Giant enhancement of exciton diffusion near an electronic Mott insulator

The experimental setup and moiré system are depicted in Fig. 1a. The system consists of an H-stacked WS₂/WSe₂ bilayer heterostructure hosting a triangular moiré lattice (see Supplementary Material). Optically exciting this system creates IX that form between spatially separated electrons and holes residing in different moiré registries  (?). The nature and dynamics of IX is determined by the state of the electron gas (Fig. 1a). The spatially resolved photoluminescence (PL) emission, which is broader than the diffraction-limited optical excitation, encodes the information of IX diffusion (Fig. 1b-c). The excitation pump is resonant with the WSe₂ intralayer exciton in all our measurements (upper panel of Fig. 1d). Upon ultra-fast electron transfer (in the order of femtoseconds  (?)), the IX form, diffuse, and optically recombine (lower panel of Fig. 1d). As mentioned earlier, in this work we focus on the ultra-low exciton density regime where exciton-exciton interactions are negligible. Hence, the system can be treated as individual bosons moving within a gas of fermions.

To identify this regime, we look at the collected PL spectrum of IX as a function of the electron filling ($\nu_{e}$) for different pump intensities. Details on the calibration of $\nu_{\rm{e}}$ are provided in Supplementary Note 1. Figure 1e-f displays the normalized PL spectra for two pump intensities (80 nW/$\mu$m² and 5 nW/$\mu$m²). For both exciton densities, the charge neutral (CN) region is dominated by a low-energy IX (X1) which transitions into a high-energy exciton (X2) at $\nu_{e}\!\sim\!1$. The transition is benchmarked by a $\sim\!46$ meV gap in the PL emission, and it has been recently demonstrated to arise from the strong on-site exciton-electron repulsions owing to the formation of an electronic Mott-insulating state at $\nu_{e}\!\approx 1$  (?, ?, ?). Although the system’s response in the Mott insulating state is similar for both pump intensities, there is a clear difference at low electron densities: in the ultra-low exciton density case (panel f) an additional gap of $\sim\!16$ meV is observed near $\nu_{e}=0$. This redshift corresponds to the transition from X1 to attractive polaron (AP), as corroborated by the diffusion measurements discussed later. The AP formation is facilitated in such an H-stacked system where the hole can bind with multiple nearest neighboring electrons  (?); more details regarding the AP formation can be found in Supplementary Note 2. Understanding the precise nature of the AP in the presence of a lattice is an open problem and will be an interesting future research direction both experimentally and theoretically. Here, we identify the AP with the dressed charge complex comprised of X1 and dopant charges. It is important to note that AP formation occurs in both cases when the system is doped, however, a complete transition into AP can only be observed when the ratio of X1 to electron density is small; in other words, when each exciton can be dressed by surrounding electrons. This observation is validated by even lower pump intensity ($<$ 5 nW/$\mu m^{2}$) measurements, as shown in Supplementary Note 3. All the presented data has been taken in this low-pump intensity regime unless stated otherwise.

In this low-pump intensity regime, we identify four distinct spectral features as a function of electron doping as shown in Fig. 1f. The extracted intensity and energy values are shown in Fig. 2a-b, and they can be distinguished as follows:

1. 1.

   Charge-neutral region: dominated by X1 emission,

2. 2.

   Moderate-doping region ($1/7<\nu_{\rm{e}}<6/7$): largely dominated by AP, except at certain fractional fillings (e.g. 1/3, 2/3, and 6/7) that host correlated states.

3. 3.

   Mott-insulating region ($6/7<\nu_{\rm{e}}<1$): dominated by X1. The strong electron localization inhibits the formation of APs in this region.

4. 4.

   High-doping region ($\nu_{e}\!>\!1$): dominated by X2 due to double occupancy of moiré sites (doublon-hole pair).

Having identified the four different doping regimes, we now measure the diffusion dynamics of the quasiparticles of each regime and their dependence on the density of the 2D electron gas.

