Abstract
The properties of twodimensional transition metal dichalcogenides arising from strong spin–orbit interactions and valleydependent Berry curvature effects have recently attracted considerable interest^{1,2,3,4,5,6,7}. Although singleparticle and excitonic phenomena related to spin–valley coupling have been extensively studied^{1,3,4,5,6}, the effects of spin–valley coupling on collective quantum phenomena remain less well understood. Here we report the observation of superconducting monolayer NbSe_{2} with an inplane upper critical field of more than six times the Pauli paramagnetic limit, by means of magnetotransport measurements. The effect can be interpreted in terms of the competing Zeeman effect and large intrinsic spin–orbit interactions in noncentrosymmetric NbSe_{2} monolayers, where the electron spin is locked to the outofplane direction. Our results provide strong evidence of unconventional Ising pairing protected by spin–momentum locking, and suggest further studies of noncentrosymmetric superconductivity with unique spin and valley degrees of freedom in the twodimensional limit.
Main
Monolayer transition metal dichalcogenide (TMD) of the hexagonal structure consists of a layer of transition metal atoms sandwiched between two layers of chalcogen atoms in the trigonal prismatic structure^{8} (Fig. 1a). It possesses outofplane mirror symmetry and broken inplane inversion symmetry. The presence of 4d transition metal also gives rise to large spin–orbit interactions (SOIs). The mirror symmetry restricts the crystal field (ɛ) to the plane. The SOIs split the spin states at finite momentum k in the absence of inversion symmetry. They manifest as an effective magnetic field H_{SO}(k) along the direction of k × ɛ, which is out of plane for the restricted twodimensional (2D) motion of electrons in the plane. The electron spin is thus oriented in the outofplane direction and in opposite directions for electrons of opposite momenta^{1,2,3} (Fig. 1a). Such spin–momentum locking is destroyed in the bulk where inversion symmetry and spin degeneracy are restored^{1,2,7} (Fig. 1b). New valley and spindependent phenomena including optical orientation of the valley polarization^{3,4} and the valley Hall effect^{5} have been recently demonstrated in groupVI TMD monolayers such as MoS_{2}. The effect of spin–momentum locking on collective quantum phenomena in monolayer TMDs, however, remains an open area.
In this work, we demonstrate very high inplane upper critical fields induced by spin–momentum locking in superconducting monolayers of groupV TMD niobium diselenide (NbSe_{2}). This was achieved through transport and magnetotransport studies of highquality NbSe_{2} devices of varying layer thickness. Bulk 2HNbSe_{2} is a wellstudied type II anisotropic multiband superconductor with a zerofield critical temperature T_{C0} ≈ 7 K (refs 9,10,11,12,13,14). Superconductivity in atomically thin NbSe_{2} (refs 15,16) and samples down to the monolayer thickness^{17,18} has also been observed recently. Monolayer NbSe_{2} can be viewed as heavily holedoped monolayer MoSe_{2} (refs 8,19): the Fermi surface is composed of one pocket at the Γ point and two pockets at the K and the K′ point of the Brillouin zone with each pocket spin split into two pockets by the SOIs (ref. 19; Fig. 1a); the spin splitting around the Γ point is much smaller than that at the K (K′) point^{19} and the spin–momentum locking effects are dominated by the K (K′) pockets.
In conventional superconductors, superconductivity can be quenched under a sufficiently high external magnetic field by the orbital^{11,20} and spin Zeeman effect^{21,22}. They are originated from the coupling between the electron momentum and the magnetic field (for example, vortex formation) and from spin alignment by the magnetic field, respectively. In the limit of monolayer thickness, the interlayer coupling vanishes and the orbital effect is absent for an inplane magnetic field H_{∥}. The inplane upper critical field (critical field at zero temperature) H_{c20}^{∥} is thus determined by direct spinflip, known as the Pauli paramagnetic limit H_{P} ≈ 1.84 T_{C0} (in tesla for T_{C0} in kelvin) for isotropic BCS superconductors^{21,22}. Under this field the Zeeman splitting energy matches the superconducting energy gap, or equivalently, the binding energy of a Cooper pair. Spin–momentum locking and the consequent pairing of the K and K′ electrons with spin locked to the two opposite outofplane directions (Fig. 1a), referred to as Ising pairing, is expected to significantly enhance the inplane paramagnetic field^{23,24,25} in monolayer NbSe_{2}.
In our experiment, we fabricated monolayer and fewlayer samples of NbSe_{2} by mechanical exfoliation of 2HNbSe_{2} single crystals followed by direct transfer onto SiO_{2}/Si substrates with prepatterned electrodes and encapsulation by thin layers of hexagonal boron nitride (see Methods). The sample thickness was determined with monolayer accuracy by optical spectroscopy, particularly, the frequency shift of the interlayer shear mode^{17}. The crystal symmetry is characterized by optical secondharmonic generation. (See Supplementary Section 1 for details on optical characterization.) Figure 1c is an optical image of a representative device (bilayer in this case). The temperature dependence of the normalized fourpoint resistance R(T)/R(300 K) at zero magnetic field is shown in Fig. 1d from 2 to 300 K for a typical bulk, bilayer and monolayer device. (See Supplementary Section 2 for original data and more devices.) All samples show a metallic behaviour with phononlimited transport at high temperature (with R ∝ T) and disorderlimited transport at low temperature (with R approaching a constant value) before reaching the superconducting state. The residual resistance ratio evaluated using the roomtemperature resistance and the normalstate resistance right above the superconducting transitionR_{n} (taken at 8 K) varies from ∼30 in the bulk to ∼10 in the monolayer device. The square resistance per layer (at 8 K) was estimated to be ≈ 200 Ω for the bilayer of Fig. 1c and similar values were estimated for other devices. These values are much smaller than h/4e^{2} ≈ 6,450 Ω, where a disorderinduced superconductor–insulator transition emerges^{26}. Here e is the electron charge and h is the Planck constant. Our samples are therefore in the lowdisorder regime compared with similar layered compounds studied previously^{16} and the disorder effects on T_{C0} are expected to be small^{26}.
Figure 2a shows the temperature dependence near the superconducting transition of the resistance normalized by R_{n} for NbSe_{2} samples of varying thickness N. Superconductivity is observed for all samples down to the monolayer thickness. A significant drop in the transition temperature accompanied with a significant broadening is observed for N < 4 (∼2.6 nm). The broadening can be attributed to enhanced thermal fluctuations in two dimensions^{20,27} when the sample thickness falls below the bulk outofplane coherence length of 2.7 nm (ref. 10). We have used the Aslamazov–Larkin formula^{28} to determine T_{C0} (solid lines, Fig. 2a), which is close to the temperature corresponding to 0.5R_{n}. We have also performed current excitation measurements to investigate the importance of phase fluctuations, and the Berezinskii–Kosterlitz–Thouless transition temperature^{27} is found to be close to T(0.01R_{n}). Figure 2b summarizes the Ndependence of T_{C0}, T(0.5R_{n}) and T(0.01R_{n}). The monotonic dependence of T_{C0} on N can be accounted for by the decreasing interlayer Cooper pairing^{29,30} (as evidenced by the linear dependence of [ln(T_{C0}(N)) − ln(T_{C0}(N = 1))]^{−1} on [cos(π/(N + 1))]^{−1} in the inset of Fig. 2b). Effects on T_{C0} from the competing chargedensitywave order, which is enhanced with decreasing N, are thus likely to be weak in NbSe_{2} (ref. 14). (See Supplementary Sections 3 and 4 for details on the study of the characteristics of 2D superconductivity in NbSe_{2}).
With the above understanding of the nature of superconductivity in 2D NbSe_{2}, we now study its magnetic response. Figure 3a–c shows the temperature dependence of the fourpoint resistance for a bulk, trilayer and monolayer sample under both outofplane (H_{⊥}) and inplane magnetic fields (H_{∥}). A second monolayer device with twopoint measurements up to 20 T is also included. For all samples, we normalize the temperature by the corresponding T_{C0}, and the resistance by R_{n}. We define the critical temperature T_{C} under a finite magnetic field H_{c2} as the temperature corresponding to 50% of R_{n}, in accordance with the zerofield convention. For twopoint measurements, as shown in Fig. 3c, we assign T_{C0} as the temperature at which a rapid resistance drop occurs while the sample is cooled from the normal state. It is clear that for all samples superconductivity is more susceptible to H_{⊥} than to H_{∥} and the magnetic anisotropy is significantly larger in atomically thin samples than in the bulk. (For the angular dependence study of the magnetic response refer to Supplementary Section 5.)
