Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

You are viewing this page in draft mode.

Deep moiré potentials in twisted transition metal dichalcogenide bilayers


In twisted bilayers of semiconducting transition metal dichalcogenides, a combination of structural rippling and electronic coupling gives rise to periodic moiré potentials that can confine charged and neutral excitations1,2,3,4,5. Here we show that the moiré potential in these bilayers at small angles is unexpectedly large, reaching values above 300 meV for the valence band and 150 meV for the conduction band—an order of magnitude larger than theoretical estimates based on interlayer coupling alone. We further demonstrate that the moiré potential is a non-monotonic function of moiré wavelength, reaching a maximum at a moiré period of ~13 nm . This non-monotonicity coincides with a change in the structure of the moiré pattern from a continuous variation of stacking order at small moiré wavelengths to a one-dimensional soliton-dominated structure at large moiré wavelengths. We show that the in-plane structure of the moiré pattern is captured by a continuous mechanical relaxation model, and find that the moiré structure and internal strain, rather than the interlayer coupling, are the dominant factors in determining the moiré potential. Our results demonstrate the potential of using precision moiré structures to create deeply trapped carriers or excitations for quantum electronics and opto-electronics.

This is a preview of subscription content

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Structure of twisted heterobilayer WSe2/MoSe2.
Fig. 2: Spectroscopic properties of moiré patterns of different wavelengths.
Fig. 3: Spectroscopic imaging of conduction and valence band edges.
Fig. 4: Quantifying the moiré potential.

Data availability

Data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.


  1. Seyler, K. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. Nature 567, 66–70 (2019).

    ADS  Article  Google Scholar 

  2. Wu, F. et al. Theory of optical absorption by interlayer excitons in transition metal dichalcogenide heterobilayers. Phys. Rev. B 97, 035306 (2018).

    ADS  Article  Google Scholar 

  3. Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).

    ADS  Article  Google Scholar 

  4. Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

    Article  Google Scholar 

  5. Wang, J. et al. Excitonic phase transitions in MoSe2/WSe2 heterobilayers. Preprint at (2020).

  6. Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).

    ADS  Article  Google Scholar 

  7. Wang, Z. et al. Evidence of high-temperature exciton condensation in two-dimensional atomic double layers. Nature 574, 76–80 (2019).

    ADS  Article  Google Scholar 

  8. Wu, F. et al. Hubbard model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett. 121, 026402 (2018).

    ADS  Article  Google Scholar 

  9. Zhang, Y. et al. Moiré quantum chemistry: charge transfer in transition metal dichalcogenide superlattices. Phys. Rev. B 102, 201115 (2020).

    ADS  Article  Google Scholar 

  10. Yu, H. et al. Moiré excitons: from programmable quantum emitter arrays to spin–orbit-coupled artificial lattices. Sci. Adv. 3, e1701696 (2017).

    ADS  Article  Google Scholar 

  11. Ruiz-Tijerina, D. A. et al. Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides. Phys. Rev. B 99, 125424 (2019).

    ADS  Article  Google Scholar 

  12. Geng, W. et al. Moiré potential, lattice corrugation and band gap spatial variation in a twist-free MoTe22/MoS2 heterobilayer. J. Phys. Chem. Lett. 11, 2637–2646 (2020).

    Article  Google Scholar 

  13. Zhang, C. et al. Interlayer couplings, moiré patterns, and 2D electronic superlattices in MoS2/WSe2 hetero-bilayers. Sci. Adv. 3, e1601459 (2017).

    ADS  Article  Google Scholar 

  14. Pan, Y. et al. Quantum-confined electronic states arising from the moiré pattern of MoS2-WSe2 heterobilayers. Nano Lett. 18, 1849–1855 (2018).

    ADS  Article  Google Scholar 

  15. Zhang, Z. et al. Flat bands in twisted bilayer transition metal dichalcogenides. Nat. Phys. 16, 1093–1096 (2020).

    Article  Google Scholar 

  16. Li, H. et al. Imaging moiré flat bands in 3D reconstructed WSe2/WS2 superlattices. Preprint at (2020).

  17. Rosenberger, M. R. et al. Twist angle-dependent atomic reconstruction and moiré patterns in transition metal dichalcogenide heterostructures. ACS Nano 14, 4550–4558 (2020).

