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Lifu Zhang, Liuxin Gu, Ruihao Ni, Ming Xie, Suji Park, Houk Jang, Rundong Ma, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), You Zhou

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[Electrical Control and Transport of Tightly Bound Interlayer Excitons in a <math display="inline">  <mrow>    <msub>      <mrow>        <mi>MoSe</mi>      </mrow>      <mrow>        <mn>2</mn>      </mrow>    </msub>    <mo>/</mo>    <mi>hBN</mi>    <mo>/</mo>    <msub>      <mrow>        <mi>MoSe</mi>      </mrow>      <mrow>        <mn>2</mn>      </mrow>    </msub>  </mrow></math> Heterostructure](https://mdr.nims.go.jp/datasets/a3448d01-0d1c-4786-9bd8-412b1b648eca)

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1  Electrical control and transport of tightly bound interlayer excitons in a MoSe2/hBN/MoSe2 heterostructure  Lifu Zhang1,*, Liuxin Gu1,*, Ruihao Ni1, Ming Xie2, Suji Park3, Houk Jang3, Rundong Ma1, Takashi Taniguchi4, Kenji Watanabe5, You Zhou1,6,†  1Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA 2Condensed Matter Theory Center, University of Maryland, College Park, MD 20742, USA  3Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA 4Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 5Research Center for Materials Nanoarchitectonics, National Institute for Materials Science,  1-1 Namiki, Tsukuba 305-0044, Japan 6Maryland Quantum Materials Center, College Park, Maryland 20742, USA  †To whom correspondence should be addressed: youzhou@umd.edu *These authors contributed equally to this work.    mailto:youzhou@umd.edu2  Abstract Controlling interlayer excitons in van der Waals heterostructures holds promise for exploring Bose-Einstein condensates and developing novel optoelectronic applications, such as excitonic integrated circuits. Despite intensive studies, several key fundamental properties of interlayer excitons, such as their binding energies and interactions with charges, remain not well understood. Here we report the formation of momentum-direct interlayer excitons in a high-quality MoSe2/hBN/MoSe2 heterostructure under an electric field, characterized by bright photoluminescence (PL) emission with high quantum yield and a narrow linewidth of less than 4 meV. These interlayer excitons show electrically tunable emission energy spanning ~180 meV through the Stark effect, and exhibit a sizable binding energy of ~81 meV in the intrinsic regime, along with trion binding energies of a few millielectronvolts.  Remarkably, we demonstrate the long-range transport of interlayer excitons with a characteristic diffusion length exceeding 10 µm, which can be attributed, in part, to their dipolar repulsive interactions. Spatially and polarization-resolved spectroscopic studies reveal rich exciton physics in the system, such as valley polarization, local trapping, and the possible existence of dark interlayer excitons. The formation and transport of tightly bound interlayer excitons with narrow linewidth, coupled with the ability to electrically manipulate their properties, open exciting new avenues for exploring quantum many-body physics, including excitonic condensate and superfluidity, and for developing novel optoelectronic devices, such as exciton and photon routers.          3  Spatially indirect excitons (IXs) with long lifetimes and electric dipole moments can form macroscopic coherent quantum phases, such as exciton superfluidity, at high densities and low temperatures1-4. The quantum degeneracy temperature of IXs, which scales proportionally to the exciton binding energy, is limited in traditional III-V double–quantum well systems4, 5. Transition metal dichalcogenide (TMD) heterostructures, predicted to host tightly bound IXs4, have recently emerged as an exciting new platform for exploring coherent high-temperature condensate and superfluid states6-9.  Furthermore, these IXs exhibit spin-valley coupling10, 11, strong dipolar interactions12, 13, and electrical tunability via the Stark effect14-16, thereby offering a system rich for exploring exciton physics. To date, optical studies of interlayer excitons have primarily focused on multilayer TMD hetero- and homo-junctions10-22. In heterostructures, local variations in the twist angle and strain can induce substantial inhomogeneous broadening of IXs, for instance, by lattice reconstruction, introducing a disorder potential atop the periodic moiré lattice17-20, 23, 24. On the other hand, although disorder potential can be greatly reduced in natural homo-layered TMDs, these systems become momentum-indirect16, 21, 22, which feature significant