## Introduction

Excitons in semiconductor systems can be optically excited and read out, thereby encoding and storing optical signals into the excitons’ spin, valley, and orbital degrees of freedom1,2,3,4. In analogous to electronic circuits, circuits with excitons as the active information carriers have been envisioned, which transport and manipulate excitonic states with applied electrical and magnetic fields5,6. Transducing between photons and solid-state media, such excitonic circuits can directly process optical signals and regenerate light without additional optical-electrical conversions so that they can be very efficient6. However, unlike electrons or holes, charge-neutral excitons experience no net force under uniform electric fields. They can also be dissociated by a moderately strong in-plane electric field if the binding energy is small, for example, in GaAs quantum well systems7,8. Therefore, diffusion has been one of the main mechanisms utilized for exciton motion in GaAs9,10 and 2D materials11,12,13. Particularly, in GaAs systems, the high mobility of exciton has enabled 10 s of microns diffusion length at low temperature6. To actively transport excitons with controlled directionality, surface acoustic waves (SAW) have been employed to effectively transport excitons in GaAs quantum wells14,15,16,17,18. Over 100 s of microns transport distance is achieved at a temperature below < 4K15,16. However, the low exciton binding energy of only a few meVs in GaAs prohibits operation at higher temperatures19.

Compared to GaAs quantum wells, excitons in 2D transition metal dichalcogenides (TMDCs) (e.g., MoS2, MoSe2, WS2, WSe2) have binding energy on the order of hundreds of meVs3,4 with strong resonances even at room temperature (RT)20,21,22,23, making them promising for a plethora of optoelectronic and quantum applications24,25,26,27,28. Particularly, the indirect excitons (IXs) in bilayers and heterobilayers of TMDCs have additional desirable properties27,29,30,31,32. Because these IXs consist of electrons and holes separated in different layers and valleys, their population lifetimes at low temperature are up to 100 s of nanoseconds4, facilitating long-range transport before relaxation. Importantly, IXs have a permanent perpendicular dipole moment so that their energy can be tuned with an out-of-plane electric field33,34, thus they can be driven by a lateral gradient of electric-field35. Indeed, prototypical TMDC excitonic transistors have been demonstrated based on field-controlled IX diffusion in MoSe2/WSe2 heterobilayers35,36,37. The application of an out-of-plane static electric field creates an energy barrier or trap for IX, so switching on and off the electric field can suppress or enable exciton diffusion37. However, due to limited exciton mobility38, diffusive and repulsive transport of IXs can only achieve a transport distance of a few µm using milliwatts of optical excitation power at a low temperature39. More importantly, such diffusive transport is non-directional, thus challenging to realize functional excitonic circuits, which entail transporting excitons in a controlled direction over a long distance.

## Results

Here, we demonstrate the long-range and directional transport of IXs in a bilayer WSe2 using SAW at temperatures up to RT. The IXs in a bilayer WSe2 are momentum indirect with a long lifetime of up to 10 ns at 10 K and ~1 ns even at RT29,37. It was reported recently that the exciton energy of IXs in bilayer WSe2 could be modulated by an out-of-plane static field29. Consider a bilayer WSe2 placed directly on a piezoelectric substrate (Fig. 1a). SAW is excited and propagates through the area where a bilayer WSe2 is transferred to. On a piezoelectric substrate such as LiNbO3, the propagating SAW will generate a near-field piezoelectric field with a large out-of-plane field (Ez) amplitude on the order of 107 V/m at 1 mW/µm of acoustic power density. As such, the Ez component will periodically modulate the IX energy in space and time, creating a dynamic trapping potential for the IXs in the extrema of Ez. As the SAW propagates, the IXs will drift along the time-varying gradient of Ez and thus be carried by the SAW—like surfing on a wave—to travel a long distance before they recombine (Fig. 1b). With a SAW velocity of ~3.0 × 103 m/s and an exciton population lifetime > 10 ns, the IXs can travel >30 µm, an order of magnitude longer than the exciton diffusion length.

