Abstract
Interactions between outofplane dipoles in bosonic gases enable the longrange propagation of excitons. The lack of direct control over collective dipolar properties has so far limited the degrees of tunability and the microscopic understanding of exciton transport. In this work we modulate the layer hybridization and interplay between manybody interactions of excitons in a van der Waals heterostructure with an applied vertical electric field. By performing spatiotemporally resolved measurements supported by microscopic theory, we uncover the dipoledependent properties and transport of excitons with different degrees of hybridization. Moreover, we find constant emission quantum yields of the transporting species as a function of excitation power with radiative decay mechanisms dominating over nonradiative ones, a fundamental requirement for efficient excitonic devices. Our findings provide a complete picture of the manybody effects in the transport of dilute exciton gases, and have crucial implications for studying emerging states of matter such as Bose–Einstein condensation and optoelectronic applications based on exciton propagation.
Main
Exciton transport has been proposed as a potential basis for realizing scaled optical interconnects and modulators in chipscale optical processing systems^{1}. Strongly bound and longlived propagating excitons can act as information carriers within a semiconductor—a desirable prospect for photonic circuits^{2,3}. In particular, spatially indirect excitons can propagate with micrometrescale diffusion lengths. They can be controlled via the quantumconfined Stark effect, enabling tuning of their potential energy by an applied vertical electric field^{4,5}. Van der Waals heterostructures of twodimensional (2D) materials have been used as platforms for manipulating spatially indirect interlayer excitons (IXs)^{6}. In particular, typeII transition metal dichalcogenide (TMDC) heterostructures have been employed to realize excitonic devices^{7} and circuits^{8}. Heterostructure devices showing roomtemperature switching of exciton currents^{7}, tunable valleypolarized emission^{9} and micrometrescale transport of polarized exciton currents^{10} have been demonstrated.
The spatial separation of charges comprising IXs gives rise to fixed outofplane dipole moments^{11,12}. Repulsive Coulomb interactions between exciton populations of outofplane dipolar ensembles induce anomalous transport dynamics that deviate from the standard diffusive propagation that is characteristic of bosonic gases^{13,14,15,16}. Instead, excitons generated in single TMDC layers, also called intralayer excitons, are mainly influenced by quantummechanical exchange interactions^{17}. The interplay between dipolar and exchange interactions strongly depends on the outofplane exciton dipole length. Layerhybridized states with intra and interlayer components are therefore expected to show variable effective outofplane dipole lengths^{18}. The ability to control the degree of exciton hybridization is highly desirable as a means to modify the concurrent manybody interactions and tune the anomalous diffusion of exciton ensembles. While layerhybridized states in twisted moiré heterostructures and their Stark shifts with applied electric fields have been previously investigated^{19,20,21}, such typeII heterostructures do not allow tuning of exciton–exciton interactions because the dipole moment of the emitting species is intrinsically fixed by layer arrangement. Moreover, the moiré superlattice that is formed in stacked bilayers has been shown to induce periodic potential traps that dramatically reduce the effective diffusivity of outofplane excitons^{22,23}. This effect hampers reliable longrange exciton transport, making moiréless structures preferable for manipulating longrange propagating dipolar gases in excitonic devices.
Here we exploit concurrent intervalley transitions in natural WSe_{2} homobilayers to control the layer hybridization of exciton states by applying a vertical electric field. 2HWSe_{2} homobilayers are moiréless structures that have been indicated to be a natural platform for Bose–Einstein condensation of IX states^{24}. However, an indepth study of the dynamics and transport of layerhybridized tunable exciton states in this platform is still lacking. Furthermore, we achieve electrostatic control over hybrid IX (hIX) transport in a structure with no moiré potential by varying the interplay between Coulombic dipolar repulsions and attractive exchange interactions. Our work sheds light on the influence of dipole length and hybridization on longrange IX transport and is supported by a microscopic theory. Moreover, we show that the propagating exciton species in this platform are characterized by a quantum yield that is constant in power with dominant radiative recombination channels, independently of the layer hybridization. The study of electrically tunable dipolar ensembles with constant quantum yield and micrometrescale transport opens the way to realizing efficient excitonic devices based on van der Waals heterostructures of 2D materials.
