Abstract
The recent discovery of artificial phase transitions induced by stacking monolayer materials at magic twist angles represents a paradigm shift for solid state physics. Twistinduced changes of the singleparticle band structure have been studied extensively, yet a precise understanding of the underlying Coulomb correlations has remained challenging. Here we reveal in experiment and theory, how the twist angle alone affects the Coulombinduced internal structure and mutual interactions of excitons. In homobilayers of WSe_{2}, we trace the internal 1s–2p resonance of excitons with phaselocked midinfrared pulses as a function of the twist angle. Remarkably, the exciton binding energy is renormalized by up to a factor of two, their lifetime exhibits an enhancement by more than an order of magnitude, and the excitonexciton interaction is widely tunable. Our work opens the possibility of tailoring quasiparticles in search of unexplored phases of matter in a broad range of van der Waals heterostructures.
Introduction
To tackle the formidable manybody scenario of Coulomb correlations in solids, Lev Landau introduced the concept of quasiparticles. These fictitious entities, which consist of particles dressed by their manybody interactions, are usually characteristic of material composition. Yet van der Waals layered materials provide an efficient tuning knob: the twist angle between adjacent layers can turn the semimetal graphene into a Mott insulator^{1}, superconductor^{2}, or ferromagnet^{3}; topological phases^{4,5,6,7}, spinpseudospin coupling^{8}, and shear solitons^{9} have been discussed for twisted bilayers of transition metal dichalcogenides (TMDs).
TMDs form a unique laboratory to explore such effects because quantum confinement in two dimensions and reduced dielectric screening give rise to exceptionally strong Coulomb interactions. In the monolayer limit, this situation allows hydrogenlike electronhole pairs—excitons—to bind with energies on the order of hundreds of meV (refs. ^{10,11,12}). In van der Waals heterostructures, the relative orientation between adjacent layers strongly influences the electronic structure because of a subatomic variation of the orbital overlap of adjacent layers^{1,2,3,4,5,6,9,13,14}. Hybridization effects between the electronic states of the constituent layers allow the band structure to be externally controlled by the twist angle θ (refs. ^{13}^{,}^{15,16,17,18,19,20}). Interband photoluminescence combined with theoretical calculations have provided valuable insight into these phenomena^{16,21,22}, but cannot resolve the internal excitonic structure which reveals the Coulomb correlations within a bilayer (BL).
Interband spectroscopy can only probe optically bright excitons, whereas the formation and decay dynamics of excitons in prototypical monolayer TMDs such as WSe_{2}—and especially in twisted BLs—are often dominated by dark states^{23,24}. Midinfrared (MIR) photons, in contrast, may directly interrogate Lymanlike 1s–2p transitions of excitonic species, irrespective of interband dipole moments, centerofmass momenta, spin, or spatial separation of electron and hole. This concept^{25} has been employed to probe Coulomb correlations^{26} in quantum wells^{27}, TMD monolayers^{28,29} and heterostructures^{30}, where electronic correlations are dominated by chemical composition and quantum confinement.
By exploiting the internal 1s–2p resonance of excitons in homobilayers of WSe_{2}, we demonstrate how the twist angle renormalizes the exciton binding energy by a factor of up to two and enhances their lifetime by more than an order of magnitude, whereas the exciton–exciton interaction is widely tunable at a certain angle. This fascinating scenario can be quantitatively explained with our microscopic manybody theory, which identifies the wavefunction overlap, interlayer hopping, and hybridization effects at the atomic interface as effective means to synthesize customtailored hybrid exciton species.
Results
Rydberg spectroscopy of excitons in twisted WSe_{2} BLs
The WSe_{2} BL samples are produced by mechanical exfoliation and subsequent deterministic transfer onto a diamond substrate (see Methods). Monolayer flakes with diameters exceeding 70 µm (Fig. 1a) are stamped on top of each other. Before the transfer, the crystallographic orientations of the layers are determined by polarizationresolved secondharmonic generation^{15,31}. The component of the second harmonic I_{2ω,} polarized parallel to the nearinfrared pump laser indicates the armchair directions of each monolayer (inset Fig. 1a, blue/orange flower patterns), revealing the relative twist angle θ between both layers (see Methods). We fabricated a series of BLs with θ ranging from 0° to 60° (see Supplementary Figs. 1 and 2).
