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Disorder-induced multifractal superconductivity in monolayer niobium dichalcogenides


The interplay between disorder and superconductivity is a subtle and fascinating phenomenon in quantum many-body physics. Conventional superconductors are insensitive to dilute non-magnetic impurities, known as Anderson’s theorem1. Destruction of superconductivity and even superconductor–insulator transitions2,3,4,5,6,7,8,9,10 occur in the regime of strong disorder. Hence, disorder-enhanced superconductivity is rare and has been observed only in some alloys or granular states11,12,13,14,15,16,17. Owing to the entanglement of various effects, the mechanism of enhancement is still under debate. Here, we report a well-controlled disorder effect in the recently discovered monolayer NbSe2 superconductor. The superconducting transition temperatures of NbSe2 monolayers are substantially increased by disorder. Realistic theoretical modelling shows that the unusual enhancement possibly arises from the multifractality18,19 of electron wavefunctions. This work provides experimental evidence of the multifractal superconducting state.

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Fig. 1: Epitaxial NbSe2 monolayer.
Fig. 2: Disorder-enhanced superconductivity.
Fig. 3: Theoretical modelling.
Fig. 4: Spatially resolved spectra near the Fermi level.

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All relevant data are available from the corresponding authors on reasonable request.

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All relevant codes or algorithms are available from the corresponding authors on reasonable request.


  1. 1.

    Anderson, P. W. Theory of dirty superconductors. J. Phys. Chem. Solids 11, 26–30 (1959).

    ADS  Article  Google Scholar 

  2. 2.

    Strongin, M., Thompson, R. S., Kammerer, O. F. & Crow, J. E. Destruction of superconductivity in disordered near-monolayer films. Phys. Rev. B 1, 1078 (1970).

    ADS  Article  Google Scholar 

  3. 3.

    Imry, Y. & Strongin, M. Destruction of superconductivity in granular and highly disordered metals. Phys. Rev. B 24, 6353 (1981).

    ADS  Article  Google Scholar 

  4. 4.

    Graybeal, J. M. & Beasley, M. R. Localization and interaction effects in ultrathin amorphous superconducting films. Phys. Rev. B 29, 4167 (1984).

    ADS  Article  Google Scholar 

  5. 5.

    Maekawa, S. & Fukuyama, H. Localization effects in two-dimensional superconductors. J. Phys. Soc. Jpn 51, 1380–1385 (1982).

    ADS  Article  Google Scholar 

  6. 6.

    Maekawa, S., Ebisawa, H. & Fukuyama, H. Theory of dirty superconductors in weakly localized regime. J. Phys. Soc. Jpn 53, 2681–2687 (1984).

    ADS  Article  Google Scholar 

  7. 7.

    Ma, M., Halperin, B. I. & Lee, P. A. Strongly disordered superfluids: quantum fluctuations and critical behavior. Phys. Rev. B 34, 3136 (1986).

    ADS  Article  Google Scholar 

  8. 8.

    Haviland, D. B., Liu, Y. & Goldman, A. M. Onset of superconductivity in the two-dimensional limit. Phys. Rev. Lett. 62, 2180 (1989).

    ADS  Article  Google Scholar 

  9. 9.

    Hsu, J. W. P., Park, S. I., Deutscher, G. & Kapitulnik, A. Superconducting transition temperature in a Nb/NbxSi1−x bilayer system. Phys. Rev. B 43, 2648 (1991).

    ADS  Article  Google Scholar 

  10. 10.

    Jisrawi, N. M. et al. Reversible depression in the T c of thin Nb films due to enhanced hydrogen adsorption. Phys. Rev. B 58, 6585 (1998).

    ADS  Article  Google Scholar 

  11. 11.

    Kammerer, O. F. & Strongin, M. Superconductivity in tungsten films. Phys. Lett. 17, 224 (1965).

    ADS  Article  Google Scholar 

  12. 12.