We extract the diffusion length from the spatially-resolved PL emission by employing a Gaussian fitting routine and analyze it as a function of $\nu_{e}$. Considering that the measured PL profile corresponds to a convolution between quasiparticle diffusion and laser intensity distribution, the diffusion length is given by $L_{\rm{d}}=\sqrt{\sigma_{\rm{X}}^{2}-\sigma_{\rm{L}}^{2}}$, where $\sigma_{\rm{X}}$ and $\sigma_{\rm{L}}$ correspond to the half-widths of the PL and laser profiles, respectively. More details about the fitting procedure are given in SM. The obtained values for the diffusion length are displayed as blue markers in Fig. 2c. In the CN region, the diffusion length is less than 200 nm; over an order of magnitude lower than previously reported measurements in structures without a moiré lattice  (?, ?). Such a suppression in diffusion length indicates that the IX kinetic energy is quenched by the presence of a strong moiré potential in our system. Further suppression of $L_{\rm{d}}$ is observed upon doping the system, i.e. in the region dominated by AP. Remarkably, a significant enhancement of $L_{\rm{d}}$ is noticed as the system is further doped to $\nu_{\rm{e}}\!\sim\!1$, followed by a significant drop in the regime dominated by X2. For further clarity, spectrally resolved diffusion is shown in Supplementary Note 4.

A complete analysis of the IX mobility requires the measurement of its lifetime ($\tau$), as the modulations in diffusion can originate either from the tunneling rate variation or the quasiparticle’s lifetime. We measure this quantity by using a femtosecond pulsed laser with a repetition rate of $\sim\!500$ kHz while the signal is collected in a superconducting single photon detector (see Methods). The observed lifetime as a function of $\nu_{\rm{e}}$ is shown with green markers in Fig. 2c. We notice a trend of lifetime reduction with doping, and modulations at some fractional fillings. In agreement with previous reports  (?), $\tau$ decreases by over an order of magnitude, from $\sim\!1$ $\mu$s in the CN region to $\sim\!30$ ns in the Mott region. The reduction in $\tau$ and enhancement of $L_{\rm{d}}$ in the Mott region suggests the presence of highly mobile particles with low effective mass.

Assuming that the dynamics is diffusive, we can use the diffusion coefficient ($D=L_{\rm{d}}^{2}/\tau$) to study different regimes. We observe a remarkable three-orders of magnitude enhancement of the mobility at the Mott-insulating region, see Fig. 2d. This intriguing behavior can be counter-intuitive at first, as high mobility contrasts with the insulating nature of the electron gas. To comprehend this anomalous behavior, we analyze how the changes in $\nu_{\rm{e}}$ lead to the renormalization of the diffusing particle’s effective mass ($m^{*}$). This effect originates from the Coulomb potential ($V_{\rm{Coul}}$) exerted by the electron gas on the diffusing particles. The addition of $V_{\rm{Coul}}$ and the moiré potential, together with changes in the mass of the diffusing particles determine the large variations in $m^{*}$. An effective model to describe this behavior  (?) and the extracted values of $m^{*}$ for each doping region are presented in Supplementary Note 5.

The four physical situations depicted in Fig. 2e describe how the electrons affect $D$. With the mentioned model, we estimate the energy dispersion of the diffusing particles for each case (Fig. 2f). For the CN region, as $V_{\rm{Coul}}\approx 0$, the electron and hole experience a similar potential landscape in their respective layers, favoring their motion together as a bound particle. As expected for moiré-trapped excitons, we observe a diffusion coefficient several orders of magnitude lower than the values reported for IX in the absence of moiré potential  (?). In the AP region, where $V_{\rm{Coul}}$ remains comparatively negligible, we detect an increase in the effective mass $m^{*}_{\rm{AP}}$. The formation of heavier quasiparticles resulting from the dopant electrons dressing the injected optical excitons leads to the observed reduction in mobility. The decreasing trend in $D$ reverses at $\nu_{\rm{e}}=1/2$. This phenomenon is discussed in more detail in the next section.

