Electronic structure of a persistent excitonic fluid far beyond the Mott limit

Main

In semiconductors, Coulomb interactions bind electrons and holes into excitons, giving rise to sharp absorption peaks below the electronic bandgap and efficient radiative recombination. In the dilute regime, excitons behave as well-defined, charge-neutral quasiparticles. They have been extensively investigated, particularly in two-dimensional materials, where reduced dielectric screening strongly enhances their binding energy and oscillator strength [1, 2].

The fate of excitons at high densities remains debated. In the conventional Mott picture, increasing carrier density enhances screening and eventually destabilizes excitons, leading to their ionization into unbound electron–hole pairs [3, 4]. The optical response changes dramatically as the system reaches population inversion: the exciton absorption peak progressively disappears while optical gain emerges [5]. THz spectroscopy has also revealed the coexistence of bound excitons and unbound carriers across the Mott transition [5]. This exciton–plasma crossover is gradual and depends on exciton properties, including dimensionality, binding energy and Bohr radius. In semiconductors with strongly anisotropic screening, excitonic signatures can persist up to photoexcited carrier densities of ($10^{14}\ \mathrm{cm}^{-2}$) [5, 6].

However, the conventional Mott picture assumes thermally distributed carrier populations, which need not always be realized following photoexcitation. For nonthermal distributions, such as those generated by resonant pumping at the optical gap, theory predicts that excitons can persist to remarkably high carrier densities through the emergence of exceptional points in the spectrum of the non-Hermitian Bethe–Salpeter Hamiltonian [7, 8]. This topological mechanism gives rise to a quantum fluid with spectral features reminiscent of a BCS condensate, in which bound electron–hole pairs remain protected against ionization even when embedded in the electron-hole continuum.

In Time- and angle-resolved photoelectron spectroscopy (tr-ARPES), the quantum exciton fluid at high density is predicted to manifest itself through a characteristic band reconstruction with a Mexican-hat dispersion [7, 8]. Such a dispersion has so far been observed only in the coherent regime and interpreted in terms of many-body Floquet dressing [9]. Within this picture, the reconstructed bands are expected to disappear as the optical coherence decays. However, theory based on T-matrix diagrams predicts that the excitonic band reconstruction can survive well beyond this timescale, sustained by the exciton population rather than the macroscopic polarization [8].

Here, we investigate this long-living state using tr-ARPES [10, 11] on the two-dimensional hybrid perovskite $(\mathrm{BA})_2(\mathrm{MA})_2\mathrm{Pb}_3\mathrm{I}_{10}$, a Ruddlesden–Popper compound in which inorganic quantum wells are separated by organic layers [12, 13] (Figure 1a). The reduced dimensionality stabilizes bound electron–hole states up to room temperature, making this material a suitable platform for investigating high-density excitonic phenomena [12, 2, 1]. Beyond resolving their dispersion, tr-ARPES enables the tomographic reconstruction of excitonic wavefunctions and has emerged as a powerful probe of excitons in two-dimensional (2D) transition metal dichalcogenides [14], organic crystals [15, 16], topological insulators [17], moiré heterostructures [18, 19] and hybrid structures [20].

By mapping the photoelectron distribution over a wide range of resonant pump fluences, we reveal a characteristic Mexican-hat band reconstruction that survives at excitation fluences three times the optical saturation threshold and persists well beyond the optical coherence time. These findings provide evidence for a stable quantum exciton fluid in which bound electron–hole correlations remain robust against screening-induced ionization.

Figure 1 shows tr-ARPES data acquired on a $(\mathrm{BA})_2(\mathrm{MA})_2\mathrm{Pb}_3\mathrm{I}_{10}$ crystal at 50 K. Energy- and momentum-resolved ARPES maps [21] are first recorded at very early delays (50 fs for panel c–d and 100 fs for panel e) using a weak pump fluence ($\sim3\ \mu\mathrm{J}/\mathrm{cm}^2$), for three representative pump photon energies. As shown schematically in Figure 1b, optical transitions at 2.4 eV, 2.22 eV and 2.07 eV lie above the conduction band minimum just below it, and at the exciton resonance, respectively. Excitation with 2.4 eV photons generates free carriers in the conduction band (dashed white line in Figure 1c), along with a weak spectral signature at the exciton resonance. At 2.22 eV (Figure 1d), the response is dominated by population at the conduction band minimum (CBM), with only a faint excitonic contribution. In contrast, resonant pumping at 2.07 eV selectively prepares a coherent exciton population (Figure 1e). The energy separation between the CBM and the exciton resonance yields an exciton binding energy of $E_B = 0.18$ eV, in excellent agreement with prior all-optical measurements [1]. Notice in Figure 1e, that the spectral weight is strongly peaked at the center of the Brillouin zone and exhibits the characteristic downward dispersion previously reported in transition-metal dichalcogenides [19, 22]. As theoretically expected at low excitation density [8], the overall shape of the ARPES intensity map closely follows the curvature of the calculated valence-band replica.