non-radiative broadening of IX. As a result, IXs in TMDs typically manifest much larger linewidths than their intralayer counterpart in monolayers, which hinders the formation of exciton superfluidity and limits our understanding of their critical properties, such as their binding energies and their interactions with free carriers. Here, we employ a homo-bilayer TMD system, with two MoSe2 monolayers separated by an atomically thin layer of hBN. The thin hBN spacer preserves the direct bandgap in bilayers while allowing for carrier tunneling and IX emission. Although IX PL has been observed in similar structures, earlier works focused on the hybridization of IXs and intralayer excitons25, 26. In this work, we demonstrate a low-disorder system with degenerate intralayer exciton energies and sharp IX PL linewidth. The high optical quality allows us to directly measure the interlayer exciton and trion binding energies and observe the long-range transport and trapping of IXs, creating a new platform for studying exciton transport, photophysics, and condensates. We first establish the electric-field and doping control of interlayer excitons in a MoSe2/hBN/MoSe2 device.  We align the two monolayers close to zero degree to promote the carrier tunneling and IX emission (Fig. 1a, see Table S1 for hBN thickness in different devices). Under zero electric field, the sample’s PL spectrum at 6 K is dominated by intralayer neutral exciton X0 and charged excitons XT (Fig. 1b). The two monolayers have degenerate X0 energies, suggesting minimal strain difference. Next, we apply an electric field while keeping both layers intrinsic (𝑉𝐵𝐺 = −𝛼𝑉𝑇𝐺, where 𝛼 is the thickness ratio of the top and bottom hBN layers, Fig. S1). Under a large electric field, an additional peak, IX, emerges at a lower energy of 1.596 eV with a narrow linewidth of ~3.7 meV, accompanied by a reduction in X0 and XT intensities. Remarkably, the quantum yield of IX is comparable to that of the intralayer excitons in monolayer MoSe2 under zero electric field (Fig. S2c). The electric-field dependent PL spectra in Figs. 1c show a clear Stark effect of IX, from which we estimate a permanent out-of-plane dipole of 𝑢 = 𝑒 ∗ 0.81 𝑛𝑚 for IX (SI [27]), consistent with the thickness of the hBN spacer (two layers) plus the finite thickness of MoSe2. From reflectance measurements, we verify that both layers are intrinsic, and the XT emission likely originates from in-gap states in MoSe2 (Fig. S2a). We attribute IXs to the momentum-direct interlayer exciton at the K-K transition and extract their binding energies. In particular, we calculate the IX energy under zero electric field to be 1.731 eV from the Stark effect, which is given by the difference between the quasiparticle bandgap and the 4  interlayer binding energy. By measuring the Rydberg states of X0, we estimate the quasiparticle bandgap of monolayer MoSe2 to be ~1.812 eV using a screened Keldysh potential model33, 34 (SI & Fig. S2a [27]). Therefore we estimate the binding energy of IX to be ~81 meV, consistent with theoretical calculations4, 35 (see Table S1 for its dependence on hBN thickness). We note that, different from other studies25, we did not observe clear signatures of IX or their anti-crossings with X0 in reflectance (Fig. S2b), suggesting their low oscillator strength and weaker carrier tunneling. We further investigate the doping dependence of IX. Under symmetric gating with VBG = 𝛼VTG, the measured doping-dependent PL resembles that of monolayers without any IX emission, as the carriers are evenly distributed across the two layers (Fig. S3a). To make IX energetically favorable and detectable through PL, we vary the total doping under a large electric field, following VBG =𝛼VTG + δ (Fig. 1d). The X0 intensity reduces upon doping but remains finite. This is because the applied field polarizes carriers into one layer and makes the other layer intrinsic, which is further verified by measuring X0 reflectance (Figs. 1e, S2d, and S4). In contrast, the intrinsic IX0 vanishes upon doping one layer, with an additional peak appearing below it, which we identify as charged interlayer excitons, i.e., interlayer trions or polarons (IXT in Fig. 1d). The trion binding energy (in the low doping regimes) is estimated to be ~3 meV for negatively charged and ~5 meV for positively charged IXs, with the difference likely due to the varying effective masses of electrons and holes (𝑚𝑒∗ > 𝑚ℎ∗ )36, 37.  