The dramatic three-orders-of-magnitude increase in mobility near the Mott region can be attributed to the stronger effective potential in the electron layer and the suppressed potential in the hole layer, as depicted in Fig. 2e. This significant variation in effective potential results from the crystallization of electrons, which generates $V_{\rm{Coul}}$ with the same period and phase as the electron moiré lattice but out of phase with the hole moiré lattice as holes occupy different atomic registry. Additionally, due to the absence of any vacancies in the electron layer, the hole in the other layer is free to hop into any moiré site. Hence, in the Mott region, the hole is not bound to a specific electron and can recombine with any electron in the lattice in a non-monogamous way. This non-monogamous hopping, combined with the reduced effective potential depth, leads to a decrease in $m^{*}$, accounting for the orders-of-magnitude enhancement in $D$. However, beyond $\nu_{\rm{e}}=1$, the excited electrons become mobile again, allowing the electron and hole to move together, in contrast to the previous non-monogamous hole dynamics. This is pictorially shown in the last panel of Fig. 2e, where an additional electron on top of the electron lattice and a hole in the other layer form a diffusing exciton. Interestingly, in this X2 region, both the hole and the electron encounter shallower potentials due to the out-of-phase $V_{\rm{Coul}}$ in each layer. This shallower potential results in orders-of-magnitude increased mobility of X2 compared to X1. However, the monogamous motion manifests as a reduction in mobility compared to the holes in the Mott region. This discussion elucidates the observed strong variation in $D$ across the four different fermionic regions.

The remarkable sensitivity of the mobility to the fermionic state of the system is further manifested at certain fractional fillings that host generalized Wigner crystals due to the long-range repulsive interactions. Before focusing on these specific filling factors, we reiterate that the general trend of $D$ is reversed at $\nu_{\rm{e}}\!=\!1/2$. This observation can be understood by analyzing the occupation of the nearest neighbor (NN) sites of the injected IX below and above $\nu_{\rm{e}}\!=\!1/2$. Figure 3 illustrates how, in the two regimes, the NN filling determines the interplay between the two diffusion mechanisms. At $\nu_{\rm{e}}\!=\!0$, where all the NN sites are vacant, only monogamous motion can take place (Fig. 3a), while for $\nu_{\rm{e}}\!\sim\!1$, non-monogamous motion exclusively determines the exciton mobility due to the complete occupation of the moiré sites (Fig. 3b). For any other filling, the system experiences an interplay between the two diffusion mechanisms.

To elucidate, we make a direct comparison of the available diffusion channels at $\nu_{\rm{e}}\!=\!1/3$ and $\nu_{\rm{e}}\!=\!2/3$. The monogamous motion of the exciton, allowed for $\nu_{\rm{e}}\!=\!1/3$, is blocked by the filled NN sites in the electron layer at $\nu_{\rm{e}}\!=\!2/3$ (Fig. 3c and e). Hence, one would expect lower mobility at $\nu_{\rm{e}}\!=\!2/3$ compared to $\nu_{\rm{e}}\!=\!1/3$. However, the experimental results (Fig. 2d) show a local minimum but an overall increase in $D$ at $\nu_{\rm{e}}\!=\!2/3$ compared to $\nu_{\rm{e}}\!=\!1/3$. To understand this, we note that while the occupied NN sites hinder the monogamous motion of electron and hole, they simultaneously create non-monogamous channels that facilitate the diffusion of holes, as shown by the arrows in the bottom layer of Fig. 3e. Therefore, the increasing trend in $D$ is a consequence of the dominant non-monogamous hole diffusion. This effect gets stronger at $\nu_{\rm{e}}\!=\!6/7$ as the occupation of both NN and next-nearest neighbor (NNN) sites (Fig. 3f) increases the number of diffusion channels in the hole layer. It should be noted that the local minima in $D$ at $\nu_{\rm{e}}\!=\!2/3$ and $6/7$ result from the inhibition of the monogamous motion of IXs because the electrons’ wavefunctions are more localized.

The interplay of the two discussed diffusion mechanisms determines the non-trivial dependence of $D$ in the explored range of $\nu_{\rm{e}}$. In this picture, the dominant mechanism changes at $\nu_{\rm{e}}=1/2$. The monogamous motion of quasiparticles (X1 or AP) determines the mobility in the region 0 $<\nu_{\rm{e}}<$ 1/2, while it is the non-monogamous hole diffusion for 1/2 $<\nu_{\rm{e}}<$ 1. $\nu_{\rm{e}}=1/2$ (Fig. 3d) acts as the turning point, exhibiting the lowest mobility as both processes are effectively suppressed.

