Abstract

Atomically thin layers of two-dimensional materials can be assembled in vertical stacks that are held together by relatively weak van der Waals forces, enabling coupling between monolayer crystals with incommensurate lattices and arbitrary mutual rotation1,2. Consequently, an overarching periodicity emerges in the local atomic registry of the constituent crystal structures, which is known as a moiré superlattice3. In graphene/hexagonal boron nitride structures4, the presence of a moiré superlattice can lead to the observation of electronic minibands5,6,7, whereas in twisted graphene bilayers its effects are enhanced by interlayer resonant conditions, resulting in a superconductor–insulator transition at magic twist angles8. Here, using semiconducting heterostructures assembled from incommensurate molybdenum diselenide (MoSe2) and tungsten disulfide (WS2) monolayers, we demonstrate that excitonic bands can hybridize, resulting in a resonant enhancement of moiré superlattice effects. MoSe2 and WS2 were chosen for the near-degeneracy of their conduction-band edges, in order to promote the hybridization of intra- and interlayer excitons. Hybridization manifests through a pronounced exciton energy shift as a periodic function of the interlayer rotation angle, which occurs as hybridized excitons are formed by holes that reside in MoSe2 binding to a twist-dependent superposition of electron states in the adjacent monolayers. For heterostructures in which the monolayer pairs are nearly aligned, resonant mixing of the electron states leads to pronounced effects of the geometrical moiré pattern of the heterostructure on the dispersion and optical spectra of the hybridized excitons. Our findings underpin strategies for band-structure engineering in semiconductor devices based on van der Waals heterostructures9.

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Data availability

The datasets used for Figs. 25 and Extended Data Figs. 1, 49 are provided as Source Data. All other data is available from the corresponding authors upon reasonable request.

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Acknowledgements

We acknowledge financial support from the European Graphene Flagship Project under grant agreement 696656, EC Project 2D-SIPC and EPSRC grant EP/P026850/1. E.M.A. and A.I.T. acknowledge support from EPSRC grants EP/M012727/1 and the European Union’s Horizon 2020 research and innovation programme under ITN Spin-NANO Marie Sklodowska-Curie grant agreement 676108. D.A.R.-T. and V.I.F. acknowledge support from ERC Synergy Grant Hetero2D, EPSRC EP/N010345 and the Lloyd Register Foundation Nanotechnology grant. K.S.N. acknowledges financial support from the Royal Society, EPSRC, US Army Research Office and ERC Synergy Grant Hetero2D. H.S.S. acknowledges a research fund (NRF-2017R1E1A1A01074493) from the National Research Foundation by the Ministry of Science and ICT, South Korea. M.R.M. acknowledges support from the National Science Centre (UMO-2017/24/C/ST3/00119). R.V.G. acknowledges financial support from the Royal Society Fellowship Scheme and EPSRC CDT Graphene-NOWNANO EP/L01548X. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST.

Author information

Affiliations

  1. Department of Physics and Astronomy, University of Sheffield, Sheffield, UK

    • Evgeny M. Alexeev
    •  & Alexander I. Tartakovskii
  2. School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, UK

    • David A. Ruiz-Tijerina
    • , Mark Danovich
    • , Matthew J. Hamer
    • , Daniel J. Terry
    • , Maciej Koperski
    • , Kostya S. Novoselov
    • , Roman V. Gorbachev
    •  & Vladimir I. Fal’ko
  3. National Graphene Institute, University of Manchester, Manchester, UK

    • David A. Ruiz-Tijerina
    • , Mark Danovich
    • , Matthew J. Hamer
    • , Daniel J. Terry
    • , Maciej R. Molas
    • , Maciej Koperski
    • , Kostya S. Novoselov
    • , Roman V. Gorbachev
    •  & Vladimir I. Fal’ko
  4. Centro de Nanociencias y Nanotecnología, Universidad Nacional Autónoma de México, Ensenada, Mexico

    • David A. Ruiz-Tijerina
  5. Department of Energy Engineering and Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea

    • Pramoda K. Nayak
    • , Seongjoon Ahn
    •  & Hyeon Suk Shin
  6. Department of Physics, Indian Institute of Technology Madras, Chennai, India

