Holstein polaron in a valley-degenerate two-dimensional semiconductor

Abstract

Two-dimensional (2D) crystals have emerged as a class of materials with tunable carrier density1. Carrier doping to 2D semiconductors can be used to modulate many-body interactions2 and to explore novel composite particles. The Holstein polaron is a small composite particle of an electron that carries a cloud of self-induced lattice deformation (or phonons)3,4,5, which has been proposed to play a key role in high-temperature superconductivity6 and carrier mobility in devices7. Here we report the discovery of Holstein polarons in a surface-doped layered semiconductor, MoS2, in which a puzzling 2D superconducting dome with the critical temperature of 12 K was found recently8,9,10,11. Using a high-resolution band mapping of charge carriers, we found strong band renormalizations collectively identified as a hitherto unobserved spectral function of Holstein polarons12,13,14,15,16,17,18. The short-range nature of electron–phonon (e–ph) coupling in MoS2 can be explained by its valley degeneracy, which enables strong intervalley coupling mediated by acoustic phonons. The coupling strength is found to increase gradually along the superconducting dome up to the intermediate regime, which suggests a bipolaronic pairing in the 2D superconductivity.

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Fig. 1: Holstein polaron of MoS2.
Fig. 2: Spectral function of Holstein polarons.
Fig. 3: Doping dependence of polarons.
Fig. 4: Strength of e–ph coupling and superconductivity.

References

  1. 1.

    Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).

  2. 2.

    Li, L. J. et al. Controlling many-body states by the electric-field effect in a two-dimensional material. Nature 529, 185–189 (2016).

  3. 3.

    Landau, L. D. On the motion of electrons in a crystal lattice. Phys. Z. Sowjet. 3, 664–665 (1933).

  4. 4.

    Fröhlich, H. Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954).

  5. 5.

    Holstein, T. Studies on polaron motion. Ann. Phys. 8, 343–389 (1959).

  6. 6.

    Alexandrov, A. S. Polarons in Advanced Materials (Springer, Dordrecht, 2007).

  7. 7.

    Hulea, I. N. et al. Tunable Fröhlich polarons in organic single-crystal transistors. Nat. Mater. 5, 982–986 (2006).

  8. 8.

    Ye, J. T. et al. Superconducting dome in a gate-tuned band insulator. Science 338, 1193–1196 (2012).

  9. 9.

    Lu, J. M. et al. Evidence for two-dimensional Ising superconductivity in gated MoS2. Science 350, 1353–1357 (2015).

  10. 10.

    Saito, Y. et al. Superconductivity protected by spin-valley locking in ion-gated MoS2. Nat. Phys. 12, 144–149 (2016).

  11. 11.

    Costanzo, D., Jo, S., Berger, H. & Morpurgo, A. F. Gate-induced superconductivity in atomically thin MoS2 crystals. Nat. Nanotech. 11, 339–344 (2016).

  12. 12.

    Engelsberg, S. & Schrieffer, J. R. Coupled electron–phonon system. Phys. Rev. 131, 993–1008 (1963).

  13. 13.

    Meden, V., Schönhammer, K. & Gunnarsson, O. Electron-phonon interaction in one dimension: exact spectral properties. Phys. Rev. B 50, 11179–11182 (1994).

  14. 14.

    Wellein, G. & Fehske, H. Polaron band formation in the Holstein model. Phys. Rev. B 56, 4513–4517 (1997).

  15. 15.

    Bonča, J., Trugman, S. A. & Batistić, I. Holstein polaron. Phys. Rev. B 60, 1633–1642 (1999).

  16. 16.

    Hohenadler, M., Aichhorn, M. & von der Linden, W. Spectral function of electron-phonon models by cluster perturbation theory. Phys. Rev. B 68, 184304 (2003).

  17. 17.

    Goodvin, G. L., Berciu, M. & Sawatzky, G. A. Green’s function of the Holstein polaron. Phys. Rev. B 74, 245104 (2006).

  18. 18.

    Berciu, M. & Sawatzky, G. A. Light polarons and bipolarons for a highly inhomogeneous electron–boson coupling. Eur. Phys. Lett. 81, 57008 (2008).

  19. 19.

    Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).

  20. 20.

    Perfetti, L. et al. Mobile small polarons and the Peierls transition in the quasi-one-dimensional conductor K0.3MoO3. Phys. Rev. B 66, 075107 (2002).

  21. 21.

    Shen, K. M. et al. Angle-resolved photoemission studies of lattice polaron formation in the cuprate Ca2CuO2Cl2. Phys. Rev. B 75, 075115 (2007).

  22. 22.

    Moser, S. et al. Tunable polaronic conduction in anatase TiO2. Phys. Rev. Lett. 110, 196403 (2013).

  23. 23.