Figure 1b depicts the scenario when a bilayer WSe2 is under both optical pumping and SAW modulation. Since the bilayer WSe2 is inversion symmetric, there are two energy degenerate IXs with dipole moment p pointing along the +z and −z directions, respectively. For simplicity, hereon, we use +z IX to explain the transport process. The electric field E(r,t) induced by the SAW modulates the IX energy by $$\varDelta U=-p\cdot E$$ and thus creates a dynamic potential well moving at the acoustic velocity. At low temperatures, the optically excited IXs will quickly relax to the energy minimum created by the SAW in real space and travel with the propagating SAW. Figure 1c shows the optical image of our device. We designed a focusing IDT structure that focuses the acoustic wave into the WSe2 region (Fig. 1d) to concentrate the acoustic power density and enhance the piezoelectric field E. The IDT excites a strong SAW mode at 1.237 GHz with an acoustic wavelength of 2.832 µm, assuming an acoustic velocity of 3.5 × 103 m/s for the z-propagating Rayleigh mode in a y-cut LiNbO3 substrate (see S.I). Figure 1d shows the simulated electric field profile of the focused beam of the acoustic wave, which at the focal point has a waist of ~3.0 µm. To reduce the inhomogeneous broadening and exciton trapping from spatial variation of surface potential, we encapsulated the bilayer WSe2 with ~10 nm hexagonal boron nitride (h-BN) flakes using the standard pick-up method and transferred onto the LiNbO3 substrate with pre-patterned IDTs40. A thin layer of indium-tin-oxide (ITO) was deposited on the heterostructure region as a top transparent electrode. This top electrode plays an important role by efficiently suppressing the in-plane piezoelectric field component (Ex), which may cause exciton dissociation14,41,42,43, while maintaining a relatively strong out-of-plane field component Ez. The finite-element method (FEM) simulation (Fig. 1e, f) compares the piezoelectric fields in the situations of with (Fig. 1e) and without (Fig. 1f) the top ITO layer, showing that the ITO layer suppresses Ex by about two orders of magnitude.

We first demonstrate efficient SAW transport of IXs by performing spatially resolved photoluminescence (PL) measurement. Without SAW and at a low pump power, the diffusive IX flux density can be described by the diffusion equation: $$j=-D\nabla N$$, where D is the temperature-dependent diffusion coefficient and N is the exciton density (inset, Fig. 2a). Due to the low D in TMDC, the IX diffusion is relatively weak. In comparison, when SAW is turned on, the IX population is strongly modulated by the piezoelectric field and drifts along the SAW propagation direction (inset, Fig. 2b). As the velocity of the SAW (~3,500 m/s) is much lower than the thermal velocity of excitons (104 – 105 m/s in our experimental condition, see S.I.), the IXs can be treated as an exciton gas in quasi-equilibrium. With sufficiently high SAW amplitude, the IX gas will be trapped in the energy minimum of SAW and drift with a center-of-mass velocity identical to the SAW velocity.

### Interlayer exciton transport

Figure 2a, b compare the PL images measured at 100 K with SAW off and on, respectively. The IXs were excited with a He-Ne laser at 633 nm, having a power of Pp = 20 µW, and focused to a diffraction-limited spot size of 1 µm. The excitation spot is placed near the edge of the WSe2 flake close to the IDT and in the middle of the acoustic wave beam (Fig. 1d). Without SAW (Fig. 2a), the IXs diffuse by only 1–2 µm, consistent with previous measurement results. When SAW is turned on with power Ps = 6 mW, as shown in Fig. 2b, we observe strong exciton emission at two spots on the far edge of the WSe2 flake, along the SAW propagation direction and ~20 µm away from the pump spot. The two separate emission spots are at the corners (one convex and one concave) of the flake edge, where the acoustic wave is most focused (Fig. 1d). The IXs are transported at the acoustic velocity of 3500 m/s so the traveling time is less than 6 ns, shorter than their lifetime29. The results reveal SAW-driven transport of IXs over a device size limited distance, setting the lower bound of the propagation length to ~20 µm.