    • Pramoda K. Nayak
  7. Department of Engineering Science, University of Oxford, Oxford, UK

    • Sangyeon Pak
    • , Juwon Lee
    •  & Jung Inn Sohn
  8. Division of Physics and Semiconductor Science, Dongguk University-Seoul, Seoul, South Korea

    • Jung Inn Sohn
  9. Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland

    • Maciej R. Molas
  10. National Institute for Materials Science, Tsukuba, Japan

    • Kenji Watanabe
    •  & Takashi Taniguchi
  11. Henry Royce Institute for Advanced Materials, Manchester, UK

    • Roman V. Gorbachev
    •  & Vladimir I. Fal’ko

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Contributions

E.M.A. carried out microscopy and optical spectroscopy experiments. E.M.A. and A.I.T. analysed optical spectroscopy data. D.A.R.-T., M.D. and V.I.F. developed the theory. P.K.N., S.A., S.P., J.L., J.I.S. and H.S.S. carried out CVD growth of the monolayers and fabricated the heterobilayer samples grown from them. M.J.H., D.J.T. and R.V.G. fabricated mechanically exfoliated monolayers and corresponding heterobilayers using glove-box techniques. M.R.M. and M.K. carried out second-harmonic generation measurements on exfoliated samples. K.W. and T.T. synthesized the hBN crystals. E.M.A., D.A.R.-T., A.I.T. and V.I.F. wrote the manuscript. E.M.A., K.S.N., V.I.F. and A.I.T. conceived the experiment. All authors participated in discussions. A.I.T. oversaw the project.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Vladimir I. Fal’ko or Alexander I. Tartakovskii.

Extended data figures and tables

  1. Extended Data Fig. 1 Integrated intensity and linewidth for room-temperature photoluminescence spectra of MoSe2/WS2 heterobilayers as a function of the interlayer twist angle θ.

    a, b, Variation of integrated intensity (a) and linewidth (b). See Methods for a description of the fitting procedure. Data acquired for two individual substrates containing MoSe2/WS2 heterobilayers made from CVD-grown monolayers of MoSe2 and WS2 are shown with light red and dark red symbols. The blue curve in b shows the results of theoretical calculations described in Supplementary Note 3. Source data

  2. Extended Data Fig. 2 Breakdown of the moiré harmonic potential approximation in MoSe2/WS2.

    a, MoSe2 A-exciton (XA, red) and interlayer exciton (iX, blue) bands in MoSe2/WS2 within the moiré Brillouin zone, for twist angle θ = 54°. Purple arrows represent a second-order virtual tunnelling process enabled by intralayer–interlayer exciton hybridization, giving rise to a moiré potential for bright (Γ-point) MoSe2 A excitons. b, Perturbation theory breaks down for MoSe2 Γ-point A excitons at θ = 58°, owing to the exciton band crossing. As a result, the effective moiré potential diverges for this twist angle, indicating that the harmonic potential approximation is invalid. c, Perturbative parameter \(|\mathop{T}\limits^{ \sim }|/|{{\mathscr{E}}}_{{m{i}}{m{X}},\downarrow }^{+}(-{m{\Delta }}{\boldsymbol{K}})-{{\mathscr{E}}}_{{m{X}},\downarrow }(0)|\) (top; see Supplementary Note 2) and line energy of the photoluminescence P1 peak (bottom). The vertical grey lines divide the plot into the 8° ≤ θ ≤ 52° plateau and the two modulation regions, showing that the latter is always outside the region of validity of the harmonic approximation.

