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# Formation of moiré interlayer excitons in space and time

## Abstract

Moiré superlattices in atomically thin van der Waals heterostructures hold great promise for extended control of electronic and valleytronic lifetimes1,2,3,4,5,6,7, the confinement of excitons in artificial moiré lattices8,9,10,11,12,13 and the formation of exotic quantum phases14,15,16,17,18. Such moiré-induced emergent phenomena are particularly strong for interlayer excitons, where the hole and the electron are localized in different layers of the heterostructure19,20. To exploit the full potential of correlated moiré and exciton physics, a thorough understanding of the ultrafast interlayer exciton formation process and the real-space wavefunction confinement is indispensable. Here we show that femtosecond photoemission momentum microscopy provides quantitative access to these key properties of the moiré interlayer excitons. First, we elucidate that interlayer excitons are dominantly formed through femtosecond exciton–phonon scattering and subsequent charge transfer at the interlayer-hybridized Σ valleys. Second, we show that interlayer excitons exhibit a momentum fingerprint that is a direct hallmark of the superlattice moiré modification. Third, we reconstruct the wavefunction distribution of the electronic part of the exciton and compare the size with the real-space moiré superlattice. Our work provides direct access to interlayer exciton formation dynamics in space and time and reveals opportunities to study correlated moiré and exciton physics for the future realization of exotic quantum phases of matter.

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## Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 432680300/SFB 1456, project B01, 217133147/SFB 1073, projects B07 and B10, and 223848855/SFB 1083, project B9. G.S.M.J. acknowledges financial support by the Alexander von Humboldt Foundation. A.A. and S.H. acknowledge funding from EPSRC (EP/T001038/1, EP/P005152/1). A.A. acknowledges financial support by the Saudi Arabian Ministry of Higher Education. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) and JSPS KAKENHI (grant numbers 19H05790, 20H00354 and 21H05233). E.M. acknowledges support from the European Union Horizon 2020 research and innovation programme under grant agreement number 881603 (Graphene Flagship) as well as Vinnova via the competence centre ‘2D-TECH’ and the Knut and Alice Wallenberg Foundation via the Grant KAW 2019.0140.

## Author information

Authors

### Contributions

D. Steil, R.T.W., S.S., G.S.M.J., S.H., M.R. and S.M. conceived the research. D. Schmitt, J.P.B. and W.B. carried out the time-resolved momentum microscopy experiments and analysed the data. W.B., D.R.L. and G.S.M.J. carried out the real-space reconstruction of the momentum fingerprints. A.A. fabricated the samples. G.M., S.B. and E.M. developed the microscopic model and analysed the results. All authors discussed the results. M.R. and S.M. were responsible for the overall project direction and wrote the manuscript with contributions from all co-authors. K.W. and T.T. synthesized the hBN crystals.

### Corresponding authors

Correspondence to Marcel Reutzel or Stefan Mathias.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Alessandra Lanzara, Simone Latini and Soren Ulstrup for their contribution to the peer review of this work.

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## Extended data figures and tables

### Extended Data Fig. 1 Fabrication and real-space imaging of the of the WSe2/MoS2/hBN heterostructure.

a–d Schematic of the WSe2/MoS2/hBN heterostructure assembly on a Si substrate with a native oxide layer. a A PDMS stamp with monolayer MoS2 flake (red) is aligned with another PDMS stamp holding a monolayer WSe2 (yellow), then the top PDMS stamp is brought in contact with bottom PDMS. b The top PDMS stamp is then withdrawn, leaving the MoS2 on top of the WSe2. c The PDMS stamp holding the WSe2/MoS2 is then brought into contact with SiO2/Si substrate with hBN (blue). d the PDMS is then withdrawn, leaving behind the final WSe2/MoS2/hBN heterostructure. e/f The WSe2/MoS2/hBN heterostructure, the uncovered SiO2/Si substrate, bulk hBN, the WSe2 and MoS2 monolayers, and bulk WSe2 are labelled in the e optical microscope and the f photoemission real-space image (ħω = 4.96 eV). Point-like structures (blisters) in the heterostructure region can be attributed to residual gas trapped either at the MoS2/hBN or the WSe2/MoS2 interface. The blisters in the monolayer WSe2 region are most likely trapped at the WSe2/hBN interface.