Results
Electrically tunable interlayer dipolar ensembles
Our devices consisted of natural WSe_{2} homobilayers, fully encapsulated in hexagonal boron nitride (hBN), with a bottom Cr/Pt gate and a top semitransparent Pt gate. The heterostacks were assembled on an SiO_{2} substrate by mechanical transfer (Methods). Figure 1c shows the image of an ultraclean encapsulated bilayer WSe_{2} device (A), acquired by a.c.mode atomic force microscopy. Images of a second device (B) are included in Supplementary Note 1. Natural 2HWSe_{2} homobilayers host momentumindirect spinbright KΛ and KΛ′ transitions as the energetically lowest states^{25,26} involving holes at the K/K′ points and electrons at the Λ/Λ′ points of the Brillouin zone (Fig. 1a). Given their indirect nature, excited states in WSe_{2} appear in the photoluminescence (PL) spectrum as their phonon replicas^{25}. Intervalley excitons in bilayer WSe_{2} are hybrid^{27} in their intra and interlayer components^{28}. Our device architecture (Fig. 1b) allowed us to modulate the PL emission from layerhybridized intervalley excitons in bilayer WSe_{2} with an increasing applied vertical electric field E_{z}, causing a shift in the loweststate transition from KΛ (K′Λ′) to KΛ′ (K′Λ) (Supplementary Table 1 and Supplementary Note 2). The states in brackets represent degenerate states with opposite dipole moments^{18}. In the presence of a positive vertical field with magnitude E_{z} = 300 mV nm^{−1}, the interlayer mixing coefficient of the energetically lowest state (KΛ′) was calculated to be \(\left {C_{\mathrm{IX}}^{{\mathrm{K}}{{\Lambda }}^\prime }} \right^2 = 0.80\). We note that hybrid excitons are expected to have higher oscillator strengths than purely interlayer species, as they exhibit higher radiative decay rates. This is a desirable feature for highertemperature operation. However, further studies are needed to evaluate the influence of nonradiative dynamics on hIX recombination at room temperature. In this work, we focused on the performance and tunability of hIX transport at a temperature of 4 K.
The potential energy U of outofplane electron–hole pairs with fixed dipole lengths d can be modulated by the linear quantumconfined Stark effect as \(\updelta U \approx dE_z\). The degree of interlayer character of hIXs is highly tunable via the application of a vertical electric field. It can then be observed through the Stark effect acting on the outofplane component of the transitions of interest. From the fielddependent PL spectra of device A (Fig. 1d,e), we could distinguish two main ranges corresponding to the favourable transitions KΛ and K′Λ′ (\(\left {E_z} \right < 200\,{{{\mathrm{mV}}}}\,{{{\mathrm{nm}}}}^{  1}\)) or KΛ′ and K′Λ (\(\left {E_z} \right > 200\,{{{\mathrm{mV}}}}\,{{{\mathrm{nm}}}}^{  1}\)). The peaks assigned to the corresponding transitions are discussed in Supplementary Note 2. From the linear Stark shift of the PL peaks with the highest intensity, we extracted different effective outofplane dipole lengths d_{eff} with respect to the vertical field on the basis of the prevalent emitting states^{29} (Fig. 1f and Supplementary Note 3). The brightest PL peak can shift between different phonon replicas. In particular, we observed a clear shift from the lowfield to the highfield dominant transition at \(\left {E_z} \right = 200\,{{{\mathrm{mV}}}}\,{{{\mathrm{nm}}}}^{1}\). The electric field ranges corresponding to specific dominant transitions are highlighted in Fig. 1f. In particular, KΛ and K′Λ′ excitons were characterized by smaller dipole lengths (\(d_{{{{\mathrm{eff}}}}} \simeq 0.1\,{{{\mathrm{nm}}}}\)) with respect to the KΛ′ and K′Λ counterparts (d_{eff} > 0.2 nm)^{29,30}. In our case, high positive and high negative vertical electric fields linked to the KΛ′ and K′Λ transitions are related to dipolar ensembles 0.41 nm and 0.24 nm long, respectively (Supplementary Note 3). Asymmetries in the fielddependent behaviour of outofplane transitions have been reported as a function of doping^{31}. In Supplementary Note 4 we show how the collective dipole moment of highd transitions can be effectively modulated by gating. In fact, effective outofplane dipole lengths of collective ensembles can be tuned by induced or intrinsic doping^{31}. In particular, d_{eff} modulations on the order of ångströms are attributed to the electric field screening of the exciton wavefunctions. Thus, we attributed the difference between the two branches in the Stark shift measurements of Fig. 1e to the presence of intrinsic doping in the WSe_{2} homobilayers used in this work.
Fieldeffect control of hybrid exciton transport
Our system hosts layerhybridized excitons characterized by different effective lengths, allowing us to determine the dipoledependent transport properties of strongly interacting exciton gases in the dilute regime. Purely IX gases with large separations between electrons and holes are characterized by negligible exchange forces^{14,17,32}. Hybrid ensembles, however, host sizeable intra and interlayer components at every given field^{33}. Thus, both Coulombic dipolar repulsions and attractive exchange interactions give rise to renormalized hybrid exciton energies. Their interplay is dictated by the level of interlayer hybridization, which we modulated by applying a vertical electric field. By tuning the hybridization and d_{eff} of the probed excitons, we achieved control over the concurrent manybody interactions in the micrometrescale transport of dilute exciton gases. We studied the tunable manybody interactions by measuring the steadystate effective diffusion area of hIXs in the presence of an applied electric field (Fig. 2a). With negligible E_{z}, shortrange exciton transport was observed due to the prevalence of the intralayer component in the energetically degenerate KΛ and K′Λ′ states. Such lowfield transitions featured randomly oriented dipoles and negligible repulsive interactions. However, with higher positive or negative fields, sizeable outofplane dipole lengths and greater interlayer mixing of hybrid states resulted in stronger collective repulsive forces. Consequently, we were able to electrostatically enhance the steadystate exciton gas expansion by a progressive transition from lowd to highd dominated ensembles with increasing interlayer components (Fig. 2b).