The wavevector distribution measured by tr-ARPES in Figure 1e (weak and resonant excitation) directly reflects the squared modulus of the exciton wavefunction envelope in reciprocal space [23]. Experimentally, this quantity (Figure 1f) is extracted from a Momentum Distribution Curve (MDC) by integrating the ARPES intensity in Figure 1e over energy. Theoretically, the wavefunction is extracted from Bethe–Salpeter calculations based on a tight-binding parametrization of the two-dimensional perovskite multiple-quantum-well structure [12]. The tight-binding model is constructed by fitting the conduction and valence bands to the ab initio electronic band structure. This approach yields an imaginary part of the dielectric function, which well agrees with the exciton resonance observed in the experimental optical absorption spectrum and predicts an exciton binding energy of $E_B = 0.186$ eV. Moreover, as shown in Figure 1f, the calculated envelope of the exciton wavefunction in wavevector space ($|\psi(\mathbf{k})|^2$) overlaps with the experimental MDC. From this analysis, we infer in-plane root-mean-square radius of $\sqrt{\langle r^2\rangle} = 4$ nm, confirming the generation of a spatially delocalized Wannier excitons (Figure 1g).

Being bound electron–hole pairs, bright excitons possess a large transition dipole moment. To demonstrate this directly, we monitor the integrated tr-ARPES intensity while tuning the pump photon energy across the absorption edge. The resulting excitation density exhibits a pronounced resonance at 2.07 eV (Figure 2a), consistent with the optical absorption spectrum and photoluminescence measured over the same spectral range (Figure 2b). Figure 2c shows the integrated photoelectron intensity as a function of pump fluence, measured under resonant excitation and 100 fs after the arrival of the pump pulse. The approximately linear dependence observed up to 50 $\mu$J/cm$^2$ allows us to calibrate the tr-ARPES intensity against the photoexcitation density, assuming an optical penetration depth of 140 nm. Above this fluence, however, the photoexcitation density deviates from the linear scaling and gradually approaches a saturation value of $5 \times10^{12}$ cm$^{-2}$. Remarkably, the pronounced excitonic resonance persists well beyond the fluence $F_s = 100$ $\mu$J/cm$^2$ at which the photoexcitation density reaches 75% of the saturation threshold. As the pump fluence increases to $F_s$ (Figure 2d), the resonance broadens only modestly. Even at 350 $\mu$J/cm$^2$, corresponding to $3.5F_s$, the resonance remains clearly visible (Figure 2e). Thus, a bound state with a large transition dipole moment can still be efficiently excited under these extreme driving conditions. Such a behavior contrasts with the conventional Mott picture, which predicts that increasing carrier density progressively screens the Coulomb interaction and ultimately drives the dissociation of excitons into unbound electron–hole pairs.

The crossover from weak to high photoexcitation density is monitored by acquiring ARPES intensity maps as a function of an increasing pump fluence. We present data for exciton densities in the range $1.9 \times10^{11}$ to $5 \times10^{12}$/cm$^2$ and acquired at pump probe delay of 100 fs. At the highest photoexcitation density, approximately 2.5 electron–hole pairs are confined within the area defined by the two-dimensional exciton wavefunction, $\pi\langle r^2\rangle$, placing the system deep in the quantum regime.

The ARPES map of the exciton gas (low density) differs markedly from that of the quantum exciton fluid (high density). Whereas the spectral weight disperses downward in the dilute regime (Figure 3a), an upward dispersion emerges in the quantum fluid and becomes clearly visible at an excitation density of $1.9 \times10^{12}\ \mathrm{cm}^{-2}$ (Figure 3b). At even higher densities, the upward dispersion becomes more pronounced, approaching that of electrons in the conduction band (white dashed line in Figure 3c-Figure 3d).

