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A monolayer transition-metal dichalcogenide as a topological excitonic insulator

Nature Nanotechnology volume 15, pages 367–372 (2020)Cite this article

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Abstract

Monolayer transition-metal dichalcogenides in the T′ phase could enable the realization of the quantum spin Hall effect1 at room temperature, because they exhibit a prominent spin–orbit gap between inverted bands in the bulk2,3. Here we show that the binding energy of electron–hole pairs excited through this gap is larger than the gap itself in the paradigmatic case of monolayer T′ MoS2, which we investigate from first principles using many-body perturbation theory4. This paradoxical result hints at the instability of the T′ phase in the presence of spontaneous generation of excitons, and we predict that it will give rise to a reconstructed ‘excitonic insulator’ ground state5,6,7. Importantly, we show that in this monolayer system, topological and excitonic order cooperatively enhance the bulk gap by breaking the crystal inversion symmetry, in contrast to the case of bilayers8,9,10,11,12,13,14,15,16 where the frustration between the two orders is relieved by breaking time reversal symmetry13,15,16. The excitonic topological insulator is distinct from the bare topological phase because it lifts the band spin degeneracy, which results in circular dichroism. A moderate biaxial strain applied to the system leads to two additional excitonic phases, different in their topological character but both ferroelectric17,18 as an effect of electron–electron interaction.

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Fig. 1: Electronic band structure of T′-MoS2.
Fig. 2: Exciton wavefunction from first principles.
Fig. 3: Topological versus excitonic order.
Fig. 4: Signatures of the topological excitonic insulator.
Fig. 5: Phase diagram versus strain and temperature.

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Data availability

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

Code availability

Many-body perturbation theory calculations were performed by means of the codes Yambo (http://www.yambo-code.org/) and Quantum ESPRESSO (http://www.quantum-espresso.org), which are both open source software. Results for the two-band model were obtained through custom Fortran codes that are available from M.R. upon reasonable request.

References

  1. Kane, C. L. & Mele, E. J. Quantum spin hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    Article  CAS  Google Scholar 

  2. Qian, X., Liu, J., Fu, L. & Li, J. Quantum spin Hall effect in two-dimensional transition metal dichalcogenides. Science 346, 1344–1347 (2014).

    Article  CAS  Google Scholar 

  3. Wu, S. et al. Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359, 76–79 (2018).

    Article  CAS  Google Scholar 

  4. Onida, G., Reining, L. & Rubio, A. Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev. Mod. Phys. 74, 601–659 (2002).

    Article  CAS  Google Scholar 

  5. Keldysh, L. V. & Kopaev, Y. V. Possible instability of the semimetallic state against Coulomb interaction. Sov. Phys. Sol. State 6, 2219 (1965); Fiz. Tverd. Tela 6, 2791 (1964).

  6. Jèrome, D., Rice, T. M. & Kohn, W. Excitonic insulator. Phys. Rev. 158, 462 (1967).

    Article  Google Scholar 

  7. Volkov, V. A. & Kopaev, Y. V. Theory of phase transitions in semiconductors of the A4B6 group. Sov. Phys. JETP 37, 1103–1108 (1974); Zh. Eksp. Teor. Fiz. 64, 2184–2915 (1973).

  8. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  CAS  Google Scholar 

  9. König, M. et al. Quantum spin hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    Article  CAS  Google Scholar 

  10. Liu, C., Hughes, T. L., Qi, X.-L., Wang, K. & Zhang, S.-C. Quantum spin Hall effect in inverted type-II semiconductors. Phys. Rev. Lett. 100, 236601 (2008).

    Article  CAS  Google Scholar 

  11. Knez, I., Du, R.-R. & Sullivan, G. Evidence for helical edge modes in inverted InAs/GaSb quantum wells. Phys. Rev. Lett. 107, 136603 (2011).

    Article  CAS  Google Scholar 

  12. Budich, J. C., Trauzettel, B. & Michetti, P. Time reversal symmetric topological exciton condensate in bilayer HgTe quantum wells. Phys. Rev. Lett. 112, 146405 (2014).

    Article  CAS  Google Scholar 

  13. Pikulin, D. I. & Hyart, T. Interplay of exciton condensation and the quantum spin Hall effect in InAs∕GaSb bilayers. Phys. Rev. Lett. 112, 176403 (2014).

    Article  CAS  Google Scholar 

  14. Du, L. et al. Evidence for a topological excitonic insulator in InAs/GaSb bilayers. Nature Commun. 8, 1971 (2017).

    Article  CAS  Google Scholar 

  15. Xue, F. & MacDonald, A. H. Time-reversal symmetry-breaking nematic insulators near quantum spin Hall phase transitions. Phys. Rev. Lett. 120, 186802 (2018).