The charged IX shows a significant blueshift with increasing total gate voltages on both electron and hole sides (also in Fig. S3b-d). In such a bilayer with layer-polarized carriers, varying the gate voltage alters both the doping level and the electric field between layers. The effective screening by the highly doped layer and weak screening by the intrinsic layer leads to a reduction in the electric field magnitude and blueshift of IX with increasing voltage (see Fig. 1f and Discussion III [27]). To separate doping and electric-field influences on this blueshift, we measure the field-dependent PL of charged IXs at various doping levels and extract their energies at zero field, from which we find that the blueshift is primarily due to the electric-field effect (Fig. S5). We then investigate the valley polarization properties of IX by exciting the sample with a circularly polarized laser at 1.95 eV and detecting the preservation of circular polarization in PL. Intriguingly, IX exhibits a ~ 20% degree of circular polarization (DOCP), defined as (Ico – Icross)/(Ico + Icross), while the DOCP of intralayer excitons is almost zero (Figs. 2a and S6).  In MoSe2, the absence of intralayer excitons DOCP is due to rapid valley mixing caused by electron-hole exchange38, 39. In contrast, the electron-hole exchange of IX can be much weaker thanks to the spatial separation of the electron-hole wavefunction10, 40, 41. The finite DOCP also suggests that the formation of IXs occurs faster than intervalley electron-hole exchange. The DOCP of IX does not vary significantly with the electric field except in the crossover region, where it hybridizes with intralayer species, as expected (Fig. 2b).  Next, we examine the optical nonlinearity of IX by measuring their PL spectra under different excitation powers (Fig. 2c). With higher power, IX blueshifts while no obvious shift of X0 and XT can be observed (Figs. S7). The stronger nonlinearity of IX compared to intralayer excitons arises from their repulsive dipole-dipole interactions due to the aligned electric dipols12, 16. The n-doped IX exhibits a similar trend but with a larger blueshift at the same excitation powers (Fig. S7). In the hole-doped case, however, we observe a shift in the XT energy, which suggests optical pumping may alter the doping levels (Fig. S8), likely due to Auger-assisted hole tunneling across hBN dielectrics42. 5  We focus our analysis of the power-dependent PL on the intrinsic and n-doped regimes to avoid photo-doping effects. From fitting, we find a ~3.1 meV blueshift of neutral IX and ~4.1 meV blueshift of n-doped IX along with linewidth broadening of ~2-3 meV when the excitation reaches 16 µW (Fig. 2d). The IX density, 𝑛𝐼𝑋 , can be estimated using the mean-field parallel plate-capacitance model15 following: 𝛿𝐸 = 𝑒𝑢𝑛𝐼𝑋/𝜀 , where 𝛿𝐸  is the energy shift, 𝑒 is the electron charge and 𝜀 is the dielectric constant (Discussion II and IV [27]). Notably, we estimate a maximum exciton density. 𝑛𝐼𝑋 of ~7.6 × 1010 cm-2 and ~1.3 × 1011 cm-2 under our highest excitation power of 16 µW and 64 µW, for the intrinsic and n-doped case, respectively. The integrated IX PL intensity has a linear dependence on 𝑛𝐼𝑋 in the n-doped case and has a nearly quadratic dependence in the intrinsic region (Fig. S7d). This suggests that radiative recombination in n-doped IX is relatively independent of exciton density, whereas in the intrinsic region, the recombination can be density-dependent due to processes like collective radiation, biexciton formation, and Auger recombination43-45. The electrical control of spectrally sharp IXs with strong dipolar interactions in our device allows us to probe the transport properties of IX. Figures 3a, 3b, and S9 show the spatial map of IX PL under different excitation powers. With the excitation laser fixed near the sample’s bottom right corner, the IX PL is visible away from the laser spot and extends further with increasing excitation power, showing distinct local bright areas. In contrast, no long-range transport is observed for intralayer excitons, even at the highest power (Fig. S9c). To quantify the IX transport, we extract the normalized, radially averaged PL intensity 𝐼𝑛𝑜𝑟𝑚(𝑟) from these spatial maps, where 𝑟 is the distance from the center of the laser spot. Since spatial inhomogeneities significantly affect diffusion profiles, we focus on data collected from another sample area with a more uniform IX distribution (Figs. 3c and S10). Exciton density near the laser resembles the Gaussian beam profile and asymptotically approaches 𝑛𝐼𝑋 ∝  𝑒−𝑟/𝐿𝐷/√𝑟 𝐿𝐷⁄   further away (see Discussion IV [27]). Fitting the density profile away from the laser (dashed lines in Fig. 3c), we extract a diffusion length 𝐿𝐷 = 17 ± 6 µm for the neutral IX, which does not vary significantly