We could extract a lowerbound estimate of purely IX densities from the measured blueshift using the parallelplate capacitor model^{34}. However, layerhybridized excitons in WSe_{2} homobilayers show sizeable attractive exchange interactions, resulting in a densitydependent redshift that counteracts the effect of dipolar repulsive Coulomb forces. To delve into the manybody picture of strongly interacting dipolar gases, we developed a microscopic theory that accounts for the two main components driving excitonic transport in hybrid forms. We studied hIX interactions by deriving a hybrid exciton–exciton interaction Hamiltonian that we used to disentangle the main contributions to the densitydependent exciton energy renormalization (Supplementary Note 5):
where \(g_{\mathrm{d  d}}\) is the dipole–dipole interaction strength, which is negligible for intralayer excitons in monolayer TMDCs and dominates for spatially separated IXs; \(g_{x  x}\) accounts for exchange interactions, which are highly dependent on the outofplane separation for interlayer states^{33}. The interactions are weighted by the valleyspecific exciton density \(n_x^\xi\), where the total exciton density n_{x} is given by \(n_x = \mathop {\sum}\nolimits_\xi {n_x^\xi }\). Equation (1) takes into account all contributions from the different intervalley transitions, with \(\xi = {\mathrm{K}}{{\Lambda }},{\mathrm{K}}^\prime {{\Lambda }}^\prime ,{\mathrm{K}}{{\Lambda }}^\prime ,{\mathrm{K}}^\prime {{\Lambda }}\) (\({\bar {\xi}}\) denotes the opposite valley, that is if \(\xi = {\mathrm{K}}{{\Lambda }}\), then \({\bar {\xi}} = {\mathrm{K}}^\prime {{\Lambda }}^\prime\)). In Fig. 2c,d, we show that the densitydependent energy normalization for hIXs in WSe_{2} homobilayers is highly dependent on the vertical electric field as the hybrid exciton–exciton interaction crucially depends on the interlayer mixing coefficients. We have calculated the energy shifts for layerhybridized excitons in undoped WSe_{2} homobilayers by solving a hybrid exciton eigenvalue problem that accounts for mixing between intra and interlayer exciton states^{18} (Supplementary Note 2).
The d_{eff} of the hIXs was extracted by fitting a linear function to the energy shift \(\Delta E^\xi = n_xd_{{{{\mathrm{eff}}}}}^\xi /{\it{\epsilon }}\), with \({\it{\epsilon }} = {\it{\epsilon }}_0{\it{\epsilon }}_ \bot\) where \({\it{\epsilon }}_0\) is the vacuum permittivity and \({\it{\epsilon }}_ \bot\) is the outofplane component of the dielectric tensor of the TMDC. The tunable effective outofplane dipole length of the exciton species is directly related to the level of layer hybridization, with d_{IX} = 0.5 nm for purely interlayer states. For the simulated ideal energy shifts for the dominant KΛ′ transition (interlayer mixing \(\left {C_{\mathrm{IX}}^{{\mathrm{K}}{{\Lambda }}^\prime }} \right^2 = 0.8\) for E_{z} = 300 mV nm^{−1}), we predicted d_{eff} = 0.32 nm. The calculated value is consistent with our measured values ranging from 0.24 nm to 0.41 nm in the presence of intrinsic doping. While excitons with a negligible interlayer component show no densitydependent shift in energy, as discussed in Supplementary Note 5, species with sizeable net outofplane dipole lengths show an increase in the magnitude of the blueshift at higher vertical electric fields. We characterized the measured renormalization shifts from highd (0.41 nm) and lowd (0.24 nm) ensembles by observing a linear blueshift at low optical excitation intensity (P_{in} < 150 μW) followed by ΔE saturation. We ascribed this saturation to lattice heating at high exciton densities, as previously observed with spatially indirect excitons in double quantum well systems (Supplementary Note 6).
