Nature of the quantum metal in a two-dimensional crystalline superconductor

Journal name:
Nature Physics
Year published:
Published online

Two-dimensional (2D) materials are not expected to be metals at low temperature owing to electron localization1. Consistent with this, pioneering studies on thin films reported only superconducting and insulating ground states, with a direct transition between the two as a function of disorder or magnetic field2, 3, 4, 5, 6. However, more recent works have revealed the presence of an intermediate quantum metallic state occupying a substantial region of the phase diagram7, 8, 9, 10, whose nature is intensely debated11, 12, 13, 14, 15, 16, 17. Here, we observe such a state in the disorder-free limit of a crystalline 2D superconductor, produced by mechanical co-lamination of NbSe2 in an inert atmosphere. Under a small perpendicular magnetic field, we induce a transition from superconductor to the quantum metal. We find a unique power-law scaling with field in this phase, which is consistent with the Bose-metal model where metallic behaviour arises from strong phase fluctuations caused by the magnetic field11, 12, 13, 14.

At a glance


  1. Environmentally controlled device fabrication.
    Figure 1: Environmentally controlled device fabrication.

    a, Schematic of the heterostructure assembly process. Boron nitride (BN)/graphite (G) on a polymer stamp (PDMS) is used to electrically contact and encapsulate NbSe2 in an inert atmosphere. The heterostructure is lithographically patterned and the edge of graphite is metallized with Cr/Pd/Au. b, Optical images of the heterostructure before (left) and after (right) device fabrication. In the (false-coloured) left panel, the bilayer NbSe2 is outlined with a dashed green line and the overlap between the graphite and bilayer NbSe2 is shaded black. Scale bar is 5μm in both images.

  2. Characterization of bilayer NbSe2 device.
    Figure 2: Characterization of bilayer NbSe2 device.

    a, Sheet resistance as a function of temperature, showing a superconducting transition at Tc = 5.26K. Temperature-dependent critical magnetic fields parallel and perpendicular to the layers are shown in the inset. The black line is a linear fit to Hc2 1 − T/Tc at high temperatures. The red line is a fit to , the Tinkham formula for 2D samples28. b, Voltage–current behaviour at different temperatures. Inset shows the exponent a versus T extracted from power-law fitting V ~ Ia near the normal state transition. a = 3 at the BKT temperature 5.01K.

  3. Magnetic-field-tuned phase transitions in 2D NbSe2.
    Figure 3: Magnetic-field-tuned phase transitions in 2D NbSe2.

    a, 2D colour plot of sheet resistance versus temperature and perpendicular magnetic field. The black line marks Hc2(T). b, Arrhenius plot of resistance for several magnetic fields shows the thermally activated regime (black lines) and saturation at low temperatures (coloured lines). c, Energy barrier versus magnetic field extracted from a linear fit to the activated region. The solid red line is an empirical fit to the formula given as an inset.

  4. Emergence of the quantum metal.
    Figure 4: Emergence of the quantum metal.

    a, Magnetoresistance below the superconducting transition for different temperatures. The data scale to a power law R ~ (HHc0)α(T) and collapse onto a single curve in the quantum metallic phase below 1K. α versus T is plotted in the inset. The grey line is a guide-to-eye linear scaling (high T) and the red line is an empirical fit to Bose-metal scaling (low T). b, Full HT phase diagram of the bilayer NbSe2 device. The red circles are Hc2(T). The purple squares, dividing the Bose metal from the thermally assisted flux flow (TAFF) regime, mark the transition from activated behaviour R ~ exp(U(H)/T) to temperature-independent resistance R = R(H), that is, the intersection of the black and coloured lines in Fig. 3b. The blue triangles denote the boundary of the superconducting phase Hc0(T). This criterion is determined by when hysteresis vanishes in VI measurements (see Supplementary Fig. 4).