with excitation power. As we move closer to the laser and increase the laser power, the interactions-driven current becomes increasingly prominent, surpassing the pure diffusion current driven by the density gradient (Fig. S11 and Discussion IV [27]). It is important to mention that we have ignored other nonlinear exciton decays such as exciton-exciton annihilation46 and exciton-phonon interactions47, since we focus on regions away from the laser excitation. However, these nonlinear processes, along with the drift and diffusion of IX, can contribute to the measured sublinear power dependence of exciton densities (Fig. S7e). The bright spots away from the laser in the diffusion maps can act as a natural trap for IX, which increases the local exciton density and the condensate temperature. Figure S12 show the PL of neutral and doped IXs collected from a local bright spot ~2.9 µm away from the laser, with a sharp linewidth of ~4-5 meV similar to Fig. 1b. A blueshift of ~1 meV, along with linewidth broadening of ~1 meV, is observed under the maximum pump power (Figs. S12c and S12d). We estimate the locally trapped exciton density to be ~1010 cm-2, only a few times less than that under direct laser excitation (Fig. 3d). Lastly, we present the intriguing observation of the periodic modulations of X0 and XT PL intensities in response to varying electric fields under fixed doping (Fig. 4a). Under positive fields, electrons become layer-polarized, resulting in predominantly emission from the top layer’s n-doped trion, XTt, and the bottom layer’s neutral exciton, X0b. IX emerges when its energy drops 6  below XTt, different from the intrinsic and p-doped cases, where IX appears when it becomes degenerate with X0 (Figs. 1 & S13). This difference stems from the more efficient hole tunneling than electrons in TMD heteorstructures6, 42: hole tunneling leads to IX-XTt hybridization in the n-doped region (Fig. 4b), as opposed to IX-X0b coupling in the p-doped and intrinsic cases (Fig. 4c). Increasing electric fields typically induce a continuous change in carrier density in each layer and a monotonic increase (decrease) in XTt (X0b) intensity. However, at three distinct electric fields, we observe reduced XTt and enhanced X0b emissions (Figs. 4d and S13b). These anomalies, marked as Ie, IIe, and IIIe, recur periodically at ~21 mV/nm intervals. Feature IIIe appears at the electric field where IX becomes degenerate with neutral exciton X0b. This energy degeneracy can lead to a resonantly enhanced IX-X0b coupling, introducing an additional decay pathway for the long-lived IX into X0b, which enhances X0b and reduces XTt emission (Fig. S13). Similarly, features Ie and IIe may arise when possible dark interlayer excitons with lower energy than IX reach degeneracy with X0b. Figure 4e shows a map of integrated X0 intensity as we vary top and bottom gates, where we observe similar anomalous features in the intrinsic and p-doped regimes. Across all doping conditions, the energy difference between bright and dark excitons is ~17 meV. The dark excitons may emerge due to misaligned MoSe2 layers creating local atomic registry variations. This moiré superlattice could give rise to IX species with different optical selection rules and oscillator strengths25, 48. Alternatively, interlayer electron–phonon coupling might generate phonon replicas of IXs, a hypothesis supported by the similar energy scales to phonon energies in hBN and MoSe249-52. Indeed, in a device with a larger twist angle (~3°), no IX emission is observed but the modulation of X0 and XT is still present, likely due to their coupling with momentum-indirect dark IXs. Detailed optical and transport investigations will be crucial to clarify the dynamics between dark and bright IXs. The electrically tunable IX with sharp linewidth forms a promising platform for exploring many-body excitonic physics such as condensate and superfluidity. Using the experimentally extracted 𝑛𝐼𝑋 in the intrinsic regime, we estimate the quantum degeneracy temperature , 𝑇𝑞 =2𝜋ℏ2𝑛𝐼𝑋𝑚∗𝑘𝐵, of ~3.3 K and a Berezinskii–Kosterlitz–Thouless (BKT) transition temperature 𝑇𝐵𝐾𝑇  of ~0.7 K (Discussion V [27]), under a direct laser excitation of 16 µW. Since non-resonant excitation may raise their temperature above that of the lattice, IXs trapped in the bright spots allow for more efficient IX cooling and offer a more viable pathway for achieving an exciton condensate. Our experiments suggest IXs confined in such traps have quantum degeneracy and superfluid temperatures of ~1.0 K and ~0.2 K, respectively, already accessible in an optical dilution refrigerator. Nevertheless, to enhance the critical temperatures, it is desirable to further increase the IX densities. The upper limit of 𝑛𝐼𝑋 imposed by the Mott transition is well above our experimental values, given their large binding energy (~81 meV). We estimate maximum quantum degeneracy 𝑇𝑞𝑚𝑎𝑥  and superfluid temperatures 𝑇𝐵𝐾𝑇𝑚𝑎𝑥 to be ~94 K and ~19 K, near the Mott transition (see Discussion V and Fig. S14 [27]). Experimentally realizing such high densities necessitates further device optimization, including reducing gate current through resonant excitation and increasing the hBN gate thickness (Fig. S7f, sample D2). Another intriguing approach is to electrically inject IX, which may reduce heating in the bilayer system7, 53, 54. Crucially, disorder may disrupt the macroscopic coherence of the superfluid when its strength ∆ exceeds the thermal energy at the critical temperature, ∆> 𝑘𝐵𝑇𝐵𝐾𝑇8. Using the measured PL 7  linewidth (~3.7 meV) as a proxy for the maximum disorder strength ∆, we find ∆  already significantly lower than the thermal energies at 𝑇𝑞𝑚𝑎𝑥 and comparable to that at 𝑇𝐵𝐾𝑇𝑚𝑎𝑥. Another important consideration is the flavors of excitons, such as valley degeneracy and dark excitons. Although valley degeneracy in IXs would halve the quantum degeneracy temperature, leveraging the optical polarization of IXs shown here, possibly enhanced through resonant excitation, can substantially raise the condensate temperature. Likewise, dark IXs, having higher energies than bright ones in the intrinsic regime, might have a limited impact on the exciton condensate. In addition to quantum manipulation of bosonic particles, the electrical control and transport of robust interlayer excitons could form a basis for novel optoelectronics and valleytronics, such as excitonic transistors14, 55 and manipulation of spin currents.   8   Fig. 1 Electrical control of IX. a, Schematic of the designed device. b, Representative PL spectrum under no electric field and -0.17 V/nm (negative defined as pointing upward). c, Electric field dependence of PL spectra when both layers are intrinsic. Dashed lines indicate where the spectra in b are collected. d, Doping dependence of PL spectra (VBG = 1.12VTG + 12 V). e, A 2D map of reflection contrast at X0 as a function of top and bottom gates. f, Left: schematic of the electric field across the device under an asymmetric doping scan (VBG = 1.12VTG + δ and δ > 0). A larger bottom gate voltage induces layer-polarized doping. Right: the magnitude of doping and E-field across the two layers versus VTG. 9   Fig. 2 Valley polarization and dipolar interactions of IX. a, Polarization resolved PL emission from MoSe2/hBN/MoSe2 under a circularly polarized laser excitation. b, DOCP as a function of the electric field under hole doping. c, Power-dependent PL emission from IX under an electric field of -0.2 V/nm in the intrinsic regime. A long-pass filter is used to block the intralayer exciton signals. d, Blueshift of the neutral and n-doped IX as a function of the laser power.  10   Fig. 3 Spatial diffusion and trapping of IX. a-b, Spatial map of IX emission under different excitation powers under an electric field of -0.2 V/nm, with fixed laser location (black dashed circle). Scale bar: 2 µm. c, Radially averaged IX PL intensity a function of the distance from laser center r under varying excitation powers. Dashed lines: fitted curves. d, Estimated IXs density in a natural trap as a function of laser power. 11   Fig. 4 Coupling between intra- and inter-layer excitons. a, PL spectra of a MoSe2/hBN/MoSe2 device as a function of electric field with a fixed electron doping. b, c, Hole tunneling  facilitates (b) IX-XT coupling under electron doping and (c) IX-X0 under hole doping. d, X0 and XT intensities as a function of electric field. Ie, IIe, and IIIe donate three anomalies, corresponding to the vertical dashed lines in a. e, A 2d map of X0. The dashed cyan line depicts the voltage conditions where a is collected.   12  Acknowledgements: This research is primarily supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences Early Career Research Program under Award No. DE-SC-0022885. The fabrication of samples is supported by the National Science Foundation CAREER Award under Award No. DMR-2145712. This research used Quantum Material Press (QPress) of the Center for Functional Nanomaterials (CFN), which is a U.S. Department of Energy Office of Science User Facility, at Brookhaven National Laboratory under Contract No. DE-SC0012704. K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan for hBN synthesis.  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