    Abeles, B., Cohen, R. W. & Cullen, G. W. Enhancement of superconductivity in metal films. Phys. Rev. Lett. 17, 632 (1966).

    ADS  Article  Google Scholar 

  13. 13.

    Naugle, D. G. The effect of very thin Ge coating on the superconducting transition of thin Sn and Tl films. Phys. Lett. A 25, 688 (1967).

    ADS  Article  Google Scholar 

  14. 14.

    Garland, J. W., Bennemann, K. H. & Mueller, F. M. Effect of lattice disorder on the superconducting transition temperature. Phys. Rev. Lett. 21, 1315 (1968).

    ADS  Article  Google Scholar 

  15. 15.

    Tsuei, C. C. & Johnson, W. L. Superconductivity in metal-semiconductor eutectic alloys. Phys. Rev. B 9, 4742 (1974).

    ADS  Article  Google Scholar 

  16. 16.

    Parashar, R. S. & Srivastava, V. Superconducting T c enhancement in weakly disordered Ge-covered tin films. Phys. Rev. B 32, 6048 (1985).

    ADS  Article  Google Scholar 

  17. 17.

    Osofsky, M. S. et al. New insight into enhanced superconductivity in metals near the metal-insulator transition. Phys. Rev. Lett. 87, 197004 (2001).

    ADS  Article  Google Scholar 

  18. 18.

    Feigel’man, M. V., Ioffe, L. B., Kravtsov, V. E. & Yuzbashyan, E. A. Eigenfunction fractality and pseudogap state near the superconductor–insulator transition. Phys. Rev. Lett. 98, 027001 (2007).

    ADS  Article  Google Scholar 

  19. 19.

    Burmistrov, I. S., Gornyi, I. V. & Mirlin, A. D. Superconductor–insulator transitions: phase diagram and magnetoresistance. Phys. Rev. B 92, 014506 (2015).

    ADS  Article  Google Scholar 

  20. 20.

    Ugeda, M. M. et al. Characterization of collective ground states in single-layer NbSe2. Nat. Phys. 12, 92–97 (2016).

    Article  Google Scholar 

  21. 21.

    Tsen, A. W. et al. Nature of the quantum metal in a two-dimensional crystalline superconductor. Nat. Phys. 12, 208–212 (2016).

    Article  Google Scholar 

  22. 22.

    Xi, X. X., Berger, H., Forró, L., Shan, J. & Mak, K. F. Gate tuning of electronic phase transitions in two-dimensional NbSe2. Phys. Rev. Lett. 117, 106801 (2016).

    ADS  Article  Google Scholar 

  23. 23.

    Xi, X. X. et al. Ising pairing in superconducting NbSe2 atomic layers. Nat. Phys. 12, 139–143 (2016).

    Article  Google Scholar 

  24. 24.

    Zhou, B. T., Yuan, N. F. Q., Jiang, H. L. & Law, K. T. Ising superconductivity and Majorana fermions in transition-metal dichalcogenides. Phys. Rev. B 93, 180501 (2016).

    ADS  Article  Google Scholar 

  25. 25.

    Ma, M. & Lee, P. A. Localized superconductors. Phys. Rev. B 32, 5658 (1985).

    ADS  Article  Google Scholar 

  26. 26.

    Evers, F. & Mirlin, A. D. Anderson transition. Rev. Mod. Phys. 80, 1355 (2008).

    ADS  Article  Google Scholar 

  27. 27.

    Chalker, J. T. & Daniell, G. J. Scaling, diffusion, and the integer quantized Hall effect. Phys. Rev. Lett. 61, 593 (1988).

    ADS  Article  Google Scholar 

  28. 28.

    Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958).

    ADS  Article  Google Scholar 

  29. 29.

    Abrahams, E., Anderson, P. W., Licciardello, D. C. & Ramakrishnan, T. V. Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673 (1979).

    ADS  Article  Google Scholar 

  30. 30.

    Cuevas, E. & Kravtsov, V. E. Two-eigenfunction correlation in a multifractal metal and insulator. Phys. Rev. B 76, 235119 (2007).