To confirm the transport is indeed from IXs, we performed spatial and energy-resolved measurements. We aligned the slit of the spectrometer with the SAW propagation direction (y-axis) and acquired spectral PL images with SAW off and on (Fig. 2c, d). The emission of the transported IXs is predominantly at the energy around 1.56 eV, which agrees with the IX energy of bilayer WSe229,44,45. The result is consistent with our expectation that only IXs with perpendicular dipole moment are efficiently transported by SAW. At the pump spot (Fig. 2e), the IX emission slightly decreases when SAW is turned on, presumably due to the transport of IXs by the SAW. In contrast, at the far edge of the flake (Fig. 2f), the application of SAW increased the IX emission intensity by more than two orders of magnitude. In control devices without the top ITO layer, no IX emission (see Figure 11 in S.I.) can be observed beyond the IX diffusion distance. It agrees with the expectation that the in-plane piezoelectric field, if not screened by the ITO (Fig. 1e), will dissociate the IXs to free carriers46. These free carriers will have a very low recombination rate in the SAW because they are spatially separated by half the acoustic wavelength (~1.4 µm in our device). Therefore, we can confirm that the emission at the edge of the flake is from SAW transported IXs.

### Power-dependent exciton transport

We next characterize the IX transport at different SAW powers Ps. We find that the transported exciton density increases monotonically with the SAW power, but the trend is highly nonlinear with an activation behavior. For Ps < 3.0 mW, the IX transport is negligible, and the IX emission is localized near the laser excitation spot (Fig. 3a). When Ps increases to 4.5 mW, two emission spots appear at the flake edge (Fig. 3b), suggesting activation of the transport process. When Ps is further increased to 6 mW, the exciton transport becomes so efficient that the emission intensity at the flake edge is already comparable to that at the pump spot (Fig. 3c). Figure 3d plots the integrated emission intensity at the flake edge, which is proportional to the transported exciton density nT, as a function of Ps. The experimental result agrees with our theoretical model of SAW activated transport in which the transported exciton density nT is proportional to $${e}^{\sqrt{\frac{{P}_{s}}{{P}_{t}}}}$$ at a given temperature (see S.I.). Fitting the result at 100 K gives a relatively small threshold power of Pt ~ 0.1 mW, beyond which exciton transport is activated by the SAW. Similar exponential behavior and power law have also been observed in coupled GaAs quantum wells14, where exciton transport is impeded by disorder and defect-induced potential variations at the low SAW power limit. In TMDC such as bilayer WSe2, the intrinsic defect density can be >7.0 × 1010 cm−2, along with the strain-induced potential variation caused by the transfer process40. For efficient IX transport to happen, the SAW modulation of IX energy ΔU needs to overcome the defect and strain induced potential variation (see S.I.), and the efficiency of exciton transport is sensitive to the material quality.

### Temperature-dependent Exciton transport

The activation behavior motivates us to further measure SAW-driven IX transport at different temperatures. Figure 4 shows the spectral PL mapping at temperatures from 6 K to 200 K with fixed pump power Pp = 20 µW and SAW power Ps = 6 mW. In this wide temperature range, we observe rich features showing the roles of exciton localization, SAW-driven transport, recombination, and phonon scattering. Note that the measurement at each temperature is normalized to its respective maxima for clarity. At 6 K and with SAW off, the emission spectrum shows a series of sharp resonances (Fig. 4a), which are attributed to defect emission and phonon-assisted recombination of the IXs. When SAW is on, the sharp resonances disappear and the emission intensity decreases significantly (Fig. 4b and SI). Meanwhile, IX transport at this low temperature is very weak. The suppression of IX emission can be explained by that the SAW reduces the coupling between excitons and recombination centers such as defects. When the temperature increases to 30 K, the sharp exciton resonances disappear (Fig. 4c). At this elevated temperature range, the excitons are thermalized into exciton gas, leading to weaker couplings with the local defects. As a result, the SAW-driven transport starts to make the PL at the flake’s edge observable (Fig. 4d). When the temperature reaches 100 K, the IX emission becomes more than two times brighter at the pump spot (Fig. 4e) than at 30 K. This is because indirect IX recombination becomes more efficient when the thermal phonon density is high. The SAW-driven transport is also the most prominent at this temperature (Fig. 4f) because the defects and disorders have a less trapping effect on the thermalized excitons and the exciton population lifetime is still sufficiently long. At 200 K (above the Debye temperature), the thermal phonon population is high. As a result, IX emission has a broad spectrum and high intensity without SAW (Fig. 4g). However, the IX transport is impeded due to strong exciton-phonon scattering and decreased exciton population lifetime (Fig. 4h). Overall, our systematic measurements have revealed three regimes of IX behaviors and the transition between them. At the lowest temperatures, the IXs are highly localized and their transport is diffusion-limited. At intermediate temperatures, the IXs are bright with phonon-assisted recombination. At the same time, they still have a long lifetime, allowing for efficient SAW-driven transport over a long distance. At further elevated temperatures, strong phonon scattering decreases exciton mobility and lifetime, preventing efficient transport. Nevertheless, thanks to the strong SAW modulation, even at room temperature, we still observe a SAW-driven transport distance of ~2.0 µm (see Supplementary Fig. 8).