The samples were precharacterized using photoluminescence spectroscopy with an excitation wavelength of 488 nm (see Methods). The resulting spectra (Fig. 1b) feature a single emission peak at an energy of ~1.54 eV for the pristine BL with θ = 60° (2H stacking), which we attribute to the phononassisted recombination of KΛ excitons^{32}. For θ = 59°, this peak blue shifts by 20 meV, whereas a second maximum emerges at an energy of ~1.65 eV, corresponding to the direct KK transition. This trend continues for θ = 53° until the two spectral features merge for θ = 30°. When the twist angle is further reduced to θ = 5° and θ = 0° (i.e., the 3R stacking) the two peaks are discernible again and move towards energies of 1.52 eV and 1.66 eV, respectively. Similar photoluminescence characteristics of BLs have been reported in the literature^{16,17,21}.
To explore the actual microscopic origin and the role of Coulomb correlations in the photoexcited twisted BLs, we study the internal structure and binding energy of the excitons. To this end, we generate 1s A excitons in the WSe_{2} BLs with a 100fs nearinfrared (NIR) laser pulse centered at an energy of 1.68 eV (Fig. 1c, orange intensity envelope). After a variable pumpprobe delay time t_{pp}, the exciton ensemble is probed by a phaselocked MIR pulse (Fig. 1c, red wave), the electric field of which is monitored by electrooptic sampling (see Methods and Supplementary Fig. 3). The intraexcitonic 1s–2p transition (Fig. 1c, red arrows) imprints a characteristic change on the MIR waveform, from which information about the excitonic correlations of the nonequilibrium system can be extracted. A Fourier domain Fresnel analysis^{26,27,28,30} directly reveals the pumpinduced change of the full dielectric response, including the change of the real part of the optical conductivity, Δσ_{1}, and the dielectric function, Δε_{1}, at variable delays t_{pp}.
Fig. 2 summarizes the changes in dielectric function measured for seven WSe_{2} BLs with varying twist angles θ, at t_{pp} = 5.1 ps, where the photoexcited state has thermalized (see Supplementary Fig. 4). For all twist angles, Δσ_{1} (Fig. 2a) exhibits a distinct maximum with a corresponding dispersive feature in Δε_{1} (Fig. 2b), signifying a resonant response of the photoexcited sample. As we demonstrate below, these resonances relate to the internal 1s–2p transition of BL excitons. Between θ = 60° and 30°, the resonance blue shifts with decreasing angle (Fig. 2a dashed line, blue arrow). Strikingly, for θ = 0° the resonance exhibits the opposite behavior: it shifts substantially to the red and gains oscillator strength (red arrow). A phenomenological analysis of the data with a twocomponent model (see Methods) confirms a systematic blue shift of the resonance (Fig. 2c) from 100 meV for θ = 60° to 121 meV for θ = 30°. Importantly, the resonance at θ = 0° is red shifted to 67 meV—corresponding to an energy reduction by a factor of two. This strong asymmetry of the 1s–2p transition energy—a direct measure of the exciton binding energy^{28,30} (see Methods)—for θ = 0° and θ = 60° is in stark contrast to the almost degenerate interband photoluminescence (see Fig. 1b).
Microscopic manybody theory
To explain these surprising observations, we have to go beyond singleparticle band structure effects and include Coulomb correlations, by a combination of ab initio density functional theory (DFT) calculations with density matrix theory (see Methods). In WSe_{2} BLs, the energetic minimum of the conduction band lies at the Λ and Λ′ points. Thus, following creation of excitons at the Κ point (Fig. 3a, red arrow), electrons scatter via phonons into the global energetic minima^{33} (Fig. 3a, blue arrow), corroborated by a transient shift of the excitonic 1s–2p resonance (see Supplementary Fig. 4). The conduction band states in the Λ and Λ′ valleys are composed of selenium p orbitals, whose overlap with the corresponding wavefunctions of the neighboring layer leads to efficient interlayer hopping^{34} and hybridization of the excitonic wavefunctions (X_{hyb}). We study this scenario by tracing excitons with their hole fixed in one layer—without loss of generality assigned to the top layer—whereas the electrons are located in Λlike valleys of either the top (Λ_{top}) or the bottom (Λ_{bot}) layer (see Fig. 3b and Methods), forming intra (X_{intra}) or interlayer excitons (X_{inter}). The energy of both exciton species is influenced by the electronhole Coulomb attraction. The interlayer exciton energy depends further on the spinorbit splitting of the conduction band in the bottom layer. The parabolic dispersion relations of the excitons, E_{intra} and E_{inter}, are mutually offset in momentum space by θ (see Fig. 3b). Depending on the wavefunction overlap of X_{intra} and X_{inter} in momentum space (Fig. 3c, blue/orange grid), these states can hybridize effectively. The overlap (Fig. 3c, violet surfaces) depends on θ and the orbital quantum number of the excitonic states, which defines the momentum spread of the wavefunctions. Upon hybridization, bonding and antibonding exciton states are formed. As the energy splitting between bonding and antibonding 1s states is as large as ~500 meV, the antibonding exciton state is pushed outside of our MIR probe spectrum. Therefore, we only consider the energetically lower 1s and 2phybrid states (Fig. 3d, magenta lines). As the energy reduction differs for these two states, the 1s–2p transition energy varies with θ (Fig. 3d).
For θ = 60°, X_{intra} and X_{inter} feature very different energies (Fig. 3d). This situation is characteristic of the 2H order, where the momentum overlap is strongest for intra and interlayer exciton states containing electrons from the Λ and Λ′ valley, respectively. As hybridization can only occur for spinlike electron bands and spinorbit splitting is of opposite sign for the two valleys, the nonhybridized excitons are offset by the strong spinorbit splitting. Despite this energy mismatch, the energy of X_{hyb} is clearly reduced. For increasing misalignment (θ < 60°) with respect to 2H stacking, the 1s states, which are spread out over a broad range of momenta, continue to effectively hybridize. Conversely, the 2p wavefunctions are more strongly localized in momentum space. Hence misalignment towards θ = 30° rapidly quenches hybridization in the 2p state and therefore reduces the energy gain. The combination of these mechanisms results in a distinct blue shift of the 1s–2p resonance, in excellent agreement with the experiment (Fig. 2c, blue circles). For 3R stacking, X_{intra} and X_{inter} are almost degenerate because both states contain electrons from the same valley and are therefore not offset by spinorbit splitting. In particular, the 2p states are almost perfectly degenerate, which enhances the excitonic hybridization as compared to the pristine bilayer (θ = 60°) and leads to an even smaller 1s–2p transition energy (see Fig. 3d).
Tailoring the excitonic lifetime
The variable degree of interlayer hybridization of excitons allows us to tune the exciton dynamics as well as the exciton–exciton interaction (Fig. 4). We trace the temporal evolution of the exciton population by mapping out the pumpinduced change of the MIR electric field ΔE at a fixed electrooptic sampling time, t_{eos} = 0 fs (see ref. ^{28} and Methods), for a variable pumpprobe delay time t_{pp}. Fig. 4a shows a rapid appearance of exciton density at t_{pp} = 0 ps, for all twist angles, whereas the decay dynamics depend sensitively on θ. An exponential fit (Fig. 4a, dashed lines) shows a systematic reduction of the lifetime of the hybrid excitons (Fig. 4b) from τ = 54 ps to 8 ps as θ is reduced from 60° to 30°. Intriguingly, for θ = 0°, the lifetime rises dramatically to a value of 148 ps. This behavior can be understood by considering the hybridized wavefunctions in real space.
To do this, we keep the position of the hole fixed in the upper TMD layer and plot the probability density of the electron (Fig. 4d, blue peaks). Because of strong interlayer hopping between the Λ and Λ′ points of the adjacent layers, the electron delocalizes within the BL for θ = 60°. A distinct probability density in the bottom layer (23%) signifies the interlayer character of the hybrid exciton. By changing the twist angle to θ = 50°, the orbital overlap of the conduction band states shrinks, reducing the probability density of the electron in the bottom layer (18%). With both electron and hole wavefunctions overlapping strongly in the top layer, the lifetime of the exciton drops, as processes such as phononassisted radiative decay^{32,35} or Auger recombination^{28} likely become more efficient. For θ = 30°, the interlayer hopping becomes even weaker, which further reduces the exciton lifetime. In contrast, for θ = 0° interlayer hopping is facilitated by the energetic degeneracy of the Λvalley conduction band states discussed above. In this case, the strong delocalization of the electron over the top (58%) and bottom (42%) layers distinctly extends the exciton lifetime as the hybrid state becomes more interlayerlike. At the same time, the delocalized wavefunctions provide a larger transition dipole moment, enhancing the amplitude of Δσ_{1} for θ = 0° as compared with θ = 60° (Fig. 2a).
Shaping the interaction between excitons
The twist angle controlled interlayer delocalization can also be employed to tune the interaction between excitons. To this end, we record the position of the 1s–2p resonance as a function of the pump fluence (Fig. 4c). For θ = 0°, a dramatic red shift is observed, whereas the effect is less pronounced for θ = 5° and completely absent for θ = 60° (see Supplementary Fig. 5, for the full dielectric response). We expect that an effective renormalization of the exciton binding energy should ultimately occur for all twist angles, when the exciton densities are increased beyond the values accessible in our experiment. Microscopically, the critical exciton density at which screening effects become important is determined by the exciton wavefunction. The larger the respective Bohr radius, the lower the critical density at which the orbitals start to overlap. Hence, strong interlayer delocalization accompanied by a larger lateral extent of the wavefunction (θ = 0°, see Fig. 4d) enables excitons to effectively screen the mutual Coulomb attraction between electron and hole. In addition, the delocalization of the excitons across the two layers for θ = 0° increases their polarizability and enables an outofplane dipoledipole type interaction^{36}. Consequently, hybrid excitons experience a stronger renormalization of their internal transitions with increasing exciton density as compared to previous reports on intralayer excitons^{28,37}. Thus, for increasing pump fluences the exciton binding energy is further reduced beyond the value of the hybrid exciton with negligible manybody interactions. Therefore, the theoretically predicted transition energy (see Figs. 2c and 4c, blue circle) is in very good agreement with the experimental results only for low fluences (compare grey sphere in Fig. 2c).
Discussion
In conclusion, we have tailored Coulomb correlations by interlayer hybridization in twisted WSe_{2} BLs. By accessing internal 1s–2p transitions of the KΛ excitons, we reveal systematic changes of their binding energy and lifetime with twist angle. A microscopic model based on density functional and density matrix theory quantitatively describes the underlying hybridization effects. Furthermore, we reveal how the densitydependent exciton–exciton interaction is renormalized by the twist angle. These findings underpin a qualitatively different way of finetuning the internal structure of excitons in TMDs and controlling the electronic and optical properties in van der Waals structures based on a single material. Tailoring manybody correlations over a large energy range can now be systematically extended to the full library of twodimensional crystals, which should guide the search for manmade electronic phases in van der Waals heterostructures.
Methods
Sample fabrication and photoluminescence spectroscopy
The TMD monolayers were exfoliated mechanically from a bulk single crystal using the viscoelastic transfer method^{38}. Both constituent WSe_{2} monolayers were inspected under an optical microscope and subsequently stacked on top of each other on a diamond substrate. During the transfer process, the lateral position and rotation of the layers can be controlled accurately using micropositioning stages. To remove any adsorbates trapped between the monolayers, the BLs were annealed at a temperature of 150 °C and a pressure of 1 × 10^{−5} mbar for 5 hours. Subsequently, we recorded the microphotoluminescence intensity of the BLs with a continuouswave laser (wavelength, 488 nm) focused onto the sample using a microscope objective (numerical aperture, 0.6). The radiation emitted by the BLs is then detected in backscattering geometry, dispersed in a spectrometer, and read out by a chargecoupled device camera.