    Article  CAS  Google Scholar 

  16. Zhu, Q., Tu, M. W.-Y., Tong, Q. & Yao, W. Gate tuning from exciton superfluid to quantum anomalous Hall in van der Waals heterobilayer. Sci. Adv. 5, eaau6120 (2019).

    Article  CAS  Google Scholar 

  17. Portengen, T., Östreich, T. & Sham, L. J. Theory of electronic ferroelectricity. Phys. Rev. B 54, 17452 (1996).

    Article  CAS  Google Scholar 

  18. Fei, Z. et al. Ferroelectric switching of a two-dimensional metal. Nature 560, 336–339 (2018).

    Article  CAS  Google Scholar 

  19. Jia, Z.-Y. et al.Direct visualization of a two-dimensional topological insulator in the single-layer \(1{T}^{\prime}-{\rm{WT}}{{\rm{e}}}_{2}\). Phys. Rev. B 96, 041108 (2017).

    Article  Google Scholar 

  20. Fei, Z. et al. Edge conduction in monolayer WTe2. Nat. Phys. 13, 677–682 (2017).

    Article  CAS  Google Scholar 

  21. Sajadi, E. et al. Gate-induced superconductivity in a monolayer topological insulator. Science 362, 922–925 (2018).

    Article  CAS  Google Scholar 

  22. Varsano, D. et al. Carbon nanotubes as excitonic insulators. Nat. Commun. 8, 1461 (2017).

    Article  CAS  Google Scholar 

  23. Li, J. I. A., Taniguchi, T., Watanabe, K., Hone, J. & Dean, C. R. Excitonic superfluid phase in double bilayer graphene. Nat. Phys. 13, 751–755 (2017).

    Article  CAS  Google Scholar 

  24. Liu, X., Watanabe, K., Taniguchi, T., Halperin, B. I. & Kim, P. Quantum Hall drag of exciton condensate in graphene. Nat. Phys. 13, 746–750 (2017).

    Article  CAS  Google Scholar 

  25. Kogar, A. et al. Signatures of exciton condensation in a transition metal dichalcogenide. Science 358, 1314–1317 (2017).

    Article  CAS  Google Scholar 

  26. Lu, Y. F. et al. Zero-gap semiconductor to excitonic insulator transition in Ta2 NiSe5. Nat. Commun. 8, 14408 (2017).

    Article  CAS  Google Scholar 

  27. Werdehausen, D. et al. Coherent order parameter oscillations in the ground state of the excitonic insulator Ta2NiSe5. Sci. Adv. 4, aap8652 (2018).

    Article  CAS  Google Scholar 

  28. Seradjeh, B., Moore, J. E. & Franz, M. Exciton condensation and charge fractionalization in a topological insulator film. Phys. Rev. Lett. 103, 066402 (2009).

    Article  CAS  Google Scholar 

  29. Hou, Y. et al. Emergence of excitonic superfluid at topological-insulator surfaces. Preprint at https://arxiv.org/abs/1810.10653 (2018).

  30. Wang, R., Erten, O., Wang, B. & Xing, D. Y. Prediction of a topological p + i p excitonic insulator with parity anomaly. Nat. Commun. 10, 210 (2019).

    Article  CAS  Google Scholar 

  31. Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Optical spectrum of MoS2: many-body effects and diversity of exciton states. Phys. Rev. Lett. 111, 216805 (2013).

    Article  CAS  Google Scholar 

  32. Berkelbach, T. C. & Reichman, D. R. Optical and excitonic properties of atomically thin transition-metal dichalcogenides. Ann. Rev. Condens. Matter Phys. 9, 379–396 (2018).

    Article  CAS  Google Scholar 

  33. Wang, G. et al. Excitons in atomically thin transition metal dichalcogenides. Rev. Mod. Phys. 90, 021001 (2018).

    Article  CAS  Google Scholar 

  34. Cao, T. et al. Valley-selective circular dichroism of monolayer molybdenum disulphide. Nat. Commun. 3, 1882 (2012).

    Google Scholar 

  35. Fatemi, V. et al. Electrically tunable low-density superconductivity in a monolayer topological insulator. Science 362, 926–929 (2018).

    Article  CAS  Google Scholar 

  36. Giannozzi, P. et al. Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).

    Article  Google Scholar 

  37. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).