### Extended Data Fig. 2 Static band mapping of the monolayer WSe2 and the heterobilayer WSe2/MoS2.

a,b Energy–momentum representation of the static photoemission intensity obtained in the momentum microscopy experiment along the K’W-Γ-KW direction (see inset). The important spectroscopic features are labelled in the figure: (1, 2) spin-split valence bands of WSe2; (3) valence band of MoS2; (4, 5) valence bands at the ΓW,Mo valley; (6) valence band of hBN. c, d Energy-distribution curves taken around the KW and ΓW,MoW) valley indicated by the coloured boxes in a and b.

### Extended Data Fig. 3 Inhomogeneous broadening of the photoemission spectra.

The energy-distribution-curve is obtained in a ± 0.10 Å−1 region-of-interest centred at the K’W valley of WSe2. Gaussian fitting of the valence band maximum centred at E-EVBM = 0 eV yields a full-width at half-maximum of 280 ± 10 meV.

### Extended Data Fig. 4 Determination of the twist angle Θ of the WSe2/MoS2 heterostructure.

The momentum maps in a and c show the photoemission fingerprint of the AW- and AMo-excitons when excited resonantly with 1.7 eV and 1.9 eV photons, respectively, at 0 fs pump–probe delay. Since the MoS2 AMo-exciton and the ILX are, within our energy resolution, spectrally degenerate, faint signatures of the ILX are already visible c at 0 fs delay. The dashed lines indicate the Γ -KW (orange) and Γ-KMo (dark red) direction. From their misalignment, the twist angle is extracted to Θ = 9.8 ± 0.8°. b, d At 1-ps pump–probe delay, the ILX momentum fingerprint can be identified, as described by the mBZ (red). The dashed lines indicate the relation of the ILX momentum fingerprint and the Γ -KW and Γ-KMo directions. Note the distinctly different intensity distribution of the combined spectral weight of the AMo-exciton and the faint ILX in c vs. the pure signature of the ILX at 1-ps delay in d. For each momentum-map, the photoelectron energy with respect to the valence band maximum of WSe2 and the pump–probe delay are noted in the lower left and right corner, respectively.

### Extended Data Fig. 5 Above-band-gap excitation of monolayer WSe2 and monolayer MoS2.

Photoemission yield from bright intralayer excitons and charge carriers in the conduction can be discriminated based on above-band-gap excitation data collected on a–c monolayer WSe2 (orange circle in Fig. 1c) and d–f monolayer MoS2 (real-space image not shown). a and d show energy–momentum cuts along the K-Σ direction measured on WSe2 and MoS2, respectively, at 0 fs and 300 fs (respectively 250 fs) pump–probe delay. At 0 fs, a parabolic signature with positive dispersion is detected at the K valley (noted by the black dashed parabolic line). At 300 fs (250 fs), the signature becomes more spherical. b and e show energy-distribution curves at the K valleys (momentum-integration region based on the boxes in a, d). The peak maxima is indicated by grey horizontal lines. c and f show the corresponding energy-distribution curves when excited on resonance with the AW- and AMo-exciton, respectively.

### Extended Data Fig. 6 Additional trARPES data of the ILX formation.

a E(k) cut along the KW- Γ-K’W direction integrated in the k-region shown by the black dashed box in b, 0 fs. The arrowheads on the right side of the figure indicate the photoelectron energies where the momentum maps in b and c are centred. b Within the energy window of the k-map (E- EVBM = 1.7 eV) and increasing pump–probe delay, spectral weight from the bright AW-excitons (orange hexagon) is transferred via exciton–phonon scattering to form dark ΣW-excitons (grey hexagon). c Interlayer charge transfer via the Σ-valleys forms the ILX, which is observed at E-EVBM = 1.1 eV. The Brillouin zone of MoS2 is indicated with a dotted dark red hexagon and the mBZ with a red hexagon. Spectroscopic signatures of the AW-exciton, the ΣW-exciton, and the ILX are indicated by orange, grey, and red circles, respectively, in the 1 ps data. The pump–probe delay and the binding energy of the k-maps are noted in the top and bottom left corner, respectively. Note that at 0 fs, the strong signal in c is mainly caused by LAPE. In addition, in b (0 fs), LAPE leads to photoemission yield at the Γ point.

### Extended Data Fig. 7 Selected regions of interest for the analysis of the AW-, ΣW-, and ILX formation dynamics shown in Fig. 2c.

a and b show momentum maps integrated over all measured pump–probe delays in the energy intervals of E-EVBM = 1.5–2.4 eV and 0.8–1.3 eV, respectively. The regions-of-interest that are used for filtering the excitonic photoemission signatures in momentum space are indicated by orange, black, and red circles for the AW-exciton, the ΣW-exciton, and the ILX, respectively. The grey shaded areas indicate artefacts of the detector. c (top and middle panel) Pump–probe delay evolution of the energy- distribution curves filtered for the orange and red regions of interest. c (bottom panel) Intensity vs. pump–probe delay plots for the energy boxes indicated by the bluish arrows in the middle panel.

### Extended Data Fig. 8 Correction of space-charge/photovoltage-induced shifts and exclusion of photoinduced band renormalizations.

a The rigid band shift ΔE of maximal 70 meV is corrected for each pump–probe delay. ΔE is obtained by fitting (red) momentum-integrated energy-distribution curves (black) for each delay, as exemplary shown for the −2000 fs measurement in the inset. The error bars are standard deviations obtained in each fit. b We fit the pump–probe delay-dependent energetic peak position of the valence band maximum of WSe2 at the KW valley with Gaussian distributions (red, inset) and calculate the energy difference ΔE with respect to the −2000 fs measurement that is plotted in the inset. Within the scattering of the data, ΔE does not upshift with pump–probe delay, excluding a dominant contribution of photoinduced band renormalization such as discussed in refs. 49,53.

### Extended Data Fig. 9 Charge-transfer and charge recombination times of the AW exciton (orange), the ΣW exciton (grey), and the ILX (red).

a The short-time dynamics (symbols) is fitted with error functions (dashed lines), from which the delayed onset times ti are extracted, as detailed in the text. The solid lines reproduce the model calculations initially shown in Fig. 2c of the main text. b The charge recombination time is extracted by performing single-exponential fits to the data for pump–probe delays larger than 1 ps.

### Extended Data Fig. 10 Sketch of the moiré mBz for small and large twist angles.

The twist angle Θ defines the size of the moiré mBz (red hexagon) and the related moiré reciprocal lattice vectors $${{\boldsymbol{G}}}^{{1},{2}}\,{}_{{M}}$$ (black arrows). a For twist angles larger than a few degree, $${{\boldsymbol{G}}}^{{1},{2}}\,{}_{{M}}$$ is larger than the momentum width of a single κ valley (dark reddish filled circles). All three κ valleys (and higher-order umklapp processes) are detected in the momentum microscopy experiment (Fig. 3c, d). The electronic contribution to the ILX wavefunction is spread across several moiré potential wells (Fig. 4). b For sufficiently small twist angles, $${{\boldsymbol{G}}}^{{1},{2}}\,{}_{{M}}$$ can become smaller than the width of a single κ valley, and the ILX wavefunction can be confined in a single moiré potential well. If the momentum microscopy experiment is performed with a finite momentum resolution, the photoelectron signal from the ILX can appear as a single peak, as reported in Ref. 13 for a 2° twisted WSe2/MoS2 heterostructure.

## Supplementary information

### Supplementary Information

This file contains supplementary information on the microscopic modelling of the ILX formation dynamics, as described in the sections ‘Microscopic Model – Overview’, ‘Microscopic Model – Detailed description’ and ‘Microscopic Model – Analysis’. It contains Supplementary Fig. 1 illustrating hybrid exciton dispersion and scattering rates and Tables I and II quantifying energies, mixing coefficients and scattering rates. In addition, Supplementary refs. 1–16 are provided.

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Schmitt, D., Bange, J.P., Bennecke, W. et al. Formation of moiré interlayer excitons in space and time. Nature 608, 499–503 (2022). https://doi.org/10.1038/s41586-022-04977-7

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• DOI: https://doi.org/10.1038/s41586-022-04977-7