott, trion, or excitonic insulator transitions—under ambient conditions.

---

References:

• "Theoretical phase diagram for the room temperature Electron-Hole Liquid in photo-excited quasi-2D monolayer MoS$_2$" (Rustagi et al., 2017)
• "Room-Temperature Electron-Hole Liquid in Monolayer MoS2" (Yu et al., 2017)
• "Valley engineering electron-hole liquids in TMDC monolayers" (Ray et al., 2022)
• "Room temperature electron-hole liquid phase in monolayer MoSi$_2$Z$_4$ (Z = pinctogen)" (Yadav et al., 2023)
• "Quantum Phase Transition of the Electron-Hole Liquid In the Coupled Quantum Wells" (Babichenko et al., 2016)
• "Electron-Hole Liquids in Transition Metal Oxide Heterostructures" (Millis et al., 2010)
• "Phase diagram of electron-hole liquid in monolayer heterostructures based on transition metal dichalcogenides" (Pekh et al., 2021)
• "Electron-Hole Liquid in Monolayer Transition Metal Dichalcogenide Heterostructures" (Pekh et al., 2020)
• "Dielectric electron-hole liquid in monolayer heterostructures based on transition metal dichalcogenides" (Ratnikov, 2023)