In the high fluence regime, the remnant hybridization with the valence band replica shapes the faint spectral weight at large momenta. To aid visualization, the tr-ARPES spectra in Figure 3e-Figure 3h are obtained normalizing Energy Distribution Curves (EDCs) to their maximum intensity. At intermediate fluences (Figure 3f,Figure 3g), the resulting maps reveal a characteristic Mexican-hat-like dispersion, reminiscent of that expected for a steady-state exciton liquid in a BCS-like superfluid regime. Upon further increasing the carrier density (Figure 3h), the shadow replica of the valence band shifts beyond the momentum window of $[-0.1, 0.1]\ \text{\AA}^{-1}$. Out of this momentum range, the signal becomes too weak to be resolved above the noise level, even after normalization.

We simulate the experimental results using the TR-ARPES formula developed in Ref. [8]. The nonequilibrium excitonic spectrum entering the formula is calculated within the exceptional-exciton framework [7], going beyond the first-order expansion in excitation density adopted in Ref [8]. Specifically, exciton energies and wavefunctions are obtained by diagonalizing the pseudo-Hermitian Bethe–Salpeter Hamiltonian (PH-BSH) for electronic populations characteristic of resonant pumping, rather than the thermal Fermi–Dirac distribution assumed in the conventional Mott picture. This choice is justified by the slow transfer from the resonantly pumped bright exciton to the lowest-energy spin-dark exciton, which occurs on a timescale of approximately 1 ns [24].

Consequently, the carrier distribution retains the characteristic profile of resonant pumping even a few picoseconds after excitation, when the system has entered the incoherent regime. A key consequence of these distributions is that the entire PH-BSH spectrum remains real up to excitation densities at which population inversion occurs (in our material near the $\Gamma$ point). At the critical inversion density, the conventional exciton evolves into an exceptional point corresponding to a bound state embedded in the electron–hole continuum. This topological property provides a natural protection mechanism against hybridization with the continuum, offering a physical basis for the persistence of bound excitons at these high excitation densities. As shown in Figure 3(i-Figure 3p), the simulated tr-ARPES spectra are in close agreement with the experimental data. At low fluence, spectral weight can be understood as a sideband replica of the valence band, with a momentum-dependent weight given by the squared excitonic wavefunction, $|\psi(\mathbf{k})|^2$. Instead, when the photoexcitation density is higher than the critical value, the hybridization between conduction band and valence-band replicas gives rise to the characteristic Mexican-hat-like structure.

As shown in Figure 4a, the dynamics of the quantum electron fluid reveal two clearly distinguishable timescales. The integrated tr-ARPES intensity initially decreases by approximately 10% on a timescale of 100 fs, followed by a much slower decay with a characteristic timescale of 20 ps. The initial femtosecond dynamics occurs on a timescale comparable to the loss of optical coherence measured in 2D perovskites by four-wave mixing [25]. Notably, the ultrafast component emerges only at pump fluences between 25 and 30 $\mu\mathrm{J}/\mathrm{cm}^{2}$ (Figure 4b), coinciding with the onset of the Mexican-hat dispersion (Figure 3b). Thus, the fast dynamics originates from decoherence, its signature in the ARPES signal appears to become measurable only above the critical density required to form a BCS-like fluid.

Importantly, the ultrafast dynamics primarily affects the overall photoelectron intensity, while the spectral-weight distribution and its energy–momentum dispersion remain largely unchanged (Figure 4c-Figure 4d). These features are instead governed by the exciton population, which decays predominantly through recombination processes on a fluence-dependent timescale approximately two orders of magnitude longer than that of the decoherence. Figure 4c–h shows the corresponding ARPES intensity maps following an initial photoexcitation density of $3.2 \times10^{12}\ \mathrm{cm}^{-2}$. The upward dispersion characteristic of the quantum exciton liquid persists for up to 10 ps. By 30 ps, the exciton density has fallen below the critical point, and the excitonic states are no longer embedded in the particle–hole continuum.

In conclusion, we provide experimental evidence for the formation of a high-density quantum exciton fluid in a two-dimensional hybrid perovskite. At the high excitation densities required to reach this state, the characteristic Mexican-hat dispersion represents a substantial reconstruction of the electronic structure. Remarkably, this reconstruction persists for several picoseconds, deep into the incoherent regime, demonstrating that exciton populations can reshape the electronic structure of a material well beyond the lifetime of the macroscopic polarization [9, 26, 27, 28]. Sustaining a comparably strong light-induced reconstruction over such timescales would require intense and prolonged optical excitation, with the inevitable consequence of sample damage.

The observation of stable excitons at densities exceeding the conventional Mott limit, extending into the population-inverted regime and up to the saturation density, challenges the established picture of exciton ionization at high carrier densities and supports recent theoretical predictions based on topological protection against hybridization with the electron–hole continuum. These findings highlight internal excitations as a route for controlling electronic properties of quantum materials.

Methods

Time resolved ARPES experiments have been acquired with a probe beam energy of 4.7 eV and a variable pump energy provided by optical parametric amplifier. In all reported data the sample temperature is kept at 50 K. Both probe and pump photon beams are $P$ polarized. The cross correlation between pump and probe beam has Full width half maximum of $\sim100$ fs. Single crystals of $(\mathrm{BA})_{2}(\mathrm{MA})_{2}\mathrm{Pb}_{3}\mathrm{I}_{10}$ have been cleaved at room temperature and base pressure of $2 \times10^{-10}$ mbar. Tr-ARPES maps have been collected along the $\Gamma- X$ direction by an electrostatic spectrometer. We estimate an overall energy resolution of 60 meV and wavevector resolution of $0.1\ \mathrm{nm}^{-1}$.

The electronic structure of a single inorganic Pb–I layer is described by the symmetry-based $sp^{3}$ tight-binding model of Boyer-Richard et al. [29], with one Pb $(s,p)$ and three I $(s,p)$ sites per unit cell. Exciton energies, binding energies, radii, and absorption line shapes have been computed from the Bethe–Salpeter equation (BSE) scheme, following the approach of Cho and Berkelbach [29] for layered hybrid perovskites. The agreement with available data supports the accuracy of the modelling. In particular, we find that the lowest-energy excited state is a twofold-degenerate spin-dark exciton, lying 20 meV below the first bright exciton, which is also twofold degenerate. Each of these four excitons involves a single conduction band and a single valence band, with distinct band combinations for the four excitons. Since the distinction is primarily due to spin, their spatial wavefunctions are essentially identical. The single-band character of the populated exciton allows us to simulate the nonequilibrium ARPES spectrum using the formula [7, 8] $I(k,\omega) \propto f_k\delta(\omega- E_k)$, where $f_k$ is the population of electrons in the conduction band after resonant excitation and $E_k$ is the hybridized replica of the valence band. Both quantities can be expressed in terms of the nonequilibrium valence and conduction dispersions, $\epsilon_{v/c}$, and the corresponding valence–conduction excitonic coupling, $\delta\Sigma_X \propto\sqrt{n}$, where $n$ is the excitation density. As shown in Ref. [7],

and

In these expressions, the nonequilibrium exciton energy $E_X$ is the lowest-energy discrete eigenvalue of the BSE Hamiltonian for $n < n_c$. At the critical density $n_c$, this eigenvalue merges into the electron–hole continuum and becomes an exceptional point for $n > n_c$. Remarkably, the corresponding eigenfunction remains localized in real space. These nontrivial topological solutions, known as exceptional excitons, underlie the characteristic Mexican-hat band reconstruction.

Data availability

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

Code availability

Custom analysis code is available from the corresponding author upon reasonable request.

Acknowledgements

We acknowledge financial support of the 2D-HYPE project from the Agence Nationale de la Recherche (ANR, Nr. ANR 26-CE30-7340), of the Région Ile de France via (FemtoARPES2.0 project N. EX079194), of the INFN through project TIME2QUEST and of the Horizon Europe research and innovation program of the European Union under the Marie Sklodowska-Curie Grant Agreement No. 101118915 (TIMES). Weiyan Qi acknowledges the China Scholarship Council (CSC) for funding his PhD. Computational resources were provided by GENCI (Project No. 544).

Author contributions

W.Q., S.P., A.P. and S.F.U. performed tr-ARPES experiments and data analysis, V. G., G. S. and E. P. did the Bethe Salpeter calculations, L.P. conceived the experiment, wrote the article and assured funding, K.Z. and T.P. provided the sample. E. D., S. M., A. T. and M. M. contributed to the funding. All co-authors contributed to the scientific discussions and redaction of the manuscript.

Competing interests

The authors declare no competing interests.

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