    ADS  Article  Google Scholar 

  31. 31.

    Guillamón, I. et al. Superconducting density of states and vortex cores of 2H-NbS2. Phys. Rev. Lett. 101, 166407 (2008).

    ADS  Article  Google Scholar 

  32. 32.

    Staley, N. E. et al. Electric field effect on superconductivity in atomically thin flakes of NbSe2. Phys. Rev. B 80, 184505 (2009).

    ADS  Article  Google Scholar 

  33. 33.

    Beasley, M. R., Mooij, J. E. & Orlando, T. P. Possibility of vortex–antivortex pair dissociation in two-dimensional superconductors. Phys. Rev. Lett. 42, 1165 (1979).

    ADS  Article  Google Scholar 

  34. 34.

    Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low Temp. Phys. 36, 599–616 (1979).

    ADS  Article  Google Scholar 

  35. 35.

    Fiory, A. T., Hebard, A. F. & Glaberson, W. I. Superconducting phase transitions in indium/indium-oxide thin-film composites. Phys. Rev. B 28, 5075 (1983).

    ADS  Article  Google Scholar 

  36. 36.

    Kadin, A. M., Epstein, K. & Goldman, A. M. Renormalization and the Kosterlitz–Thouless transition in a two-dimensional superconductor. Phys. Rev. B 27, 6691 (1983).

    ADS  Article  Google Scholar 

  37. 37.

    Hsu, J. W. P. & Kapitulnik, A. Superconducting transition, fluctuation, and vortex motion in a two-dimensional single-crystal Nb film. Phys. Rev. B 45, 4819 (1992).

    ADS  Article  Google Scholar 

  38. 38.

    Benfatto, L., Castellani, C. & Giamarchi, T. Broadening of the Berezinskii–Kosterlitz–Thouless superconducting transition by inhomogeneity and finite-size effects. Phys. Rev. B 80, 214506 (2009).

    ADS  Article  Google Scholar 

  39. 39.

    König, E. J. et al. Berezinskii–Kosterlitz–Thouless transition in homogeneously disordered superconducting films. Phys. Rev. B 92, 214503 (2015).

    ADS  Article  Google Scholar 

  40. 40.

    Castellani, C. & Peliti, L. Multifractal wavefunction at the localization threshold. J. Phys. A 19, L429 (1986).

    ADS  Article  Google Scholar 

  41. 41.

    Mayoh, J. & García-García, A. M. Global critical temperature in disordered superconductors with weak multifractality. Phys. Rev. B 92, 174526 (2015).

    ADS  Article  Google Scholar 

  42. 42.

    Richardella, A. et al. Visualizing critical correlations near the metal-insulator transition in Ga1–xMnxAs. Science 327, 665–669 (2010).

    ADS  Article  Google Scholar 

  43. 43.

    Sacépé, B. et al. Localization of performed Cooper pairs in disordered superconductors. Nat. Phys. 7, 239–244 (2011).

    Article  Google Scholar 

  44. 44.

    Sacépé, B. et al. Disorder-induced inhomogeneities of the superconducting state close to the superconductor–insulator transition. Phys. Rev. Lett. 101, 157006 (2008).

    ADS  Article  Google Scholar 

  45. 45.

    Ramakrishnan, T. V. Superconductivity in disordered thin films. Phys. Scr. T27, 24–30 (1989).

    ADS  Article  Google Scholar 

  46. 46.

    Feigel’man, M. V., Ioffe, L. B., Kravtsov, V. E. & Cuevas, E. Fractal superconductivity near localization threshold. Ann. Phys. 325, 1390–1478 (2010).

    ADS  Article  Google Scholar 

  47. 47.

    Chhabra, A. & Jensen, R. V. Direct determination of the f(α) singularity spectrum. Phys. Rev. Lett. 62, 1327 (1989).

    ADS  MathSciNet  Article  Google Scholar 

  48. 48.