Our physical picture of the diffusion mechanisms is limited to a regime of low exciton density at cryogenic temperatures. Additional physical processes affecting the dynamics are expected to emerge for higher densities and temperatures. To gain insights into these more complex regimes, we map the dependence of the diffusion length as a function of $\nu_{\rm{e}}$ and pump intensity. As illustrated in Figure 4a, the enhancement of the mobility near the Mott region is evident for pump intensities less than 100 nW/$\mu m^{2}$, indicated by the white dashed line. Beyond this pump intensity, the sharp peak associated with the Mott insulating region becomes less distinctive and shifts to lower $\nu_{\rm{e}}$, indicating a regime where additional processes such as double exciton occupancies and exciton-exciton scattering become relevant. The monotonic increase in $L_{\rm{d}}$ with pump intensity at charge neutrality (CN) suggests that these scattering processes are enhanced with the exciton density. For pump intensities greater than 1 $\mu$W$/\mu$m², the system enters a non-linear repulsive regime where exciton-exciton dipole repulsion dominates the diffusion dynamics. Figure 4a highlights these different regimes.

The reduction of mobility due to the formation of AP is more sensitive to exciton density. Hence, the features in this region start to fade away at around 50 nW/$\mu$m². In this case, the exciton to polaron transition is blurred out due to the larger population of excitons compared to the dopant electrons in the lattice.

Another crucial parameter for $L_{\rm{d}}$ is the temperature ($T$). Figure 4b shows that the diffusion peak at the Mott insulating state vanishes for $T\!\geq\!200$ K. Although the Mott gap persists over the full measured temperature range (see Supplementary Note 6), no significant dependence of $L_{\rm{d}}$ with $V_{\rm{g}}$ is observed beyond 200 K. This is attributed to increased phonon-assisted scattering mechanisms that dominate the diffusion dynamics at higher temperatures  (?). Additionally, the signatures of AP in the voltage regime 1 V $\!\lesssim\!V_{\rm{g}}\!\lesssim$ 2 V are suppressed above 60 K, as the AP cannot form at elevated temperatures due to its low binding energy.

In conclusion, we unveil the two mechanisms affecting the mobility of a dilute exciton population in an electron-doped moiré system. The demonstrated breakdown of conventional monogamous exciton diffusion and clear signatures of electronic correlations evidence the potential of this technique to probe complex states of matter.

In perspective, a broad range of physical effects can be explored by measuring the mobility of embedded particles in van der Waals structures. For example, the nature of exotic phases of matter, including kinetic magnetism  (?, ?), exciton condensation  (?), and anomalous quantum Hall regimes  (?, ?), could be studied with this technique. In particular, incorporating polarization resolution and fields into this technique offers the possibility to study fractional Chern insulators  (?), Mott-moiré excitons  (?), spin polarons  (?), and tunable electron-exciton interactions  (?, ?). Moreover, the microseconds-long exciton lifetimes allow one to conceive time-resolved diffusion experiments, capable of extracting valuable insights into the temporal dynamics of fermionic correlated states.

The authors acknowledge fruitful discussions with Ming Xie, Ajit Srivastava, and Angel Rubio. YZ acknowledges support from the National Science Foundation under Award No. DMR-2145712. M.K. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy–EXC–2111–390814868 and from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant Agreement No. 851161).

The authors declare no competing interests.

All of the data that support the findings of this study are reported in the main text and Supplementary Material. Source data are available from the corresponding authors on reasonable request.

We fabricate a transition metal dichalcogenide (TMD) hetero-bilayer sample for diffusion measurements. WS₂ and WSe₂ are the two monolayer TMDs used for this purpose. After aligning their edges, they are stacked to achieve 0^o twist angle, i.e., 2H stacking order. The sample is encapsulated with hBN ($\sim$ 35 nm) and gated on both sides for independent control of the out-of-plane electric field and doping. Figure S1a shows an optical micrograph of the sample. We characterize the device using reflection and photoluminescence (PL) measurements. Figures S1b and c present the reflection contrast spectrum of WSe₂ intralayer exciton and PL map of the interlayer exciton (IX) measured on the bilayer region, respectively.

