Exciton Condensation in an Atomically-thin MoS2 Semiconductor


 Condensation of a dilute Bose gas of excitons (coupled electron-hole pairs) in a direct bandgap semiconductor was first theoretically predicted in 19681. This exotic state of matter is expected to exhibit spectacular non-linear properties, such as superradiance and superfluidity. However, direct experimental observation of condensation of optically active excitons in conventional semiconductors has been hindered by their short lifetimes and weak collective excitonic interactions. Here, we have experimentally realized the condensation of short-lived excitons in a direct-bandgap, atomically-thin MoS2 semiconductor. The signature is the anomalous transport of the fast-expanding exciton density, originating from a thermalized dilute gas generated under the laser spot. Below the critical temperature Tc~150 K, the exciton liquid propagates over ultra-long distances (at least 60 micrometers) with record speed in a solid-state system of 1.8*10^7 m/s (~6% the speed of light), fuelled by the unconventionally strong repulsions among excitons. The condensation is controlled by many-body interactions in the gas mixture of excitons (bosons) and free-carriers (fermions) via an electrical backgate. Our results demonstrate electrostatic doping as a simple approach for the investigation of correlated states of matter at high-temperatures, excitonic circuitry and spin-valley Hall devices mediated by exciton superfluids in semiconducting monolayers.

micrometers) with record speed in a solid-state system of ∼1.8×10 7 m/s (∼6% the speed of light), fuelled by the unconventionally strong repulsions among excitons. The condensation is controlled by many-body interactions in the gas mixture of excitons (bosons) and free-carriers (fermions) via an electrical backgate. Our results demonstrate electrostatic doping as a simple approach for the investigation of correlated states of matter at high-temperatures, excitonic circuitry and spin-valley Hall devices mediated by exciton superfluids in semiconducting monolayers.
Similar to ultracold atomic gases 2, 3 , a dilute gas of excitons optically injected in a semiconductor below a critical temperature (T c ) is well-described as a weakly interacting Bose gas that can spontaneously condense (in momentum space) into a liquid, via many-body repulsive interactions among electron-hole pairs 1, 4 . Despite the efforts over the last five decades, experimental observation of condensation of optically-active (short-lived) excitons in direct-bandgap semiconductors has not been realized. To stabilize the condensate in these systems, the repulsive interactions among excitons must compensate the fast decay of the population. Furthermore, the formation of other complexes via attractive forces, such as charged states (trions), molecules (biexcitons) or even electron-hole droplets 5 , are competing effects against exciton condensation. To overcome the limitations in direct-bandgap semiconductors with weakly-interacting excitons, top-down approaches have been taken. For instance, by using microcavities, polaritons with ultra-light effective masses are obtained, allowing for metastable Bose condensation at high densities 6 . Alternatively, quantum fluids have been reported in double quantum-wells (DQWs) by using optically-forbidden (long-lived) dipolar excitons, whose electron and holes resides in opposite QW layers (spatially-indirect) [7][8][9][10][11][12][13] . In DQWs, the long lifetimes and the tailored repulsive interactions among indirect dipolar excitons drive the condensation.
Excitons in direct-bandgap two-dimensional semiconductors, such as monolayer transi-tion metal dichalcogenide (TMD) MoS 2 , have short lifetimes and stable exciton complexes [14][15][16] , which a priori may preclude the formation of exciton liquids. Here, by using spatially-, time-and energy-resolved photoluminescence (PL) and reflectivity measurements, we have experimentally realized the condensation of a dilute exciton gas in MoS 2 monolayer until T c ∼ 150 K, evidenced by the anomalous transport over long distances of at least 60 µm (limited by the sample size) with record speed in a solid-state system of ∼1.8×10 7 m/s (∼6% the speed of light). The condensation is driven by unconventionally strong repulsive interactions among excitons, directly revealed by ∼3 meV blueshift of the exciton energy. We also present the phase diagram by using an electrical backgate and temperature as relevant thermodynamic parameters that control the interplay between attractive (exciton-electron) and repulsive (exciton-exciton) many-body interactions present in the Bose-Fermi gas mixture in the monolayer semiconductor.
In our PL imaging experiments, we used a laser as excitation source with a well-defined Gaussian spatial distribution with full width at half-maximum (fwhm) of σ Laser ∼1.5 µm. By using near-resonant excitation ( Fig. 1(A)), a low density gas of excitons (up to n X ∼ 1.1 × 10 10 cm −2 , well below the Mott density 17 n Mott ∼ 10 13 cm −2 ) is photogenerated with minimal momentum (hk X ∼ 0), kinetic energy (E X (k X ) = E g +hk 2 X /2m * X ) and group velocity (v X (k X ) = ∂E X /(h∂k X )), whereh is reduced Planck constant and m * X is exciton effective mass. In this way, the exciton gas reaches thermal equilibrium with the semiconductor lattice right after photoexcitation (E X (k X )-E X (0) ∼ k B T , where k B is the Boltzmann constant and T is temperature), persisting within the radiative lifetime of the state.
Monolayer MoS 2 has large intrinsic concentration of electrons that markedly affects the optical response of the two-dimensional semiconductor. We control the electron density (n e ) and enhance the excitonic response in our samples by incorporating a backgate (V g ). At large positive V g (i.e., high n e ), the reflectance contrast in Fig. 1(B) is dominated by a single reso-3 nance, denoted as X − , that corresponds to the so-called charged exciton or trion state, or also described as attractive polaron. In this metallic regime (V g > 20 V), the neutral exciton, also referred to as repulsive polaron, is absent. Decreasing V g (lower n e ), the excitonic resonance X starts appearing at 38 meV above X − . A further decrease in V g leads to a strong increase in the exciton absorption and a monotonic shift to lower energies. The X − feature disappears below V g = -30 V. At this insulating regime (V g < -30 V), only the neutral exciton resonance The change in the character from metallic to insulating of the MoS 2 monolayer drives an instability in the excitonic system that results in an anomalous excitonic transport and nonlinear optical effects. Therefore, to accurately study this, we have produced high-quality large area (A ∼ 1200 µm 2 ) MoS 2 monolayers. The longest dimensions of our samples of 40-60 µm The low-temperature (T = 20 K) PL emission at V g = 0 V is localized around the laser spot, positioned at the center of the sample ( Fig. 1(D)). Remarkably, maintaining the same excitation conditions but setting V g = -60 V, we observe light-emission with homogeneous intensity distribution over the entire crystal ( Fig. 1(E)) outside the laser spot. Such PL profile suggests that excitons have homogeneously spread over large distances away from the excitation source (see Fig. S5 for reproducibility in other devices). Given the negligible kinetic energy and low mobility of the exciton gas [18][19][20][21][22][23][24] , such a long-range transport is striking and has not been theoretically predicted for TMDs. The PL profiles exhibit a tail that extends further into the sample beyond region-I, which suggests a weakly interacting exciton gas at V g = 0 V. Its intensity follows an exponential distribution (region-II). Intriguingly, when decreasing V g , the amplitude and decay length (L X ) of the exponential distribution increase nonlinearly (Figs. S7-S8). In particular, at V g = 0 V, region-II represents the ∼25% of the PL intensity that ultimately increases to ∼90% at V g = -60 V. The Moreover, the PL pattern at region-II also varies non-linearly with excitation power as shown by Fig. 2(B) at V g = -60 V and T = 20 K. With increasing power, the exponential decay becomes longer and asymptotically approaches a step-like shape at the highest injected exciton density. Remarkably, keeping P = 1000 µW and V g = -60 V, the step-like profile is observed until exceptionally high lattice temperatures of ∼150 K ( Fig. 2(C)). Further increase of T leads to a rapid decay of the exponential component of the PL profile that becomes negligible (<10%) above 180 K (Fig. S6).
Such striking instabilities of the excitonic density in region-II are attributed to a thermodynamic phase transition, where a fraction of the dilute gas of excitons has spontaneously 5 condensed into a liquid. The formation of the exciton liquid is revealed by the anomalous transport behaviour imprinted in the exponential PL distribution. Therefore, we use the exponential decay constant (L X ), defining the transport length of the exciton fluid, as a characteristic macroscopic parameter to closely monitor the phase transition, which originates from microscopic many-body interactions, as a function of V g , P and T .
The phase diagram of L X (V g , T ) is shown in Figure 2 Microscopically, what distinguishes the exciton liquid from the gas is the competition between repulsive (exciton-exciton) and attractive (exciton-electron) interactions within the mixture of excitons (boson) and electrons (fermion) (illustration Fig. 2(D)). In the gas phase, the short transport length suggests that excitons are spatially localized by the strong attractive interactions induced by the dense electron gas. In this metallic-like regime, the many-body repulsive exciton-exciton interactions required for their condensation are screened, regardless of the exciton density, which implies attractive forces are dominant to the leading order n e /n X > 1.
Nominally, interactions between electrons and excitons remain attractive for all n e 26 , 6 which prevents for the formation of the exciton liquid. The short-range attractive interactions reduce when gradually decreasing n e to become smaller, or of order of n X (n e /n X ≤ 1). In the insulating regime, the low electron density (i.e., negligible screening) allows for the repulsive exciton-exciton interactions 27 to take over, giving rise to the spontaneous condensation of the dilute exciton gas.
Moreover, the fraction of condensed excitons is also markedly affected by temperature.