    Article  CAS  Google Scholar 

  38. Marini, A., Hogan, C., Grüning, M. & Varsano, D. Yambo: an ab initio tool for excited state calculations. Comput. Phys. Commun. 180, 1392–1403 (2009).

    Article  CAS  Google Scholar 

  39. Sangalli, D. et al. Many-body perturbation theory calculations using the Yambo code. J. Phys. Condens. Matter 31, 325902 (2019).

    Article  CAS  Google Scholar 

  40. Godby, R. W. & Needs, R. J. Metal-insulator transition in Kohn-Sham theory and quasiparticle theory. Phys. Rev. Lett. 62, 1169–1172 (1989).

    Article  CAS  Google Scholar 

  41. Rozzi, C. A., Varsano, D., Marini, A., Gross, E. K. U. & Rubio, A. Exact Coulomb cutoff technique for supercell calculations. Phys. Rev. B 73, 205119 (2006).

    Article  CAS  Google Scholar 

  42. Kan, M. et al. Structures and phase transition of a MoS2 monolayer. J. Phys. Chem. C 118, 1515–1522 (2014).

    Article  CAS  Google Scholar 

  43. Luttinger, J. M. & Kohn, W. Motion of electrons and holes in perturbed periodic fields. Phys. Rev. 97, 869 (1955).

    Article  CAS  Google Scholar 

  44. Cudazzo, P., Tokatly, I. V. & Rubio, A. Dielectric screening in two-dimensional insulators: implications for excitonic and impurity states in graphane. Phys. Rev. B 84, 085406 (2011).

    Article  CAS  Google Scholar 

  45. Kane, C. L. & Mele, E. J. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    Article  CAS  Google Scholar 

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Acknowledgements

M.P. thanks G. Cicero for illuminating discussions in the early stages of this project and acknowledges Tor Vergata University for financial support through the mission sustainability project 2DUTOPI. This work was supported in part by the MaX (‘MAterials design at the eXascale’, www.max-centre.eu) European Centre of Excellence funded by the European Union H2020-INFRAEDI-2018-1 programme, grant number 824143, project 453. This work was also supported by MIUR-PRIN2017 number 2017BZPKSZ ‘Excitonic insulator in two-dimensional long-range interacting systems (EXC-INS)’. We acknowledge PRACE for awarding us access to the Marconi system based in Italy at CINECA.

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Authors and Affiliations

  1. CNR-NANO, Modena, Italy

    Daniele Varsano, Elisa Molinari & Massimo Rontani

  2. INFN, Dipartimento di Fisica, Università degli Studi di Roma Tor Vergata, Roma, Italy

    Maurizia Palummo

  3. Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, Modena, Italy

    Elisa Molinari

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  1. Daniele Varsano
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Contributions

D.V. and M.P. developed the first-principles many-body perturbation theory calculations and analysis; M.R. developed the self-consistent mean-field model and wrote the paper; and all authors initiated this project, contributed to the analysis of data and critically discussed the paper.

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Correspondence to Massimo Rontani.

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The authors declare that they have no competing financial interests.

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Peer review information Nature Nanotechnology thanks Onur Erten and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Screened Coulomb interaction.

Comparison between first-principles (black dots) and model (empty red circles) screened Coulomb interaction, W(q), shown as a function of qx, with qy = 0. The plotted data include form factors due to the wave function overlap between Bloch states: the long-wavelength value (q = 0) is extracted from Bloch states located at the Λ point.

Extended Data Fig. 2 Exciton wavefunction within the effective-mass approximation.

Contour plot of the exciton wavefunction square modulus in k space within the electron–hole centre-of-mass frame. This is the lowest-energy exciton responsible for the instability that drives the transition into the QSHX phase. The plot compares well with its first-principles counterpart (Fig. 2b).

Extended Data Fig. 3 Excitonic hybridization and band structure of the trivial ferroelectric excitonic insulator.

a,b, Contour plot in k space of the real (panel a) and imaginary (panel b) part of the self-consistent excitonic hybridization, ΔX(k). c,d, Energy bands along directions kx = 0 (panel c) and ky = Λ (panel d). Black and red curves label opposite spin projections. Since the two-fold screw-axis symmetry along y is broken, Kramers degeneracy is lifted throughout the Brillouin zone. Here T = 0 K and δ = –100 meV.

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Varsano, D., Palummo, M., Molinari, E. et al. A monolayer transition-metal dichalcogenide as a topological excitonic insulator. Nat. Nanotechnol. 15, 367–372 (2020). https://doi.org/10.1038/s41565-020-0650-4

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