Secondharmonic generation
The twist angle θ was determined for each BL by polarizationresolved secondharmonic generation (SHG) under rotation of the plane of polarization^{31}. Isolated monolayer regions of the structure were excited by a linearlypolarized femtosecond modelocked Ti:sapphire laser and the intensity of the SHG was recorded. Characteristic sixfold patterns, whose maxima indicate the armchair directions of the monolayers, are then obtained. For the representative structure in Fig. 1a, the intensity maxima span an angle ~7°. As the method is not sensitive to the electric field of the second harmonic but only to its intensity, an ambiguity remains between twist angles of θ = 7° and θ = 60°7°. This ambiguity can be resolved by recording the sixfold pattern of the SHG polarization anisotropy of the bilayer area of the sample (see Supplementary Fig. 2). The strong suppression of the SHG emission of the bilayer region attests to the fact that the two layers are almost perfectly antialigned^{39,40} corresponding to an angle close to the natural 2H stacking. Thus, the twist angle of the sample presented in Fig. 1a can be determined as θ = 53°. For the other BLs, an identical procedure is performed (see Supplementary Note 1).
Ultrafast NIR pump–MIR probe spectroscopy
A schematic of the experimental principle is depicted in Fig. 1c. 1s A excitons are resonantly created in the WSe_{2} BL by photoexcitation with 100fs pump pulses centered at an energy of 1.69 eV (Supplementary Fig. 3a, inset) from a homebuilt Ti:sapphire laser amplifier (repetition rate, 400 kHz). The phaselocked MIR probe pulses are generated by optical rectification in a GaSe crystal (thickness, 10 µm) using a fraction of the laser output. After a variable delay time t_{pp}, the probe pulses are transmitted through the photoexcited sample and the MIR waveform is recorded by electrooptic sampling in a second 10 µmthick GaSe crystal as a function of the electrooptic sampling time t_{eos} (Supplementary Fig. 3a). The MIR probe pulses are centered at a frequency of 28 THz with a full width at half maximum of 14 THz and a spectral phase, which is approximately flat between 11 and 44 THz (Supplementary Fig. 3b). Any changes to the electric field, ΔE(t_{eos}), of the waveform induced by the transient dielectric response of the sample are resolved in absolute amplitude and phase by serial lockin detection as a function of the electrooptic sampling time t_{eos}. The full complexvalued dielectric response, characterized by Δσ_{1} and Δε_{1}, can be retrieved from these data by applying a transfer matrix formalism^{26,28,30}. This technique is sensitive to the total population of bound and unbound electronhole pairs, irrespective of interband selection rules, and can, thus, address optically dark and bright excitons^{41}.
Microscopic manyparticle theory
To calculate the internal quantum transitions of excitons in the twisted BL, a manyparticle Hamiltonian was diagonalized accounting for the Coulomb interaction on a HartreeFock level. For WSe_{2} BLs, ab initio calculations suggest that momentumdark KΛ excitons represent the lowest states. Therefore, the Hamiltonian was restricted to electrons and holes at these highsymmetry points, taking into account the energetically most favorable dipoleallowed spin configuration. The BL was then treated as a bipartite system, using the monolayer eigenstates as basis functions of the two subsystems. Here, the layerlocalized basis functions were treated in an effective mass approximation with parameters extracted from ab initio simulations for monolayers. The Coulomb interaction strength was implemented on a semiclassical level assuming anisotropic homogeneous slabs to account for the complex dielectric environment of the BL system. Ab initio parameters for the anisotropic dielectric tensor of WSe_{2} were used and the influence of the substrate was included. The overlap of electronic wavefunctions of the twolayer subsystem gives rise to interlayer hybridization effects. Only the hybridization of electrons at the Λ point was considered, which is known to be orders of magnitude stronger than at the K point owing to different orbital compositions. The interlayer interaction strength was extracted from ab initio calculations of the BL band structure for different stacking angles. The Hamiltonian was diagonalized by first transforming it into an exciton frame to account for the Coulomb interaction. Subsequently, eigenenergies and wavefunctions were represented in terms of intra and interlayer excitons, resulting from the Wannier equation for the BL. The overlap of electronic wavefunctions between the two layers results in an interaction of intra and interlayer excitons. Finally, the excitonic Hamiltonian was diagonalized yielding intralayerinterlayer exciton hybrids. For the numerical calculation of the bilayer eigenstates we consider the hybridization of a large number of both intra and interlayer excitons including states with angular momentum of s, p, d, and ftype. The measured transition energies were then compared with the energetically lowest dipoleallowed transition of the calculated hybrid exciton Rydberg series where we neglect higherorder corrections such as a Berry curvatureinduced splitting of the otherwise degenerate p states. For additional details see Supplementary Note 5.
Ab initio DFT calculations
The electronic band structure was calculated on the basis of density functional theory in the PerdewBurkeErnzerhof approximation, as implemented in the Quantum ESPRESSO package. The atomic positions and inplane lattice constants of twisted BL WSe_{2} were optimized with different relative twist angles and the resulting atomic geometries were used to derive an average equilibrium distance between the two tungsten sublayers. Based on the optimized geometry of a 2H stacked BL, the evolution of the electronic band structure for varying interlayer distance was simulated. The required interlayer interaction strength to calculate the exciton hybridization was extracted from the splitting of the electronic bands at the Λ point. To obtain a continuous twist angle dependence of the interaction parameter, the interlayer distances obtained for computationally feasible angles were linearly interpolated. Spinorbit interactions and quasiparticle energy corrections from the G_{0}W_{0} approximation were explicitly included to calculate the full band structure for 2H stacked WSe_{2}. Here, the YAMBO code was used to calculate the quasiparticle corrections with spinorbit effects. Subsequently, GW calculations with the BerkeleyGW code were performed, combined with a nonuniform Neck Subsampling method to derive an additional scissorshift correction for the electronic band gaps. For additional details, see Supplementary Note 6.
Twocomponent model
A phenomenological twocomponent model was used to extract the quantitative properties of the 1s–2p transition from the transient dielectric response. To this end, the pumpinduced change of the dielectric function Δε(ω) = Δε_{1}(ω) + Δσ_{1}(ω)/(ε_{0}ω) is described by the Drude–Lorentz model:
where e is the electron charge and ε_{0} denotes the vacuum permittivity. The Lorentzian resonance with oscillator strength f_{1s,2p} describes the internal 1s–2p transition of the excitons within the WSe_{2} BL as determined by the density matrix theory (see Methods). Owing to the twist angledependent hybridization, the oscillator strength equals f_{1s,2p} = 0.34, f_{1s,2p} = 0.33, f_{1s,2p} = 0.23, f_{1s,2p} = 0.18, f_{1s,2p} = 0.19, f_{1s,2p} = 0.26, and f_{1s,2p} = 0.42 for θ = 60°, 59°, 53°, 30°, 21°, 5°, and 0°, respectively. The resonance is further characterized by the exciton density n_{X}, the resonance energy ħω_{res} and the linewidth Δ, which are free fit parameters. The effective mass of the electronhole pairs μ = 0.23 m_{0} is determined with the electron and hole effective masses within the WSe_{2} BL (ref. ^{42}), where m_{0} is the freeelectron mass. The sample is treated as a thin slab in this model with a thickness d = 0.7 nm (ref. ^{28}) of each layer. The second term in Δε(ω) is a Drude model describing unbound electronhole pairs or excitonic transitions energetically below the spectral window of the MIR probe pulses^{30}. Here, the density of free carriers n_{fc} and their scattering rate Γ govern the resulting dielectric response. Strict limits on the possible values of the free parameters are set by the fact that both independently retrieved spectra Δσ_{1} and Δε_{1} have to be reproduced simultaneously. Without any further restrictions, the numerical adaption of the measured spectra yields an overall good fit quality.
Data availability
The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Martin Furthmeier for technical assistance and Jens Kunstmann for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—ProjectID, 314695032—SFB 1277 (Subprojects A05 and B03). The Chalmers group acknowledges funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 881603 (Graphene Flagship) as well as from the Swedish Research Council (VR, project number 201800734).
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R.H. and E.M. supervised the study. P.M., F.M., A.G., M.L., and R.H. carried out the experiment. P.M., F.M., A.G., K.L., L.W., M.L., C.Y., and J.M.L. prepared and precharacterized the twisted bilayers. S.B., R.G., J.M., and E.M. carried out the theoretical modeling. All authors analyzed the data, discussed the results, and contributed to the writing of the manuscript.
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Merkl, P., Mooshammer, F., Brem, S. et al. Twisttailoring Coulomb correlations in van der Waals homobilayers. Nat Commun 11, 2167 (2020). https://doi.org/10.1038/s4146702016069z
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