Given the large binding energy of MoS 2 excitons, we estimate that the density of thermally excited electrons surpasses n X at T = (160±40) K (Fig. S12). This range of temperatures depends on the intrinsic carrier concentration and agrees well with the gas-to-liquid phase transition. The critical exciton condensation temperature (T c ∼ 150 K) observed here for direct-bandgap MoS 2 monolayer is much higher than that for superfluid helium (2.17 K), indirect excitons in III-V DQWs (<10 K) 7-9, 11, 25 , III-V and II-VI polaritons 6 (<20 K) and Cu 2 O excitons 28 (< 1 K) and is similar to the estimated temperature for tightly-bound indirect excitons in TMD DQWs 29 .
The remarkably high-T c suggests unusually strong repulsions between excitons occupying the lowest-energy state and sets a record for condensation and achievement of exciton superfluidity in direct-bandgap semiconductors.
The macroscopic expansion of the exciton liquid in the atomically-thin MoS 2 surface causes the up to three-orders-of magnitude increment of the integrated PL intensity with decreasing V g (Fig S10-11) that explains for the near-unity PL quantum yield reported in neutral MoS 2 samples 30,31 . Such giant PL enhancement, or superradiance, indicates that lightemission of the liquid originates from optically-active (bright) excitons. Therefore, the longrange propagation of excitons within their ultra-short radiative lifetime, typically τ X ∼ 1-20 ps in TMDs [14][15][16] , strongly suggests the unconventionally high-speed of the exciton fluid because We demonstrate the ultrafast transport by performing spatially-and energy-resolved pump-probe reflectance measurements with 150-femtosecond time-resolution. A broadband-energy source (probe) illuminates the entire MoS 2 crystal, while the single-frequency beam (pump) excites the center of the sample (x = 0 µm) with a reduced beam size of σ Laser ≈ 2 µm.
The pump, tuned near the exciton resonance, generates an exciton gas with estimated density n X ∼ 1.7 × 10 12 cm −2 under the laser spot. After gas thermalization, as excitons condense via repulsive interactions, they are propelled away from the generation source, altering the spatial-dependence of the intensity and energy distribution of the reflectance profile through the monolayer.  The unconventionally strong repulsions among excitons causes a significant blue-shift of the exciton feature in the pump-on with respect to the pump-off raw reflectance spectra (Figs. S17-S18). We estimate the net repulsive energy ∆E X−X arising from exciton-exciton interactions, which is shown in Fig. 4(B) for ∆t = 2.4 ps. In the range 0-5 µm, ∆E X−X has the largest value of ∼ 3 meV, comparing well to that of dipolar indirect excitons 11 and polaritons 32 condensates. ∆E X−X follows the decay of the exciton population given by DR(x, E X (0)) 9 (solid-pink curve) that decreases slowly to ∼ 1 meV far from the laser spot. In and R Off (x, E) are the reflectance profiles with the pump on and off, respectively. The positive is observed at the gas phase (∆E X−X ∼ 0). ∆E X−X decreases as exciton liquid propagates in space following the decrease in the density (solid-pink curve, right axis).

Device Fabrication
Large MoS 2 monolayers with the typical long side of 40 to 60 µm and area of ∼1200

Optical Imaging and Experimental Setup
The fabricated devices were mounted on a cold finger cryostat and cooled down using a continuous flow of liquid helium. The cryostat station was mounted on the photoluminescence (PL) and transient reflectance (TR) imaging setup that operated in two different imaging configurations (see SI for details).
In the PL imaging configuration, the samples were excited with a He-Ne (632.8 nm) continuous wave (cw) laser cleaned by an interference filter resonant with the A-exciton of the MoS 2 monolayer. Using a 50× objective lens (NA = 0.55), the laser was focused to locally excite a small spot with diameter of 1.5 µm on the large sample. The objective then collected the PL emission. An ultra-steep-edge long pass filter placed in the detection path before the optical detector removes reflected laser light, allowing only PL image signal to be measured. A set of lenses were added in the detection path to change the image magnification and effective field of view on the optical detector (sCMOS sensor). This enables imaging to be optimized for samples of different sizes. The sCMOS chip has an area of 2300 × 2300 pixels, 6.5 µm each. Our detection system provides a high spatial resolution of 40 nm per pixel in either axial direction.
For TR imaging, a femtosecond pulsed pump laser (615 nm) is used to excite the sample. A pulsed broadband white light beam given a time delay ∆t with respect to the pump is used to probe the changes in the reflectance of the sample in the visible (625 -700 nm) region.
The size of the probe beam illuminating the sample surface is controlled by a series of lenses, which focuses the probe at the back focal plane of the objective lens to maximize the probe and cover the entire sample. The reflected probe beam was collected by the same objective and dispersed by a single grating (600 gr/mm) monochromator with 550 mm of focal length and the signal was detected by a scientific CCD detector cooled with liquid-nitrogen. Energy-space reflectance spectrum map images are measured for each time delay under pump on/off conditions. As the technique relays in the changes in the optical reflectance/absorption, the dynamics of the exciton population within the light-cone, which includes the spatial propagation, radiative recombination and relaxation into other states, is monitored in great detail.