  3. Extended Data Fig. 3 The moiré mini Brillouin zone in MoSe2/WS2 heterobilayers.

    a, Real-space stacking of the MoSe2 (red) and WS2 (green) lattices for twist angle θ, with lattice vectors labelled \({{\boldsymbol{a}}}_{n}^{{{m{M}}{m{X}}}_{2}}\). Red (green) circles represent Mo (W) atoms, with Se and S atoms shown as orange circles. b, The resulting alignment between the Brillouin zones. Corresponding reciprocal lattice vectors \({{\boldsymbol{G}}}_{n}^{{{m{M}}{m{o}}{m{S}}{m{e}}}_{2}}\) and \({{\boldsymbol{G}}}_{n}^{{{m{W}}{m{S}}}_{2}}\) appear misaligned by the twist angle θ. The monolayer band edges appear at the points \({K}_{{{m{MX}}}_{{m{2}}}}\) of the corresponding Brillouin zones; the valley mismatch vectors are labelled ΔK and ΔK′. c, The two MoSe2 Bragg vectors that contribute to the hopping term in equation (3) in Supplementary Note 1. d, First circle of Bragg vectors for the two TMD crystals, defining the moiré vectors \({{\boldsymbol{b}}}_{n}={{\boldsymbol{G}}}_{n}^{{{m{W}}{m{S}}}_{2}}-{{\boldsymbol{G}}}_{n}^{{{m{M}}{m{o}}{m{S}}{m{e}}}_{2}}\) for θ ≈ 0°. e, Mini Brillouin zone defined by the vectors bn, where the lattice mismatch ΔK appears at the edge of the mini Brillouin zone. The different moiré vectors can be constructed as \(({\mathscr{D}}-{\mathscr{D}}{m{^{\prime} }}){m{\Delta }}{\boldsymbol{K}}\). An example is shown, in which ΔKC3ΔK = −b3.

  4. Extended Data Fig. 4 Calculated moiré band structures of hybridized excitons in MoSe2/WS2 heterobilayers for various interlayer twist angles.

    Bright hybridized exciton band structures within the first moiré Brillouin zone for various twist angles, calculated using the parameters reported in Extended Data Table 1 for the sample fabricated from exfoliated monolayers. The red dashed curves in all panels show the uncoupled MoSe2 A-exciton dispersion, for reference. In the top panels, the blue dashed curves correspond to the interlayer exciton state iX, whereas in the bottom panels they represent the interlayer exciton state labelled iXʹ. Source data

  5. Extended Data Fig. 5 Low-temperature reflectance contrast spectra of MoSe2/WS2 heterobilayers formed from CVD-grown monolayers.

    a, Reflectance contrast spectra recorded at T = 10 K in the MoSe2/WS2 heterostructure with an interlayer twist angle of 1° (red), 31° (orange) and 59° (blue) in the spectral range around the MoSe2 A-exciton energy. Closely aligned regions show the redshift and substantial reduction in intensity of the main peak compared to the rotationally misaligned heterobilayer, as well as the emergence of additional weak features above the main peak. b, Reflectance contrast spectra measured in the vicinity of the B-exciton energy in an isolated MoSe2 monolayer (black), and in MoSe2/WS2 heterostructures with various twist angles. The B-exciton feature in the isolated MoSe2 monolayer spectrum is labelled XB. Vertical lines show the position of the maximum derivative of the XB feature in the monolayer (black) and misaligned heterobilayer (orange) spectra. The full description of the experimental procedure for the reflectance contrast measurements is given in Methods. Source data

  6. Extended Data Fig. 6 Variation of photoluminescence and reflectance contrast spectra in MoSe2/WS2 heterobilayers fabricated from exfoliated MoSe2 and WS2 monolayers encapsulated in hBN.

    a, Bright-field image of a fully encapsulated MoSe2/WS2 sample S1; the points for which we report the photoluminescence and reflectance contrast spectra in bd are marked. Scale bar, 10 μm. bd, Low-temperature photoluminescence (black) and reflectance contrast (red) spectra recorded in several regions of the sample marked in a. The two higher-energy peaks in the photoluminescence spectra, X at 1.624 and XA at 1.652 eV, correspond to trion and excition emission unintentionally collected from the single-layer MoSe2 area located at the right side of the heterobilayer region. The position of these peaks remains unchanged in all three points, while their intensity decreases gradually with the increasing spatial separation. The two lower-energy photoluminescence peaks, labelled hX and hX1, represent the emission originating in the heterostructure region and show variation in position and relative intensities across the heterostructure region, probably caused by the non-uniform strain and doping. The reflectance contrast spectra recorded at the three points are similar, with the two lower-energy peaks directly corresponding to hX and hX1 in the photoluminescence spectra, and hX2 and hX3 representing the higher-energy states. e, Comparison of low-temperature photoluminescence spectra recorded in the samples fabricated from mechanically exfoliated monolayers. Dashed lines show photoluminescence spectra of uncoupled single-layer MoSe2, recorded in the same sample, in which an uncoupled MoSe2 monolayer area was present. Samples S1–S4 were fabricated with the crystal axes of the two materials closely aligned, whereas sample S5 was made with a considerable rotational misalignment (θ = 12°). Despite the variation of exciton (XA) and trion (X) energies, all four aligned samples show a hybridized exciton peak hX1, located 20–30 meV lower in energy than the monolayer trion line. Samples S1 and S2 show an additional lower-energy line hX positioned approximately 32 meV lower in energy than hX1. Figure 4 reports data for the closely aligned sample S1 and the misaligned sample S5. Figure 5 reports data for sample S1. Source data

  7. Extended Data Fig. 7 Temperature dependence of photoluminescence and reflectance contrast spectra in a MoSe2/WS2 heterobilayer made from exfoliated MoSe2 and WS2 monolayers encapsulated in hBN.

    The data presented here are for sample S1, for which additional data are presented in Figs. 4, 5. a, Normalized photoluminescence spectra for hBN-encapsulated MoSe2/WS2 heterostructure S1 at different temperatures. At T = 10 K, the emission spectrum consists of the hX peak at 1.565 eV and the hX1 peak 1.598 eV originating in the heterostructure, and the strong trion peak (X) at 1.624 eV and a weaker neutral exciton peak (XA) at 1.652 eV from the isolated MoSe2 monolayer. hX disappears at T ≥ 105 K, whereas hX1 is the dominant photoluminescence feature visible at room temperature. b, Reflectance contrast spectra recorded in the same region of the sample and at the same temperatures as the photoluminescence spectra in a. The two lower-energy peaks visible in the low-temperature spectrum directly correspond to the hX and hX1 photoluminescence features, whereas peaks hX2 and hX3 represent the higher-energy features which are not visible in the photoluminescence spectra. The hX and hX3 peaks become weak above T = 105 K, while the hX1 and hX2 peaks persist to much higher temperatures, with the hX1 peak remaining visible at room temperature. c, Energy of hX (red), hX1 (green) and hX2 (blue) features in photoluminescence and reflectance contrast spectra as a function of temperature. The peak positions in photoluminescence (reflectance contrast) spectra are marked with a circle (triangle). d, e, Photoluminescence linewidth (d) and integrated intensity (e) of hX (red) and hX1 (green) as functions of temperature. FWHM, full width at half maximum; cts/s, counts per second. Source data

  8. Extended Data Fig. 8 Theoretical photoluminescence spectra of MoSe2/WS2 for different temperatures.

    a, Calculated activation energy for hX1 photoluminescence, as a function of twist angle. b, Normalized photoluminescence intensity in the MoSe2 A-exciton energy range, for three different temperatures: T = 60 K, 160 K and 300 K (room temperature). Photoluminescence from state hX1 produces the peak identified as P1 in measurements from samples fabricated with CVD-grown monolayers. A second peak at higher photon energies (black arrow) is thermally activated at approximately 160 K, corresponding to photoluminescence from the hX2 state, in excellent agreement with our measurements from the samples prepared from exfoliated monolayers (Fig. 5). Photoluminescence peak broadening and redshift with increasing temperature are not taken into account for these simulations. Source data

  9. Extended Data Fig. 9 Twist-angle measurements using second-harmonic generation.

    a, b, The symbols show the data for the nearly aligned (a) and misaligned (b) samples fabricated from mechanically exfoliated monolayers. Solid lines in the graph represent the fitting of the data with ISHG sin(3α + ϕ), where α is the rotation angle of the half-wave plate and is φ is a fitting parameter defining the relative orientation of TMD crystal lattices. For full details of the SHG measurements, see Methods. Source data

  10. Extended Data Table 1 Parameters of the exciton hybridization model

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