## Discussion

In conclusion, we have demonstrated that SAW is an efficient, contact-free approach to transport IXs in bilayer WSe2 over a distance far beyond the diffusion length. The SAW-driven transport happens when the SAW modulation of IX energy can overcome the local potential variation and defect traps. Since the transport distance depends on the exciton lifetime of the optically active material and the acoustic velocity of the substrate, using TMDCs with longer lifetimes and piezoelectric substrate with higher acoustic velocity can lead to a much longer transport distance. The contact-free transport driven by the acoustic wave that is launched remotely also preserves the high quality of the materials and prevents undesired effects induced by local gates35,36,37,39. Although the maximal transport distance is reached at the temperature of 100 K in the current device, further improvement of the material quality and interface cleanness will make efficient room-temperature operation possible. We note that SAW is a universal approach to control excitons with both of its piezoelectric field and strain field can be utilized to manipulate and transport excitons in many other 2D material systems. SAW can also be guided and circulated in phononic circuits and resonators47,48, which will afford rich functionality and flexibility.

## Methods

### SAW-exciton device fabrication

The WSe2 device was fabricated by the standard polymer-assisted (PC) pick-up method. The 2D flakes of hBN and WSe2 (HQ graphene) were exfoliated to a 90 nm SiO2/ Si substrate and then picked up using PC/PDMS (Sylgard 184) stamps. The bilayer WSe2 flakes were identified by their optical contrast and then confirmed with photoluminescent measurements. The IDTs were patterned on a y-cut LiNbO3 wafer (MTI) using ebeam lithography. The IDT is aligned to generate SAW propagating along the z-axis of the LiNbO3 wafer. Layers of 12 nm chromium and 120 nm gold were deposited using an ebeam evaporator under high vacuum. Aligned transfer was then done by a home-built transfer stage with high accuracy so that the WSe2/hBN heterostructure was precisely aligned at the focus region of the curved IDTs. PC residue was then removed using chloroform for 1 h and followed by a 5 min rinsing in the IPA. Finally, aligned ebeam writing was performed and 50 nm indium-tin-oxide (ITO) was deposited by a sputtering system (Evatec LLS EVO) under O2 condition. After deposition, the device was annealed in atmosphere at 300 °C for 5 min to improve the conductivity of the ITO film, which can ensure a better screening of the in-plane field.

### Measurement setup

A continuous-wave He-Ne laser (633 nm) was used to excite the excitons in bilayer WSe2. With an objective lens (NA = 0.42), the laser beam was focused with a diffraction-limited spot size of about 1 µm. The sample was mounted in a cryostat (Montana Instrument) with an optical window for optical access. Meanwhile, the RF signal was generated by a vector network analyzer (VNA) (Agilent E8362B) and then coupled to the wire-bonded device inside the cryostat. The calibration kits (Keysight 8052D) were used to de-embed the system so that the IDT resonance can be resolved. The interlayer exciton emission was acquired with a spectrometer (Princeton Instrument) after the laser line was removed with a 633-notch filter (Thorlabs). To image the spatial transport of the exciton, the center wavelength of the spectrometer was set to be 0 so it functioned like a mirror. The filtered signal was then collected by a cooled camera (Pixis 400) operating at −70 °C to improve the signal-to-noise ratio. To collect the spectrum results, we set the center wavelength to 790 nm, which covered the emission spectrum of the interlayer exciton. To perform the spectral-PL imaging measurement of the exciton transport, the transport direction was aligned with the slit of the spectrometer, while the x-axis of the CCD camera displayed the spectrum information. The y-axis signal gave the spatial emission of the bilayer devices.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.