    Article  Google Scholar 

  18. Weston, A. et al. Atomic reconstruction in twisted bilayers of transition metal dichalcogenides. Nat. Nanotechnol. 15, 592–597 (2020).

    ADS  Article  Google Scholar 

  19. Halbertal, D. et al. Moiré metrology of energy landscapes in van der Waals heterostructures. Nat. Commun. 12, 242 (2021).

    Article  Google Scholar 

  20. Maity, I. et al. Reconstruction of moiré lattices in twisted transition metal dichalcogenide bilayers. Preprint at (2019).

  21. Enaldiev, V. et al. Stacking domains and dislocation networks in marginally twisted bilayers of transition metal dichalcogenides. Phys. Rev. Lett. 124, 206101 (2020).

    ADS  Article  Google Scholar 

  22. Edelberg, D. et al. Tunable strain soliton networks confine electrons in van der Waals materials. Nat. Phys. 16, 1097–1102 (2020).

    Article  Google Scholar 

  23. Bai, Y. et al. Excitons in strain-induced one-dimensional moiré potentials at transition metal dichalcogenide heterojunctions. Nat. Mater. 19, 1068–1073 (2020).

    ADS  Article  Google Scholar 

  24. Naik, M. H. et al. Ultraflatbands and shear solitons in moiré patterns of twisted bilayer transition metal dichalcogenides. Phys. Rev. Lett. 121, 266401 (2018).

    ADS  Article  Google Scholar 

  25. Wilson, N. R. et al. Determination of band offsets, hybridization and exciton binding in 2D semiconductor heterostructures. Sci. Adv. 3, e1601832 (2017).

    ADS  Article  Google Scholar 

  26. Edelberg, D. et al. Approaching the intrinsic limit in transition metal diselenides via point defect control. Nano Lett. 19, 4371–4379 (2018).

    ADS  Article  Google Scholar 

  27. Kresse, G. et al. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    ADS  Article  Google Scholar 

  28. Kresse, G. et al. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).

    ADS  Article  Google Scholar 

  29. Perdew, J. P. et al. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    ADS  Article  Google Scholar 

  30. Klimeš, J. et al. Van der Waals density functionals applied to solids. Phys. Rev. B 83, 195131 (2011).

    ADS  Article  Google Scholar 

  31. Carr, S. et al. Relaxation and domain formation in incommensurate two-dimensional heterostructures. Phys. Rev. B 98, 224102 (2018).

    ADS  Article  Google Scholar 

  32. Grimme, S. et al. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 132, 154104 (2010).

    ADS  Article  Google Scholar 

Download references


We thank S. Carr and E. Kaxiras for providing the GSFE function for relaxation calculations. We thank C. Dean, L. Fu, M. Jain, I. Maiti, Q. Shi and Y. Zhang for useful discussions. This work is supported by the Programmable Quantum Materials (Pro-QM) programme at Columbia University, an Energy Frontier Research Center established by the Department of Energy (grant no. DE-SC0019443). STM experiments were supported by the Air Force Office of Scientific Research via grant no. FA9550-16-1-0601 (S.S. and A.N.P.). Synthesis of MoSe2 and WSe2 was supported by the National Science Foundation Materials Research Science and Engineering Centers programme through Columbia in the Center for Precision Assembly of Superstratic and Superatomic Solids (DMR-1420634). D.H. was supported by a grant from the Simons Foundation (579913). DFT calculations were performed with the support of the National Natural Science Foundation of China (grants nos. 11774084 and U19A2090, to M.C.).

Author information

Authors and Affiliations



S.S. performed STM experiments, assisted by S.L. D.H. performed relaxation calculations. W.W. performed SHG experiments. M.C. performed DFT calculations. J.H., W.Y., D.N.B., X.Z. and A.N.P. provided advice. Data analysis and manuscript preparation were performed by S.S. and A.N.P. with input from all coauthors.

Corresponding author

Correspondence to Abhay N. Pasupathy.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Brian LeRoy and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary discussion.

Source data

Source Data Fig. 2

Source data of spectroscopy for different moiré wavelengths.

Source Data Fig. 3

Source data of the position of the band edges.

Source Data Fig. 4

Source data for the moiré potential as a function of moiré wavelength, strain tensor components and maximal shear strain as a function of moiré wavelength.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shabani, S., Halbertal, D., Wu, W. et al. Deep moiré potentials in twisted transition metal dichalcogenide bilayers. Nat. Phys. 17, 720–725 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing