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.
Global superconductivity emerges in a sample when conduction electrons form Cooper pairs and condense into a macroscopic, phase-coherent quantum state. In two dimensions, the phase coherence can be disrupted even at zero temperature by increasing disorder—either by degrading sample quality or by applying magnetic fields to create vortices2. Granular or amorphous superconducting thin films, for which disorder levels can be varied during growth, have thus provided an established platform for the study of quantum phase transitions in 2D superconductors. Within the conventional theoretical framework, increasing sample disorder or magnetic field perpendicular to a strongly disordered film at T = 0 induces a direct transition to an insulating state as the normal state sheet resistance approaches the pair quantum resistance h/(2e)2 = 6.4 kΩ (refs 2,4). As film quality improved over time, however, an intervening metallic phase with resistance much lower than the normal state resistance has been observed in several systems with generally less disorder7,8,9,10. Its origin is not well understood, and the various theoretical treatments can be divided between bosonic-based models, in which Cooper pairing persists in the metallic phase but phase coherence is lost11,12,13,14, and models that also incorporate other fermionic degrees of freedom15,16,17.
Recently, mechanical exfoliation has emerged as a technique to produce ultraclean, crystalline 2D materials, with graphene being a well-known example18. Like amorphous films, the thickness of these samples can be easily controlled down to the level of individual atomic layers. In contrast to amorphous films, a 2D superconductor exfoliated from a layered, single crystal, can exist in the regime of minimal disorder, allowing new insight into the nature of the vortex state in two dimensions. In this work, we realize such a system using a clean bilayer of NbSe2, a well-known type-II superconductor with Tc ∼ 7 K in bulk form19,20. Unique to this sample, the normal state sheet resistance is two orders of magnitude below h/(2e)2 and insulating behaviour is never observed. Instead, the intermediate metallic phase emerges in an exceptionally large region of the magnetic field–temperature phase diagram. Unlike the typical exponential behaviour associated with the quantum tunnelling of fermionic quasiparticles7,15, we observe a new power-law scaling as a function of field at low temperature that is consistent with the Bose-metal scenario of the metallic phase11,12,13,14.
Earlier studies on isolated NbSe2 flakes do not observe a superconducting transition to a zero-resistance state in the atomically thin limit21,22. Recently, however, it has been shown the surfaces of metallic materials may oxidize, altering the electronic properties of thin samples23. Exfoliation and encapsulation by a protective layer in an inert atmosphere is thus crucial for preserving the intrinsic properties of the 2D material23,24. We achieve this using a mechanical transfer set-up installed inside a nitrogen-filled glove box (see Methods). In short, within the glove box, an exfoliated NbSe2 flake is electrically contacted by graphite (G), and the entire device is protected by an insulating layer of hexagonal boron nitride (BN). The graphite leads are then contacted using an edge-metallization technique in the ambient environment25,26. A schematic depicting the assembly and fabrication process is shown in Fig. 1a and optical images of the heterostructure are shown in Fig. 1b before (left) and after (right) device fabrication.
In the main panel of Fig. 2a, we show four-terminal sheet resistance as a function of temperature for a particular NbSe2 device prepared using the method described above. The NbSe2 thickness is 1.5 nm, as determined by an atomic force microscope (AFM) outside the glovebox after BN/graphite transfer, suggesting it consists of only two atomic layers. The resistance in the normal state is RN = 75 Ω. This corresponds to a residual resistivity that is ten times larger than that of bulk crystals20, yet the sheet resistance is still an order of magnitude less than that of the most conductive amorphous superconducting films10. We observe a clear superconducting transition to a zero-resistance state measured to the limit of our instrument resolution. The critical temperature is Tc = 5.26 K, as defined by where the resistance is 90% of the normal state value, which is slightly reduced from the transition temperature of bulk samples (6.8–7.2 K; refs 19,20).
To further characterize the quality and dimensionality of our sample, we measured the temperature-dependent critical fields, defined by R(Hc2, T) = 0.9RN, for field configurations both perpendicular and parallel to the layers, and the results are plotted in the inset of Fig. 2a. Close to Tc, we expect Hc2⊥ = φ0/(2πξ(0)2)(1 − (T/Tc)), where φ0 is the flux quantum and ξ(0) is the in-plane Ginzburg–Landau coherence length at zero temperature27. A linear fit shown by the black line yields ξ(0) = 8.9 nm, similar to the bulk value20. From the normal state residual resistance and carrier concentration, as determined by Hall measurements, we estimate the electron mean free path to be l = 17 nm, smaller than that in bulk crystals20, but nearly twice ξ(0), indicating that our device is a ‘clean’ superconductor. In this regime, the Bardeen–Cooper–Schrieffer (BCS) coherence length ξ0 = 1.35ξ(0) < l (ref. 27). In contrast, amorphous films are generally characterized as ‘dirty’ superconductors with ξ0 ≫ l. The critical parallel field does not exhibit the linear temperature dependence expected for anisotropic 3D superconductors27. Instead, a 2D superconductor with thickness d < ξ0 obeys (ref. 28). The red curve in the inset of Fig. 2a shows a best fit to this expression, from which we extract d = 3.4 nm, which is larger than the thickness as determined by AFM (1.5 nm). However, this fitting has been previously found to overestimate the true sample thickness in sufficiently thin systems28.
For a 2D system with d < ξ(0), when the in-plane magnetic penetration length λ2/d exceeds the lateral sample size (assuming λ = 250 nm as in the bulk19: λ2/d ∼ 40 μm), the superconducting phase transition is expected to be of the Berezinskii–Kosterlitz–Thouless (BKT) type29. In this scenario, the low-temperature, zero-resistance phase consists of bound vortex–antivortex pairs created by thermal fluctuations. On heating, the pairs dissociate and may move, inducing dissipation. The BKT temperature defines the vortex unbinding transition and can be determined using current–voltage measurements as a function of temperature T, as shown in the main panel of Fig. 2b. Current excites free-moving vortices, causing a nonlinear voltage dependence: V ∼ Ia(T). At TBKT, a 2D superconductor obeys the universal scaling relation, V ∼ I3 (solid line in Fig. 2b)29,30. In the inset, we plot a versus T, as determined by the slope of the different V –I traces on a log–log scale. An example guide-to-eye fit is marked by the dashed line in the main panel. We determine TBKT = 5.01 K from where a = 3 interpolates, only slightly less than Tc as defined above. This is consistent with the behaviour of systems with normal state resistance much less than h/(2e)2, where TBKT is expected to be very close to the mean-field transition temperature31.
The measurements performed above confirm that our device exhibits the characteristics of a true 2D superconductor. The extracted material parameters together with the low normal state sheet resistance further places the high sample quality within a previously unexplored regime. We next turn to the dependence of resistance on perpendicular magnetic field, as the effect of vortices can now be cleanly separated from the low static disorder present in the sample. Shown in Fig. 3a is a 2D colour map of the four-terminal sheet resistance in the same device as a function of both temperature and magnetic field applied perpendicular to the layers. For clarity, temperature traces for different field levels are shown in an Arrhenius plot in Fig. 3b. For fields larger than Hc2 ∼ 3 T, the sample is in the normal state for all temperatures. Previous works on strongly disordered films observed a field-tuned transition to an insulating state of bosons5,6; however, one might not expect insulating behaviour for finite samples of a highly conductive 2D system in the disorder-free limit10. We have applied perpendicular fields as large as 14.5 T and still see metallic behaviour in the temperature dependence (see Supplementary Fig. 1 main panel).
As one lowers the field to just below Hc2, a resistance drop is observed on cooling from the normal state. In this region of the H–T phase diagram, the device exhibits activated behaviour, as can been seen in the linear slope in the Arrhenius plot (black lines in Fig. 3b). Classically, dissipation in a superconductor in which resistance is less than the normal state value can be attributed to the motion of individual vortices (flux creep or flow)27. In a clean 2D system, we expect the dominant energy barrier to flux motion to be that of vortex–antivortex dissociation32. Flux resistance then becomes thermally activated when the temperature is comparable to the barrier energy. We have determined the activation energy from the linear portion of the Arrhenius plot in Fig. 3b for different magnetic fields, and the result is plotted in Fig. 3c. The functional form is expected to be U(H) = U0 ln(H0/H), where U0 ∼ (φ02d)/(256π3λ2), the vortex–antivortex binding energy, and H0 ∼ Hc2 (refs 7,32). A fit to this form yields U0 = 27.5 K and H0 = 2 T for our device. For λ = 250 nm (ref. 19), we estimate U0 ∼ 10 K, of the order of the experimental value.
At lower temperatures, the resistance saturates to a level dependent on magnetic field (see coloured lines in Fig. 3b), hallmark of the quantum metallic state that is the main subject of this report. Qualitatively similar behaviour has also been observed in amorphous MoGe and Ta films with much larger normal state sheet resistances7,9. This effect cannot be understood within a classical framework, in which we expect to recover superconductivity for T ≪ U. In the past, various theories have been advanced to explain the origin of dissipation in this temperature regime for disordered films11,12,13,14,15,16,17. One can distinguish between the different models on the basis of the magnetic field dependence of the saturated resistance at low temperatures.
Shimshoni et al. consider a disordered superconducting film as a percolating network of superconducting islands within an insulating matrix15. Field-induced vortices that tunnel across thin superconducting constrictions give rise to resistance when coupled to a fermionic bath. The resistance depends on field as
where RN is the normal state resistance and C is a dimensionless constant of order unity. This expression finds good agreement with the measurements of Ephron et al. on amorphous MoGe films, in which RN ∼ 1 kΩ (ref. 7), but evidently provides a poor fit to our data (see Supplementary Fig. 2). Furthermore, because our device is over an order of magnitude more conductive, the closest fitting requires C = 0.14, an unphysically small value.
Galitski et al. consider a ‘vortex metal’ phase where field-induced vortices interact with electrically neutral spinons17. The theory accounts for the large peak in low-temperature magnetoresistance observed in disordered InOx films before saturation to RN at higher fields10. In our device, however, we observe a monotonic dependence of resistance on field and see no peak structure in magnetoresistance up to 7 T (see Supplementary Fig. 1 inset).
Das and Doniach, as well as Dalidovich and Phillips, report that a 2D system of interacting bosons may form a gapless, nonsuperfluid state in the limit of zero temperature, a phase which they term a ‘Bose metal’11,12,13,14. They argue that the uncondensed Cooper pairs and vortices are responsible for the small resistance observed at small finite field12. A magnetic field introduces gauge fluctuations, which disrupt phase coherence and cause dissipation, a quantum analogue of that caused by thermal fluctuations in the BKT transition. In the Bose-metal model, resistance on the metallic side of the field-tuned transition can be described by: R ∼ (H − Hc0)2ν, where Hc0 is the critical field of the superconductor to Bose-metal transition and ν is the exponent of the superfluid correlation length, which diverges across the boundary as (H − Hc0)−ν (ref. 12).
In the main panel of Fig. 4a, we show a log–log plot of R versus (H − Hc0) taken at several different temperatures. Hc0 is a small temperature-dependent value which we determined using a method that shall be described below. The linear scaling observed here suggests a power-law dependence on field. We have fitted the data to the expression R ∼ (H − Hc0)α and the extracted exponent α is plotted in the inset as a function of temperature. At high temperatures, but below Tc, resistance increases roughly linearly with field, as expected for unhindered flux flow: R ∼ RNH/Hc2(α = 1) (ref. 27). The grey line shows linear scaling as a guide to eye. As temperature is lowered, the field dependence becomes increasingly nonlinear and collapses onto a single curve below 1 K with α ∼ 3. The red line is an empirical fit given by R[Ω] = 5.44(H[T] − 0.15)3.21, which shows excellent agreement with the data. This yields a critical exponent of ν = 1.61. Previous measurements on MoGe films at the lowest accessible temperatures observe mostly an exponential dependence of resistance with field7,33, which is consistent with quantum tunnelling of fermionized vortices15. At small very fields for the MoGe film, however, the scaling obeys a power law with unity exponent33. The power-law scaling we observe in our sample consistently over the entire field range thus suggests that fermionic tunnelling undergoes a crossover to Bose-metal behaviour in the limit of vanishing static disorder. We also found similar field scaling in the quantum metallic state for other ultrathin devices, whereas this phase is distinctly absent in the bulk crystal (see Supplementary Fig. 3).
The critical field of the transition out of the true zero-resistance state Hc0 is difficult to determine directly from the linear resistance, given the limited accuracy of our instruments. Recently, however, experiments by Qin et al. and Li et al. on disordered Ta films showed that Hc0 can be determined indirectly using current–voltage measurements9,34. Hysteresis is observed in the V –I characteristics on the superconducting side, similar to that seen in underdamped Josephson junctions27, which disappears at the onset of the metallic phase. We have performed V –I measurements on the same bilayer device at T = 0.5 K for increasing magnetic fields (see Supplementary Fig. 4 main panel). Clear hysteresis is seen between current sweeps up and down for low fields. In the inset, we plot the current hysteresis ΔI at which the voltage jump occurs as a function of magnetic field. ΔI vanishes close to Hc0 = 0.175 T. This then allows us to identify the true superconducting phase as that for H < Hc0. We have repeated this measurement for several different temperatures to determine the critical field as a function of temperature.
Figure 4b shows a full H–T phase diagram for our device. The blue triangles mark Hc0(T) and the red circles mark Hc2⊥(T). The boundary between Bose metal and flux flow (purple squares) is defined by the intersection of the fits to the activated resistance and saturated levels in Fig. 3b. The normal phase extends up to at least 14.5 T without the appearance of an insulating phase, in contrast to previous works on strongly disordered films. The high sample quality is made possible by our facile device assembly technique in an inert atmosphere, which demonstrates a new route for the production of 2D superconductors in the ultraclean limit.
Polycrystalline NbSe2 was made by heating stoichiometric amounts of Nb powder (99.5%) and Se shots (99.999%) in evacuated silica ampules. Single crystals of NbSe2 were grown using a vapour transfer method. 400 mg of NbSe2 powder and 80 mg of I2 were sealed in a 23-cm-long silica ampule with a 1.13 cm2 inner cross section. The charge was put in the hot zone of 850 °C and the sink in the cold zone of 750 °C. After one week, all the polycrystals became single crystals, with most of the thicker plates found in the hot zone.
Device assembly and fabrication.
We have exfoliated ultrathin NbSe2 flakes in a nitrogen-filled glove box containing less than 2 ppm oxygen. Separately, we prepared thin hexagonal boron nitride (BN) on a polydimethylsiloxane (PDMS) stamp covered with polypropylene carbonate (PPC), which we use to ‘pick up’ two closely spaced graphite flakes (separation ∼2 μm). The BN/graphite stack is then used to cover the NbSe2 flake inside the glove box. While graphite makes electrical contact to NbSe2, hBN provides an insulating oxidation barrier. Subsequent lithography may then be used to define a four-terminal device, in which the gapped region between the graphite leads form the channel. The graphite is then electrically contacted using an edge-metallization technique25,26.
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, 673–676 (1979).
Goldman, A. M. & Markovic, N. Superconductor–insulator transitions in the two-dimensional limit. Phys. Today 51, 39–44 (1998).
Haviland, D. B., Liu, Y. & Goldman, A. M. Onset of superconductivity in the two-dimensional limit. Phys. Rev. Lett. 62, 2180–2183 (1989).
Fisher, M. P. A. Quantum phase transitions in disordered two-dimensional superconductors. Phys. Rev. Lett. 65, 923–926 (1990).
Hebard, A. F. & Paalanen, M. A. Magnetic-field-tuned superconductor–insulator transition in two-dimensional films. Phys. Rev. Lett. 65, 927–930 (1990).
Yazdani, A. & Kapitulnik, A. Superconducting–insulating transition in two-dimensional α-MoGe thin films. Phys. Rev. Lett. 74, 3037–3040 (1995).
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, 1529–1532 (1996).
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).
Qin, Y. G., Vicente, C. L. & Yoon, J. Magnetically induced metallic phase in superconducting tantalum films. Phys. Rev. B 73, 100505 (2006).
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).
Das, D. & Doniach, S. Existence of a Bose metal at T = 0. Phys. Rev. B 60, 1261–1275 (1999).