  1. 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, 673676 (1979).
  2. Goldman, A. M. & Markovic, N. Superconductor–insulator transitions in the two-dimensional limit. Phys. Today 51, 3944 (1998).
  3. Haviland, D. B., Liu, Y. & Goldman, A. M. Onset of superconductivity in the two-dimensional limit. Phys. Rev. Lett. 62, 21802183 (1989).
  4. Fisher, M. P. A. Quantum phase transitions in disordered two-dimensional superconductors. Phys. Rev. Lett. 65, 923926 (1990).
  5. Hebard, A. F. & Paalanen, M. A. Magnetic-field-tuned superconductor–insulator transition in two-dimensional films. Phys. Rev. Lett. 65, 927930 (1990).
  6. Yazdani, A. & Kapitulnik, A. Superconducting–insulating transition in two-dimensional α-MoGe thin films. Phys. Rev. Lett. 74, 30373040 (1995).
  7. Ephron, D., Yazdani, A., Kapitulnik, A. & Beasley, M. R. Observation of quantum dissipation in the vortex state of a highly disordered superconducting thin film. Phys. Rev. Lett. 76, 15291532 (1996).
  8. Christiansen, C., Hernandez, L. M. & Goldman, A. M. Evidence of collective charge behavior in the insulating state of ultrathin films of superconducting metals. Phys. Rev. Lett. 88, 037004 (2002).
  9. Qin, Y. G., Vicente, C. L. & Yoon, J. Magnetically induced metallic phase in superconducting tantalum films. Phys. Rev. B 73, 100505 (2006).
  10. Steiner, M. A., Breznay, N. P. & Kapitulnik, A. Approach to a superconductor-to-Bose-insulator transition in disordered films. Phys. Rev. B 77, 212501 (2008).
  11. Das, D. & Doniach, S. Existence of a Bose metal at T = 0. Phys. Rev. B 60, 12611275 (1999).
  12. Das, D. & Doniach, S. Bose metal: Gauge-field fluctuations and scaling for field-tuned quantum phase transitions. Phys. Rev. B 64, 134511 (2001).
  13. Dalidovich, D. & Phillips, P. Phase glass is a Bose metal: A new conducting state in two dimensions. Phys. Rev. Lett. 89, 027001 (2002).
  14. Phillips, P. & Dalidovich, D. The elusive Bose metal. Science 302, 243247 (2003).
  15. Shimshoni, E., Auerbach, A. & Kapitulnik, A. Transport through quantum melts. Phys. Rev. Lett. 80, 33523355 (1998).
  16. Spivak, B., Zyuzin, A. & Hruska, M. Quantum superconductor–metal transition. Phys. Rev. B 64, 132502 (2001).
  17. Galitski, V. M., Refael, G., Fisher, M. P. A. & Senthil, T. Vortices and quasiparticles near the superconductor–insulator transition in thin films. Phys. Rev. Lett. 95, 077002 (2005).
  18. Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183191 (2007).
  19. Le, L. P. et al. Magnetic penetration depth in layered compound NbSe2 measured by muon spin relaxation. Physica C 185, 27152716 (1991).
  20. Soto, F. et al. Electric and magnetic characterization of NbSe2 single crystals: Anisotropic superconducting fluctuations above Tc. Physica C 460, 789790 (2007).
  21. Staley, N. E. et al. Electric field effect on superconductivity in atomically thin flakes of NbSe2. Phys. Rev. B 80, 184505 (2009).
  22. El-Bana, M. S. et al. Superconductivity in two-dimensional NbSe2 field effect transistors. Supercond. Sci. Technol. 26, 125020 (2013).
  23. Tsen, A. W. et al. Structure and control of charge density waves in two-dimensional 1T-TaS2. Proc. Natl Acad. Sci. USA (in the press).
  24. Cao, Y. et al. Quality heterostructures from two-dimensional crystals unstable in air by their assembly in inert atmosphere. Nano Lett. 15, 49144921 (2015).
  25. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614617 (2013).
  26. Cui, X. et al. Multi-terminal transport measurements of MoS2 using a van der Waals heterostructure device platform. Nature Nanotech. 10, 534540 (2015).
  27. Tinkham, M. Introduction to Superconductivity 2nd edn (Dover, 1996).
  28. Kim, M., Kozuka, Y., Bell, C., Hikita, Y. & Hwang, H. Y. Intrinsic spin–orbit coupling in superconducting δ-doped SrTiO3 heterostructures. Phys. Rev. B 86, 085121 (2012).
  29. Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low Temp. Phys. 36, 599616 (1979).
  30. Eley, S., Gopalakrishnan, S., Goldbart, P. M. & Mason, N. Approaching zero-temperature metallic states in mesoscopic superconductor–normal-superconductor arrays. Nature Phys. 8, 5962 (2012).
  31. Beasley, M. R., Mooij, J. E. & Orlando, T. P. Possibility of vortex–antivortex pair dissociation in two-dimensional superconductors. Phys. Rev. Lett. 42, 11651168 (1979).
  32. Feigelman, M. V., Geshkenbein, V. B. & Larkin, A. I. Pinning and creep in layered superconductors. Physica C 167, 177187 (1990).
  33. Mason, N. & Kapitulnik, A. True superconductivity in a two-dimensional superconducting–insulating system. Phys. Rev. B 64, 060504 (2001).
  34. Li, Y., Vicente, C. L. & Yoon, J. Transport phase diagram for superconducting thin films of tantalum with homogeneous disorder. Phys. Rev. B 81, 020505 (2010).

Download references

Author information

  1. Present address: Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA.

    • B. Hunt


  1. Department of Physics, Columbia University, New York, New York 10027, USA

    • A. W. Tsen,
    • B. Hunt,
    • C. R. Dean &
    • A. N. Pasupathy
  2. Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA

    • Y. D. Kim &
    • J. Hone
  3. International Center for Quantum Materials, Peking University, Beijing 100871, China

    • Z. J. Yuan &
    • S. Jia
  4. Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

    • S. Jia
  5. Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

    • R. J. Cava
  6. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • P. Kim


A.W.T., B.H., C.R.D. and A.N.P. conceived and designed the experiment; Z.J.Y. and S.J. synthesized the NbSe2 crystals; A.W.T. fabricated the devices with assistance from Y.D.K.; A.W.T. and B.H. performed the transport measurements; A.W.T., B.H., C.R.D. and A.N.P. analysed the data and wrote the paper. R.J.C., J.H., P.K., C.R.D. and A.N.P. advised.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary information (416,893 KB)

    Supplementary information

Additional data