    Chang, J. et al. Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67. Nat. Phys. 8, 871–876 (2012).

    Article  Google Scholar 

  49. 49.

    Wagner, K. E. et al. Tuning the charge density wave and superconductivity in CuxTaS2. Phys. Rev. B 78, 104520 (2008).

    ADS  Article  Google Scholar 

  50. 50.

    Sugawara, K., Yokota, K., Takemoto, J., Tanokura, Y. & Sekine, T. Anderson localization and layered superconductor 2H-NbSe2–xSx. J. Low Temp. Phys. 91, 39–47 (1993).

    ADS  Article  Google Scholar 

  51. 51.

    Straub, Th et al. Charge-density-wave mechanism in 2H-NbSe2: photoemission results. Phys. Rev. Lett. 82, 4504 (1999).

    ADS  Article  Google Scholar 

  52. 52.

    Rossnagel, K. et al. Fermi surface of 2H-NbSe2 and its implications on the charge-density-wave mechanism. Phys. Rev. B 64, 235119 (2001).

    ADS  Article  Google Scholar 

  53. 53.

    Borisenko, S. V. et al. Two energy gaps and Fermi-surface “arcs” in NbSe2. Phys. Rev. Lett. 102, 166402 (2009).

    ADS  Article  Google Scholar 

  54. 54.

    Feng, Y. J. et al. Order parameter fluctuations at a buried quantum critical point. Proc. Natl Acad. Sci. USA 109, 7224–7229 (2012).

    ADS  Article  Google Scholar 

  55. 55.

    Rubio-Verdú, C. et al. Multifractal superconductivity in a two-dimensional transition metal dichalcogenide. Preprint at (2018).

  56. 56.

    Berger, C. et al. Ultrathin epitaxial graphite: 2D electron gas properties and a route toward graphene-based nanoelectronics. J. Phys. Chem. B 108, 19912–19916 (2004).

    Article  Google Scholar 

  57. 57.

    Horcas, I. et al. WSXM: a software for scanning probe microscopy and a tool for nanotechnology. Rev. Sci. Instrum. 78, 013705 (2007).

    ADS  Article  Google Scholar 

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We thank A. M. García-García for stimulating discussions. This work is supported by the Ministry of Science and Technology of China (grant nos. 2018YFA0305600, 2016YFA0301002, 2017YFA0303302, 2013CB934600), the National Natural Science Foundation of China (grant nos. 51561145005, 11622433, 11574175, 51522212, 11774008, 11704414). K.T.L. would like to acknowledge the support of HKRGC (grants 16324216, 16309718, 6307117 and C6026-16W), Croucher Foundation and Dr. Tai-Chin Lo Foundation. L.G. is partially supported by Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDB07030200). M.S.B. and N.N. gratefully acknowledge support from the CREST, JST (no. JPMJCR16F1). N.N. is also supported by JSPS KAKENHI grant numbers 18H03676 and 26103006. M.S.B. is also supported by the Japan Society for Promotion of Science (Grant-in-Aid for Scientific Research (S) no. 24224009).

Author information




S.-H.J. and X.C. coordinated the project and designed the experiments. K.Z., H.L. and W.H. performed the molecular beam epitaxy growth and STM experiments. X.X. and K.T.L. provided the multifractal interpretation and the model calculations. W.Y., M.Y. and S.Z. contributed the ARPES measurement. Q.Z. and L.G. contributed the STEM characterization. M.S.B. calculated the electronic band by density functional theory. Z.-X.L., S.H., H.Y. and N.N. contributed part of the theoretical analysis. K.Z., H.L., X.X., K.T.L., S.-H.J. and X.C. co-wrote the paper. All authors discussed the results and commented on the manuscript.

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Correspondence to Xi Chen or Shuai-Hua Ji.

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Zhao, K., Lin, H., Xiao, X. et al. Disorder-induced multifractal superconductivity in monolayer niobium dichalcogenides. Nat. Phys. 15, 904–910 (2019).

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