    Lee, J. J. et al. Interfacial mode coupling as the origin of the enhancement of T c in FeSe films on SrTiO3. Nature 515, 245–248 (2014).

  24. 24.

    Chen, C., Avila, J., Frantzeskakis, E., Levy, A. & Asensio, M. C. Observation of a two-dimensional liquid of Fröhlich polarons at the bare SrTiO3 surface. Nat. Commun. 6, 8585 (2015).

  25. 25.

    Wang, Z. et al. Tailoring the nature and strength of electron-phonon interactions in the SrTiO3(001) 2D electron liquid. Nat. Mater. 15, 835–839 (2016).

  26. 26.

    Koschorreck, M. et al. Attractive and repulsive Fermi polarons in two dimensions. Nature 485, 619–622 (2012).

  27. 27.

    Zeng, H., Dai, J., Yao, W., Xiao, D. & Cui, X. Valley polarization in MoS2 monolayers by optical pumping. Nat. Nanotech. 7, 490–493 (2012).

  28. 28.

    Mak, K. F., He, K., Shan, J. & Heinz, T. F. Control of valley polarization in monolayer MoS2 by optical helicity. Nat. Nanotech. 7, 494–498 (2012).

  29. 29.

    Kaasbjerg, K., Thygesen, K. S. & Jacobsen, K. W. Phonon-limited mobility in n-type single-layer MoS2 from first principles. Phys. Rev. B 85, 115317 (2012).

  30. 30.

    Yu, P. Y. & Cardona, M. Fundamentals of Semiconductors 4th edn (Springer, Heidelberg, 2010).

  31. 31.

    Eknapakul, T. et al. Electronic structure of a quasi-freestanding MoS2 monolayer. Nano Lett. 14, 1312 (2014).

  32. 32.

    Zhang, Y. et al. Direct observation of the transition from indirect to direct bandgap in atomically thin epitaxial MoSe2. Nat. Nanotech. 9, 111–115 (2014).

  33. 33.

    Riley, J. M. et al. Negative electronic compressibility and tunable spin splitting in WSe2. Nat. Nanotech. 10, 1043–1047 (2015).

  34. 34.

    Kang, M. et al. Universal mechanism of band-gap engineering in transition-metal dichalcogenides. Nano Lett. 17, 1610–1615 (2017).

  35. 35.

    Brumme, T., Calandra, M. & Mauri, F. First-principles theory of field-effect doping in transition-metal dichalcogenides: structural properties, electronic structure, Hall coefficient, and electrical conductivity. Phys. Rev. B 91, 155436 (2015).

  36. 36.

    Hohenadler, M. et al. Photoemission spectra of many-polaron systems. Phys. Rev. B 71, 245111 (2005).

  37. 37.

    Verdi, C., Caruso, F. & Giustino, F. Origin of crossover from polarons to Fermi liquids in transition metal oxides. Nat. Commun. 8, 15769 (2017).

  38. 38.

    Covaci, L. & Berciu, M. Holstein polaron: the effect of coupling to multiple-phonon modes. Eur. Phys. Lett. 80, 67001 (2007).

  39. 39.

    Mishchenko, A. S., Nagaosa, N. & Prokof’ev, N. Diagrammatic Monte Carlo method for many-polaron problems. Phys. Rev. Lett. 113, 166402 (2014).

  40. 40.

    Covaci, L. & Berciu, M. Polaron formation in the presence of Rashba spin–orbit coupling: implications for spintronics. Phys. Rev. Lett. 102, 186403 (2009).

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Acknowledgements

This work was supported by the National Research Foundation of Korea (grants no. 2017R1A5A1014862 and no. 2017R1A2B3011368), Future-leading Research Initiative 2017-22-0059 of Yonsei University, and the POSCO Science Fellowship of POSCO TJ Park Foundation. This work was carried out with the support of the Diamond Light Source (beamline I05). The work at the Advanced Light Source was supported by the US Department of Energy, Office of Sciences under contract no. DE-AC02-05CH11231. M.K. acknowledges the Samsung Scholarship from Samsung Foundation of Culture. We thank A. Bostwick, C. Jozwiak and E. Rotenberg for help in the ARPES experiments, and R. Comin for helpful discussions.

Author information

M.K. conducted experiments and analysed data with help from W.J.S., Y.S., S.H.R., T.K.K. and M.H. S.W.J. performed spectral-function simulations. K.S.K. directed the project. M.K. and K.S.K wrote the manuscript with contributions from all the other co-authors.

Correspondence to Keun Su Kim.

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Kang, M., Jung, S.W., Shin, W.J. et al. Holstein polaron in a valley-degenerate two-dimensional semiconductor. Nature Mater 17, 676–680 (2018). https://doi.org/10.1038/s41563-018-0092-7

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