Radiative recombination of hIXs with a powerindependent quantum yield
Spatially separated exciton species are characterized by longer radiative lifetimes with respect to their intralayer counterparts as their electron and hole wavefunctions feature a smaller overlap and, consequently, a lower probability of recombining^{13}. Figure 3a shows the Stark shift of the main PL peaks and their measured lifetime in device B, highlighting the relationship between fielddependent lifetime and the change in the loweststate emitting intervalley species. Comparable results were obtained in device A, as reported in Supplementary Note 7. As a result of the longer effective outofplane dipole length of the hybrid excitations with higher fields, we observed increased lifetimes when the main emitting transition shifted from KΛ (K′Λ′) to KΛ′ (K′Λ). Excitons belonging to the former states in device B were characterized by average lifetimes of 0.40 ns, whereas the maximum value reached for the latter was around 0.75 ns (device A in Supplementary Note 7). The hBN thicknesses in device B were low enough to enable photoassisted tunnelling of carriers at high electric fields (E_{z} > 200 mV nm^{−1}) in the form of a photocurrent (Supplementary Note 7). We attribute the sudden drop in the hIX lifetime at large positive and negative fields to the dissociation of excitons and their tunnelling through the hBN barriers to the top and bottom gate electrodes^{35,36} (Supplementary Note 7).
It has been shown that the main nonradiative channel that affects the quantum yield of TMDCs is powerdependent exciton–exciton annihilation^{37}. In our case, we observed a singleexponential timeresolved PL decay independently of the applied vertical electric field and of the input optical power (Supplementary Note 7). Moreover, a linear relationship between the integrated PL intensity and the input pump intensity indicated that the quantum yield of the probed exciton species was constant with power (Fig. 3b). On the basis of these findings, we concluded that hIXs in WSe_{2} homobilayers do not recombine via secondorder powerdependent nonradiative channels. This is a significant difference from IXs in previously investigated typeII heterostructures. For purely IXs in typeII structures, densitydependent nonradiative terms have been shown to induce a decrease in the emission quantum yield of the dipolar ensembles at high excitation intensities^{10,13,14}. We note that, consistently with our findings, excitons in WSe_{2} homobilayers have been reported to feature suppressed exciton–exciton annihilation channels at room temperature, with a powerindependent quantum yield that does not require chemical treatment or induced strain^{38}. This can be explained by the decreased overlap between excitonic wavefunctions due to hybridization^{39}, together with the absence of higherlying states that fulfil energy and momentum conservation^{40}.
However, even though a single powerindependent decay channel is found for our hIXs, further analysis is required to determine its radiative and nonradiative composition. Jauregui et al.^{13} have established that it is possible to retrieve quantitative information about nonradiative decay channels from the fielddependent lifetime of purely IXs. A nonradiative decay rate was found even at low power for purely interlayer species in MoSe_{2}/WSe_{2}. By applying the same approach to the fielddependent lifetime of hIXs, we observed a linear increase in the maximum PL intensity with respect to lifetime (Fig. 3c). A linear trend with a positive coefficient, together with the constant quantum yield and singleexponential decays, indicates that the probed hIXs undergo mostly radiative recombination even at high excitation powers (Supplementary Note 7). Thus, in the absence of photoassisted tunnelling, powerindependent radiative decays are dominant for all electric fields.
The absolute quantum yield of hIXs will also be critically dependent on E_{z}. With respect to purely interlayer species in heterobilayers, lower absolute yields are expected from hIXs at low E_{z} and low P_{in} due to their momentumindirect nature. However, purely IXs are characterized by emission yields that are both power and fielddependent, with a strong decay in power due to secondorder effects. Thus, we expect that at high P_{in} and high E_{z}, the absolute yields of hIXs and IXs would become comparable. Future work could be dedicated specifically to quantitative comparisons between the absolute quantum yields of farpropagating outofplane excitonic species among different platforms.
These features indicate that the hybrid species in WSe_{2} homobilayers are promising for further studies of the propagation of highly interacting excitons, as platforms with constant emission quantum yields are required for the realization of efficient devices based on excitonic transport.
Timeresolved transport properties of tunable hIXs
To fully understand the nature of interactions between propagating electrically tunable dipolar ensembles (Fig. 4b), we studied timedependent hybrid exciton transport in our structure. To this purpose, we excited our sample with a picosecond pulsed laser and imaged the spatiotemporal expansion of the exciton cloud by its PL emission using a scanning avalanche photodiode system (Supplementary Note 8 and Methods)^{14,41}. The spatially resolved exciton cloud corresponding to the highd species is shown in Fig. 4a for different points in time. The effective hIX area as a function of time is reported for highd and lowd ensembles in Fig. 4d. The equation of motion for the spatially resolved IX density is derived as (Supplementary Note 9):
which takes the form of a 2D drift–diffusion equation, where D is the diffusion coefficient, μ_{m} = D/k_{B}T is the exciton mobility, k_{B} is the Boltzmann constant, T is temperature and τ is the exciton lifetime. The energy renormalization term from equation (1) is now variable in space (r) and time (t) through n(r, t). Dipolar repulsions in the hIX equation of motion (equation (2)) cause a nonlinear response to a pump excitation in the form of anomalous diffusion (Fig. 4c). We therefore introduced an effective diffusivity term D_{eff}(t), defined as the rate of change of the hIX cloud area (Supplementary Note 9). In hIX transport dynamics, we can distinguish between two main regimes at short (t < 1 ns) and long (t ≫ 1 ns) times after the laser pulse. The former regime is densitydependent anomalous diffusion, taking place when the hIX density is large. Here, D_{eff}(t) is significantly enhanced with respect to the intrinsic D due to the interacting nature of hIXs (Supplementary Note 9). At longer time delays from the laser pulse, as the hIX density decreases, transport is dominated by conventional diffusion and D_{eff}(t) converges to the intrinsic D.
Figure 4e,f shows the time evolution of the simulated hIX area and effective diffusivity for the dipole lengths of interest. The initial exciton density in transport simulations was estimated on the basis of a bestfit approach to the experimental results as \(n_0 \simeq 10^{12}\,{{{\mathrm{cm}}}}^{  {{{\mathrm{2}}}}}\) (Supplementary Note 9), below the exciton Mott transition limit in our system^{42}. As the highest exciton densities in the spot area were obtained for n(r, 0), the maximum hIX effective diffusivity was found at the limit of t→0 for all dipolar species. Increasing \(D_{{{{\mathrm{eff}}}}}^{\mathrm{MAX}}\) values were obtained for ensembles with higher d_{eff}, with \(D_{0.24{{{\mathrm{nm}}}}}^{\mathrm{MAX}} \approx 7\,{{{\mathrm{cm}}}}^2\,{{{\mathrm{s}}}}^{  1}\) and \(D_{0.41{{{\mathrm{nm}}}}}^{\mathrm{MAX}} \approx 11\,{{{\mathrm{cm}}}}^2\,{{{\mathrm{s}}}}^{  1}\) estimated from our simulations (Fig. 4e,f). We note that the simulated transport for highd ensembles produced a faster initial anomalous diffusion regime than the experimental data (t < 1 ns), thus causing an overshoot of the resulting effective diffusivity. Instead, good agreement between theoretical and measured data was reached in the lowd case. Therefore, given the trends observed in the anomalous diffusion regime, we concluded that both highd and lowd species show similar maximum effective diffusivities in the range of 5–10 cm^{2} s^{−1}. These diffusivity values correspond to an upper range of effective exciton mobility reaching \(\mu _{{{{\mathrm{eff}}}}}^{\mathrm{MAX}} \approx 10,000\,{{{\mathrm{cm}}}}^2\,{{{\mathrm{V}}}}^{  1}\,{{{\mathrm{s}}}}^{  1}\) for high hIX densities and highd transitions in the regime of anomalous diffusion.
The effective diffusivities of all probed excitons decreased monotonically in time, progressively saturating to \(D_{{{{\mathrm{eff}}}}}\left( \infty \right) = D\), towards a regime of conventional diffusion. We note that \(D_{{{{\mathrm{eff}}}}}\left( \infty \right)\) is independent of the effective dipole length as it is equivalent to the conventional diffusivity for an exciton gas at low excitation densities. The unaltered D was experimentally estimated by extracting the effective diffusivity of the hIX with a minimal interlayer character. Thus, by measuring the propagation of exciton ensembles at E_{z} = 0 mV nm^{−1}, we obtained a conventional diffusivity of \(D \simeq 0.32\,{{{\mathrm{cm}}}}^2\,{{{\mathrm{s}}}}^{  1}\) (Supplementary Note 10).
Conclusions
Outofplane dipolar ensembles of optical excitations travel long distances owing to the strength of their repulsive forces, governed by the effective interlayer dipole length. In this work we achieved control over the layer hybridization of exciton states in a van der Waals homobilayer structure, allowing us to tune the effective dipole length of exciton ensembles. We characterized the dipoledependent propagation of hIXs by modulating the interplay between attractive exchange interactions and repulsive Coulomb forces, the manybody effects governing exciton transport. The recorded lifetimes of hIXs in WSe_{2} homobilayers were lower than those reported for purely interlayer species (~1–600 ns)^{6,11,13}. However, hIXs are characterized by an intrinsic diffusivity of \(D_{\mathrm{hIX}} \simeq 0.3\,{{{\mathrm{cm}}}}^2\,{{{\mathrm{s}}}}^{  1}\), which is twice the diffusivity estimated for longdipole excitons in moiréless MoSe_{2}/hBN/WSe_{2} heterotrilayers^{14}, and significantly higher than those measured for MoSe_{2}/WSe_{2} heterobilayers at 4 K (ref. ^{23}). We also characterized the fieldtunable regime of anomalous diffusion by spatiotemporally resolved measurements. Our study revealed a peak effective diffusivity of ~10 cm^{2} s^{−1}, corresponding to an effective exciton mobility in the range \(\mu _{{{{\mathrm{eff}}}}}^{\mathrm{MAX}} \approx 10{,}000\,{{{\mathrm{cm}}}}^2\,{{{\mathrm{V}}}}^{  1}\,{{{\mathrm{s}}}}^{  1}\).
The main factors affecting the efficiency of future interconnects and circuits based on exciton transport in van der Waals heterostructures are determined by the material absorption, the exciton mobility and the emission quantum yield. We have obtained powerindependent high emission quantum yields of longrange propagating dipolar ensembles, a crucial step towards the efficient modulation of light in excitonic devices based on 2D materials. Furthermore, we exploited the spatiotemporal expansion of hybrid species to demonstrate that tunable repulsive dipolar exciton–exciton interactions can be achieved. This tunability makes WSe_{2} homobilayers highly attractive for future studies of strongly interacting bosonic systems.
Our microscopic understanding and control of the manybody effects governing the transport of dipolar exciton ensembles open avenues toward exploring exciton condensates in van der Waals structures and the realization of efficient excitonic devices based on 2D materials.
Methods
Device fabrication
All the devices used in this work comprised bottom gates fabricated using electronbeam lithography and metal evaporation (2 nm Cr/5 nm Pt) over a SiO_{2}/Si substrate with an oxide thickness of 270 nm. The heterostructures in devices B and C were fabricated with a polymerassisted wet transfer method. WSe_{2} (HQ Graphene) and hBN flakes were exfoliated on a polymer double layer, and bilayer WSe_{2} flakes were identified by atomic force microscopy measurements. The bottom polymer layer of the substrate was dissolved using a solvent, and the top polymer layer, together with the exfoliated flakes, was left freefloating. The bottom hBN layers, WSe_{2} bilayer and the top hBN layers were then carefully aligned and transferred in sequence on top of the bottom gates by using a dedicated homebuilt transfer setup with motorized micromanipulators. The heterostructure of device A was fabricated using a drytransfer technique employing polycarbonate membranes. WSe_{2} (HQ Graphene) and hBN flakes were exfoliated on a SiO_{2} substrate, identified by optical contrast and subsequently picked up using a single polycarbonate membrane. The heterostack was then released onto the Cr/Pt bottom gate by progressive adhesion following a temperature gradient above 150 °C. This drytransfer technique allowed us to obtain largearea structures by avoiding contamination with polymer residues and water droplets. The optical images and atomic force microscopy measurements of the devices are shown in Supplementary Note 1. All of the heterostructures were annealed under a high vacuum (10^{−6} mbar) for 6 h at a temperature of 340 °C. Top gates and electrical contacts were then fabricated by electronbeam lithography and evaporation of Pt (4 nm) and Ti/Au (2 nm/80 nm) layers, respectively.
Optical measurements
All optical measurements were performed in a vacuum at 4.6 K, unless stated otherwise, in a Heflow cryostat. Hybrid excitons were excited with a confocal microscope, while the emitted photons were collected through the same objective that had a working distance of 4.5 mm and a numerical aperture of 0.65. Optical pumping was achieved with a continuouswave 640 nm diode laser (PicoQuant, LDHIB640M) focused to the diffraction limit (spot fullwidth at halfmaximum of 1.2 μm) for steadystate measurements. For microphotoluminescence (μPL) spectral measurements, the emitted light was filtered by a 650 nm longpass edge filter and then acquired using a spectrometer (Princeton Instruments SpectraPro 500) and recorded with a CCD (chargecoupled device) camera (Princeton Instruments, Blaze 400HR/HRX). Spatial imaging of the IX emission was performed using a CCD camera (Andor Ixon) with an 800 nm longpass edge filter that removed both the laser line and the intralayer emission from WSe_{2}. For timeresolved measurements, the same solidstate laser was driven in a pulsed mode, achieving pulse widths lower than 160 ps at a repetition rate of 80 MHz. The collected photons were sent to an avalanche photodiode (APD, Excelitas Technologies, SPCMAQRH16) mounted on a 2D motorized translational stage. The output of the APD was connected to a timecorrelated photoncounting module with a resolution of 12 ps r.m.s. (PicoQuant, PicoHarp 300), which measured the arrival time of each photon. We set the time bin to 16 ps for the measurements presented in this work. The singlephoton timing resolution of the APD is ~350 ps, which is the main time limitation for this setup. The technical details can be found in Supplementary Note 8.
Microscopic manyparticle theory
To study hybridized exciton states and anomalous exciton transport at elevated electron–hole densities in TMDC bilayers, we derived a manybody Hamiltonian on a hybrid exciton basis that contained a kinetic part and a part due to exciton–exciton interactions. By solving the bilayer Wannier equation, we gained access to pure intra and interlayer exciton states. These were used as input to a hybrid exciton eigenvalue problem that accounted for mixing between intra and interlayer exciton states. We found that momentumdark KΛ (K′Λ′) excitons represented the energetically lowestlying exciton states in naturally stacked WSe_{2} bilayers. We included an outofplane electric field by exploiting the Stark shift of the IX resonance, allowing us to tune the exciton landscape in bilayers as a function of the electrical field (Supplementary Note 2). The obtained hybrid exciton states were then used to compute densitydependent energy renormalizations obtained via the Heisenberg equation of motion (Supplementary Note 5). By allowing the exciton density to be spatially dependent, we found that the exciton–exciton interaction acts as a source to a drift term in a drift–diffusion equation. We gained access to the spatiotemporal dynamics of hybrid excitons by solving the drift–diffusion equation for hybrid excitons with different levels of fielddriven interlayer mixing corresponding to different effective dipole moment lengths (Supplementary Note 9).
Data availability
The data that support the findings of this study are available via Zenodo at https://doi.org/10.5281/zenodo.7660668.
References
Baldo, M. & Stojanović, V. Excitonic interconnects. Nat. Photon. 3, 558–560 (2009).
Butov, L. V. Excitonic devices. Superlattices Microstruct. 108, 2–26 (2017).
PereaCausin, R. et al. Exciton optics, dynamics, and transport in atomically thin semiconductors. APL Mater. 10, 100701 (2022).
Grosso, G. et al. Excitonic switches operating at around 100 K. Nat. Photon. 3, 577–580 (2009).
Shanks, D. N. et al. Interlayer exciton diode and transistor. Nano Lett. 22, 6599–6605 (2022).
Ciarrocchi, A., Tagarelli, F., Avsar, A. & Kis, A. Excitonic devices with van der Waals heterostructures: valleytronics meets twistronics. Nat. Rev. Mater. 7, 449–464 (2022).
Unuchek, D. et al. Roomtemperature electrical control of exciton flux in a van der Waals heterostructure. Nature 560, 340–344 (2018).
Liu, Y. et al. Electrically controllable router of interlayer excitons. Sci. Adv. 6, eaba1830 (2020).
Ciarrocchi, A. et al. Polarization switching and electrical control of interlayer excitons in twodimensional van der Waals heterostructures. Nat. Photon. 13, 131–136 (2019).
Unuchek, D. et al. Valleypolarized exciton currents in a van der Waals heterostructure. Nat. Nanotechnol. 14, 1104–1109 (2019).
Rivera, P. et al. Observation of longlived interlayer excitons in monolayer MoSe_{2}–WSe_{2} heterostructures. Nat. Commun. 6, 6242 (2015).
Merkl, P. et al. Ultrafast transition between exciton phases in van der Waals heterostructures. Nat. Mater. 18, 691–696 (2019).
Jauregui, L. A. et al. Electrical control of interlayer exciton dynamics in atomically thin heterostructures. Science 366, 870–875 (2019).
Sun, Z. et al. Excitonic transport driven by repulsive dipolar interaction in a van der Waals heterostructure. Nat. Photon. 16, 79–85 (2022).
Lopriore, E., G. Marin, E. & Fiori, G. An ultrafast photodetector driven by interlayer exciton dissociation in a van der Waals heterostructure. Nanosc. Horiz. 7, 41–50 (2022).
Erkensten, D., Brem, S., PereaCausín, R. & Malic, E. Microscopic origin of anomalous interlayer exciton transport in van der Waals heterostructures. Phys. Rev. Mater. 6, 094006 (2022).
Erkensten, D., Brem, S. & Malic, E. Excitonexciton interaction in transition metal dichalcogenide monolayers and van der Waals heterostructures. Phys. Rev. B 103, 045426 (2021).
Hagel, J., Brem, S. & Malic, E. Electrical tuning of moiré excitons in MoSe_{2} bilayers. 2D Mater. 10, 014013 (2022).
Shimazaki, Y. et al. Strongly correlated electrons and hybrid excitons in a moiré heterostructure. Nature 580, 472–477 (2020).
RuizTijerina, D. A. & Fal’ko, V. I. Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transitionmetal dichalcogenides. Phys. Rev. B 99, 125424 (2019).
Tang, Y. et al. Tuning layerhybridized moiré excitons by the quantumconfined Stark effect. Nat. Nanotechnol. 16, 52–57 (2021).
Choi, J. et al. Moiré potential impedes interlayer exciton diffusion in van der Waals heterostructures. Sci. Adv. 6, eaba8866 (2020).
Li, Z. et al. Interlayer exciton transport in MoSe_{2}/WSe_{2} heterostructures. ACS Nano 15, 1539–1547 (2021).
Shi, Q. et al. Bilayer WSe_{2} as a natural platform for interlayer exciton condensates in the strong coupling limit. Nat. Nanotechnol. 17, 577–582 (2022).
Lindlau, J. et al. The role of momentumdark excitons in the elementary optical response of bilayer WSe_{2}. Nat. Commun. 9, 2586 (2018).
Deilmann, T. & Thygesen, K. S. Finitemomentum exciton landscape in mono and bilayer transition metal dichalcogenides. 2D Mater. 6, 035003 (2019).
Wilson, N. R. et al. Determination of band offsets, hybridization, and exciton binding in 2D semiconductor heterostructures. Sci. Adv. 3, e1601832 (2017).
Brem, S. et al. Hybridized intervalley moiré excitons and flat bands in twisted WSe_{2} bilayers. Nanoscale 12, 11088–11094 (2020).
Altaiary, M. M. et al. Electrically switchable intervalley excitons with strong twophonon scattering in bilayer WSe_{2}. Nano Lett. 22, 1829–1835 (2022).
Huang, Z. et al. Spatially indirect intervalley excitons in bilayer WSe_{2}. Phys. Rev. B 105, L041409 (2022).
Wang, Z., Chiu, Y.H., Honz, K., Mak, K. F. & Shan, J. Electrical tuning of interlayer exciton gases in WSe_{2} bilayers. Nano Lett. 18, 137–143 (2018).
Woźniak, T., Faria Junior, P. E., Seifert, G., Chaves, A. & Kunstmann, J. Exciton g factors of van der Waals heterostructures from firstprinciples calculations. Phys. Rev. B 101, 235408 (2020).
Kyriienko, O., Magnusson, E. B. & Shelykh, I. A. Spin dynamics of cold exciton condensates. Phys. Rev. B 86, 115324 (2012).
Ivanov, A. L. Quantum diffusion of dipoleoriented indirect excitons in coupled quantum wells. Europhys. Lett. 59, 586 (2002).
Kash, J. A., Mendez, E. E. & Morkoç, H. Electric field induced decrease of photoluminescence lifetime in GaAs quantum wells. Appl. Phys. Lett. 46, 173–175 (1985).
Sivalertporn, K., Mouchliadis, L., Ivanov, A. L., Philp, R. & Muljarov, E. A. Direct and indirect excitons in semiconductor coupled quantum wells in an applied electric field. Phys. Rev. B 85, 045207 (2012).
Lien, D.H. et al. Electrical suppression of all nonradiative recombination pathways in monolayer semiconductors. Science 364, 468–471 (2019).
Uddin, S. Z., Higashitarumizu, N., Kim, H., Rabani, E. & Javey, A. Engineering exciton recombination pathways in bilayer WSe_{2} for bright luminescence. ACS Nano 16, 1339–1345 (2022).
Yuan, L. & Huang, L. Exciton dynamics and annihilation in WS_{2} 2D semiconductors. Nanoscale 7, 7402–7408 (2015).
Erkensten, D. et al. Dark excitonexciton annihilation in monolayer WSe_{2}. Phys. Rev. B 104, L241406 (2021).
Akselrod, G. M. et al. Visualization of exciton transport in ordered and disordered molecular solids. Nat. Commun. 5, 3646 (2014).
Siday, T. et al. Ultrafast nanoscopy of highdensity exciton phases in WSe_{2}. Nano Lett. 22, 2561–2568 (2022).
Acknowledgements
We are grateful to A. Ciarrocchi for useful discussions. We acknowledge the help of Z. Benes (EPFL Center of MicroNanoTechnology (CMI)) with electronbeam lithography. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement nos. 894369 (Marie Curie Sklodowska ITN network ‘2Exciting’) and 881603 (Graphene Flagship Core 3 Phase). This work was financially supported by the European Research Council (grant no. 682332) the Swiss National Science Foundation (grant nos. 164015, 177007, 175822 and 205114). The Marburg group acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) under SFB 1083 and project 512604469. K.W. and T.T. acknowledge support from JSPS KAKENHI (grant nos. 19H05790, 20H00354 and 21H05233).
Funding
Open access funding provided by EPFL Lausanne
Author information
Authors and Affiliations
Contributions
A.K. initiated and supervised the project. E.L. and F.T. fabricated the devices, assisted by G.P. F.T. performed the optical measurements, assisted by E.L. and Z.S. K.W. and T.T. grew the hBN crystals. F.T. and E.L. analysed the experimental data with input from A.K. D.E., R.P.C., S.B., J.H. and E.M. developed the microscopic theory on exciton hybridization and transport. E.L., F.T. and A.K. wrote the paper with contributions from all authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Photonics thanks the anonymous reviewers for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Supplementary Information
Supplementary Notes 1–10, Figs. 1–11 and references.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Tagarelli, F., Lopriore, E., Erkensten, D. et al. Electrical control of hybrid exciton transport in a van der Waals heterostructure. Nat. Photon. 17, 615–621 (2023). https://doi.org/10.1038/s4156602301198w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s4156602301198w
This article is cited by

Exciton transport in atomically thin semiconductors
Nature Communications (2023)