Both measurements show distinctive features at the charge-neutral (CN) region ($\nu_{\rm{e}}$ = 0), the Mott-insulating region ($\nu_{\rm{e}}$ = 1), and the band-insulating region ($\nu_{\rm{e}}$ = 2). Identifying these regions is useful to calibrate the gate voltage ($V_{g}$) to $\nu_{\rm{e}}$. However, for better accuracy, we track the peak intensity variations of the IX species with doping as shown in Fig. 2a (Main text). Here, $\nu_{\rm{e}}\!=\!0$ is benchmarked by the emergence of the peak associated with the attractive polaron (AP). The presence of that peak coincides with a decrease in the X1 intensity. $\nu_{\rm{e}}\!=\!1$ is identified by the complete extinction of the X1 intensity. It is important to mention that a proper calibration of $\nu_{e}$ can only be performed at ultra-low pump intensities (a few nW/$\mu m^{2}$ or lower), because it relies on the features of the polaron gap, only present at these low exciton densities.

Due to the H-stacking order in this system, electrons and holes localize in different moiré registries as discussed in detail in recent literature  (?). This lateral separation of electrons and holes is the key to the formation of interlayer AP in the system. In this configuration, a hole in the WSe₂ layer can bind with three sites in the electron layer with equal probability. This allows the hole in the bottom layer (WSe₂) to pair with a cloud of electrons in the top layer (WS₂). Due to the complexity involved in understanding the precise nature of APs, specifically in the moiré system, we consider a trion-like charge complex, without affecting the conclusion of this work, to explain our observations. In Fig. S2a, we illustrate this exemplary case of AP where an embedded X1 binds with one doped electron. It is important to note that the additional electrons in AP, i.e. other than the optically excited one, should be from the opposite valley to avoid short-range repulsive interactions (Fig. S2b). This is not expected for R-stacked heterobilayers, because in that case, electrons and holes are confined in the same registries, leading to weak repulsive interaction between X1 and doped electrons  (?).

It should be further noted that AP formation can be distinctively observed in PL only for ultra-low excitation power. When the density of excitons is lower than the density of doped electrons in the system, we can have each exciton associated with at least one electron, forming AP. However, here, the density of doped electrons must be small as at higher densities close to $\nu_{\rm{e}}\sim$ 1 the repulsive interactions become strong and inhibit the formation of AP. We illustrate the dependence of AP on exciton and electron density in Fig. S3. For the case of 1 nW/$\mu\rm{m}^{2}$, we have a clear transition from X1 to AP upon doping. This gap is observed to reduce and eventually vanish with excitation pump intensity. Additionally, AP transitions to X1 again at high electron doping close to $\nu_{\rm{e}}\sim$ 1.

Based on the above discussion regarding AP formation, we determine the dilute exciton density regime. A complete polaron gap ($\sim$ 16 meV) is used as a benchmark for achieving the dilute regime. In the main text, we have shown a complete polaron gap for 5 nW/$\mu\rm{m}^{2}$. In Fig. S4, we demonstrate that any pump intensity below 5 nW/$\mu\rm{m}^{2}$ will have a comparable spectrum in the entire range of doping. This confirms our regime of operation.

As discussed in the main manuscript, the surge in diffusion length ($L_{\rm{d}}$) near the Mott-insulating region is a novel observation that warrants a detailed analysis. In Fig. S5a, we show spectrally resolved diffusion collected by using gratings and separating the spectrum into two regions: one below 860 nm (X2) and one above (X1 or AP). We apply the Gaussian fitting routine and extract the diffusion length. As shown by the red (X1 or AP) and purple (X2) markers, diffusion near the Mott region is solely from X1. Although both X1 and X2 have similar intensity in that region (Fig. S5b), X2 demonstrates negligible diffusion. However, for $\nu_{\rm{e}}\geq 1$, the diffusion is dominated by X2 and results in a peak near $\nu_{\rm{e}}$ = 2.