Das, D. & Doniach, S. Bose metal: Gauge-field fluctuations and scaling for field-tuned quantum phase transitions. Phys. Rev. B 64, 134511 (2001).
Dalidovich, D. & Phillips, P. Phase glass is a Bose metal: A new conducting state in two dimensions. Phys. Rev. Lett. 89, 027001 (2002).
Phillips, P. & Dalidovich, D. The elusive Bose metal. Science 302, 243–247 (2003).
Shimshoni, E., Auerbach, A. & Kapitulnik, A. Transport through quantum melts. Phys. Rev. Lett. 80, 3352–3355 (1998).
Spivak, B., Zyuzin, A. & Hruska, M. Quantum superconductor–metal transition. Phys. Rev. B 64, 132502 (2001).
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).
Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183–191 (2007).
Le, L. P. et al. Magnetic penetration depth in layered compound NbSe2 measured by muon spin relaxation. Physica C 185, 2715–2716 (1991).
Soto, F. et al. Electric and magnetic characterization of NbSe2 single crystals: Anisotropic superconducting fluctuations above Tc . Physica C 460, 789–790 (2007).
Staley, N. E. et al. Electric field effect on superconductivity in atomically thin flakes of NbSe2 . Phys. Rev. B 80, 184505 (2009).
El-Bana, M. S. et al. Superconductivity in two-dimensional NbSe2 field effect transistors. Supercond. Sci. Technol. 26, 125020 (2013).
Tsen, A. W. et al. Structure and control of charge density waves in two-dimensional 1T-TaS2 . Proc. Natl Acad. Sci. USA http://dx.doi.org/10.1073/pnas.1512092112 (in the press).
Cao, Y. et al. Quality heterostructures from two-dimensional crystals unstable in air by their assembly in inert atmosphere. Nano Lett. 15, 4914–4921 (2015).
Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).
Cui, X. et al. Multi-terminal transport measurements of MoS2 using a van der Waals heterostructure device platform. Nature Nanotech. 10, 534–540 (2015).
Tinkham, M. Introduction to Superconductivity 2nd edn (Dover, 1996).
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).
Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low Temp. Phys. 36, 599–616 (1979).
Eley, S., Gopalakrishnan, S., Goldbart, P. M. & Mason, N. Approaching zero-temperature metallic states in mesoscopic superconductor–normal-superconductor arrays. Nature Phys. 8, 59–62 (2012).
Beasley, M. R., Mooij, J. E. & Orlando, T. P. Possibility of vortex–antivortex pair dissociation in two-dimensional superconductors. Phys. Rev. Lett. 42, 1165–1168 (1979).
Feigelman, M. V., Geshkenbein, V. B. & Larkin, A. I. Pinning and creep in layered superconductors. Physica C 167, 177–187 (1990).
Mason, N. & Kapitulnik, A. True superconductivity in a two-dimensional superconducting–insulating system. Phys. Rev. B 64, 060504 (2001).
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).
We acknowledge helpful discussions with Z. Han, J.-D. Pillet, E. Shimsoni, O. Vafek, A. Kapitulnik, D. Xiao and D. Gopalan. We thank J. Shi, F. Zhao, D. Wang and S. Chen for assistance with device fabrication. This material is based on work supported by the NSF MRSEC Program through Columbia in the Center for Precision Assembly of Superstratic and Superatomic Solids (DMR-1420634). Salary support is provided by the NSF under grants NEB- 1124894 (A.W.T.) and DMR-1056527 (A.N.P.). Some measurements were performed at the National High Magnetic Field Laboratory, which is supported by the NSF Cooperative Agreement (DMR-0654118), the State of Florida and the Department of Energy. S.J. is supported by the National Basic Research Program of China (grants 2013CB921901 and 2014CB239302). R.J.C. is supported by the Department of Energy, Division of Basic Energy Sciences (grant DOE FG02-98ER45706). P.K. acknowledges support from the Army Research Office (grant W911NF-14-1-0638).
The authors declare no competing financial interests.
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Tsen, A., Hunt, B., Kim, Y. et al. Nature of the quantum metal in a two-dimensional crystalline superconductor. Nature Phys 12, 208–212 (2016). https://doi.org/10.1038/nphys3579
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