To model the variations in diffusivity with doping, we solve a simplified 1D lattice Hamiltonian: $\hat{H}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dz^{2}}+\frac{V}{2}\sin\left(\frac{2\pi z}{\Lambda}\right).$ Here, $m$ is the mass of the exciton, trion, or hole, depending on the regime of consideration, in the absence of the moiré potential. $V$ is the total potential resulting from adding the moiré potential ($V_{\rm{m}}$) and the Coulomb interactions ($V_{\rm{Coul}}$) which accounts for the modification of the superlattice due the charge order, and $\Lambda$ is the moiré lattice spacing. This Hamiltonian, whose eigenstates in real space are the Mathieu functions, has been successfully used to model interlayer excitons in TMD heterobilayers  (?). Although this model does not capture the geometric details of the moiré lattice, it provides the effective mass ($m^{*}$) of diffusing particles. $m^{*}$, determined by the values of $m$ and $V$ in each regime, is a critical parameter due to its inverse proportionality to the mobility. Extracting the variation in $m^{*}$ allows us to understand the rich behavior shown in Fig. 2d of the main text.

To evaluate the renormalization of the effective moiré potential ($V$) we consider certain parameters: $V_{\rm{m}}$ $\sim$ 100 meV in each layer  (?), the mass of electron ($m_{\rm{e}}$) and hole ($m_{\rm{h}}$) in WS₂ and WSe₂ is considered to be 0.5 $m_{\rm{0}}$ and 0.4 $m_{\rm{0}}$, respectively  (?). Here, $m_{\rm{0}}$ is the free electron mass. The list of $m$, $V$ and $m^{*}$ values for the diffusing particle in the corresponding fermionic region are listed in Table S1. For the charge-neutral (CN) region, $V$ for X1 is obtained by adding $V_{\rm{m}}$ of both layers. It can be noted that in the 1D picture, we do not account for the rotational motion of electrons and hence, direct addition is considered to be a valid approximation. For the AP region, $V$ can be assumed to remain the same as in the CN region. For our calculations, we consider the simplest scenario where an excited X1 binds to one doped electron (trion-like). Developing a full theoretical model for the AP is an open problem in the presence of a lattice and would be an interesting future direction. Here, we assume a trion picture where we can approximate its mass as 2$m_{\rm{e}}$ + $m_{\rm{h}}$. This increase in $m$ leads to a much stronger localization effect in the presence of the moiré lattice. This effect is captured by the increase in $m^{*}$ indicating slower diffusion of AP than X1.

In the Mott region, electrons in WS₂ crystallize due to strong electron-electron repulsion. However, holes in WSe₂ experience a weak effective potential because of the addition of an out-of-phase periodic potential ($V_{\rm{Coul}}\sim 88$ meV) exerted by the doped electron lattice (WS₂) on the holes. The obtained $V_{\rm{Coul}}$ is out-of-phase compared to $V_{\rm{m}}$ of the hole layer because the electrons and holes reside in different moiré registries, which are laterally displaced. This leads to a strong suppression in $V$ ($\sim$ 12 meV), allowing for non-monogamous hole diffusion.

For X2 diffusion, $V$ for the electron and hole is calculated separately and then added. The potential $V$ for the hole layer remains roughly the same as in the Mott region, as the hole still experiences the strong $V_{\rm{Coul}}$ due to the electron lattice. However, for the electron layer, there are strong repulsive on-site interactions between the doped electron and the excited electron. This effectively increases the energy of the excited electron, which then sees a shallower potential. We approximate $V$ as $V_{\rm{m}}$ - $U_{\rm{e-e}}$, where $U_{\rm{e-e}}$ represents the on-site electron-electron repulsions. For H-stacking, this can be approximated as the exciton-electron repulsion, which is observed from PL to be around 46 meV. Therefore, $V$ of 54 meV for the electron layer and 12 meV for the hole layer give an overall $V$ of 66 meV experienced by X2. This leads to an increase in the effective mass ($m^{*}$) compared to the Mott region, but it remains orders of magnitude smaller than in the CN region.

We perform temperature-dependent PL measurements to verify the presence of the Mott gap in the entire range of temperatures used for measurements. Figure S6 demonstrates the Mott gap even at 260 K, although the gap reduction is observed with temperature.