We summarize the H_{c2}–T_{C} phase diagram for differing sample thickness N in Fig. 4 for both H_{⊥} (open symbols) and H_{∥} (filled symbols). For comparison, we normalize the critical field H_{c2} by the BCS Pauli paramagnetic limit H_{P} and the critical temperature T_{C} by T_{C0} for each sample. For outofplane fields, our experiment shows a linear H_{c2}^{⊥}–T_{C} dependence that is largely thickness independent. The result can be explained by considering the orbital effect, that is, overlaps of vortex cores, as the major quenching mechanism. H_{c20}^{⊥} is determined by the combined effect of inplane coherence length and transport mean free path^{20}. Whereas the coherence length increases with reducing thickness (because of the reduction in T_{C0}), the mean free path decreases as the material becomes more disordered. The net effect is a weak H_{c2}^{⊥}–T_{C} dependence on N. The measured upper critical field for bulk NbSe_{2}H_{c20}^{⊥} ≈ 4 T (≪H_{P}) agrees well with the reported value^{9,10}.
In contrast, for inplane fields, the H_{c2}^{∥}–T_{C} dependence near T_{C0} in the monolayer is much steeper than in the bulk and follows a square root instead of a linear dependence. More significantly, the monolayer upper critical field H_{c20}^{∥} far exceeds its Pauli limit H_{P} whereas H_{c20}^{∥} ≳ H_{P} in the bulk. We note that large enhancements of H_{c20}^{∥} (≫H_{P}) have been observed in other systems^{9,31,32,33,34,35} including two recent experiments on MoS_{2} crystals gated by ionic liquids^{36,37}. Many mechanisms have been discussed including strong coupling^{38,39}, modified electron gfactor^{20,23}, interaction effects^{38,40,41}, remnant magnetic susceptibility at low temperatures^{20,21,22}, spin–orbit scattering from impurities^{31,32,42,43,44} and intrinsic SOIs (refs 23,24,25,35,36,37). The observed large enhancement of H_{c20}^{∥} in monolayer NbSe_{2} can be understood as a special case of intrinsic SOI effects predicted for noncentrosymmetric superconductors^{23,24,25}. In particular, the relevant spin–orbit field for monolayer NbSe_{2} with D_{3h} symmetry can be derived following refs 23,25
Here H_{0} is the magnitude of the spin–orbit field at the K (K′) point, a = 0.35 nm (ref. 8) is the inplane lattice constant, and denotes the outofplane unit vector. As mentioned above, the field at the K (K′) pocket far exceeds that around the Γ pocket^{19} and dominates the magnetic field response of superconductivity. Neglecting trigonal warping, the spin–orbit field at the Fermi surface for the K (or K′) pocket is approximately a constant (or ) with H_{SO} ≲ H_{0}.
In the absence of the orbital effect, H_{c20}^{∥} is determined by the alignment of spin by the external field. The upper critical field can be estimated by noting that the inplane component of the spin magnetic moment is reduced to ∼(H_{∥}/H_{SO})μ_{B} with μ_{B} denoting the Bohr magneton. The ratio H_{∥}/H_{SO} originates from a competition between H_{SO}, which locks the electron spin to the outofplane direction, and H_{∥}, which tilts the spin towards the inplane direction^{25}. Pair breaking occurs when the modified Zeeman energy ∼(H_{∥}^{2}/H_{SO})μ_{B} (known as van Vleck paramagnetism^{24,25}) overcomes the superconducting gap (that is, (H_{∥}^{2}/H_{SO})μ_{B} ∼ H_{P}μ_{B}). It thus yields the following estimate for , which can greatly exceed H_{P} if H_{SO} ≫ H_{P} for strong SOIs.
To analyse the entire H_{c2}^{∥}–T_{C} phase diagram, we introduce the pair breaking equation^{20} for monolayer NbSe_{2} with the spin–momentum locking effect incorporated
Here ψ(x) is the digamma function. Near T_{C0}, equation (2) can be reduced to (1 − (T_{C}/T_{C0})) = ((H_{c2}^{∥})^{2}/H_{SO}H_{P}), which describes well the observed squareroot dependence of . We fit the experimental H_{c2}^{∥} − T_{C} dependence to the solution of equation (2) with H_{SO} as a free parameter. The best fit (blue line, Fig. 4) gives the spin–orbit field H_{SO} ≈ 660 T, or equivalently, the total spin splitting energy 2Δ_{SO} = 2μ_{B}H_{SO} ≈ 76 meV. This value agrees well with the values from ab initio calculations for the Fermi surface around the K (K′) point of the Brillouin zone (∼70–80 meV; ref. 19), where the effect of SOIs on Cooper pairing is the strongest. The upper critical field is determined from equation (2) to be H_{c20}^{∥} ≈ 35 T, which is more than six times H_{P}. To verify this value, we have performed independent differential conductance measurements of a monolayer at 0.3 K up to H_{∥} = 31.5 T. (See Supplementary Section 6 for details.) A zerobias peak originated from Andreev reflections at the normal metal–superconductor contact was observed. With increasing magnetic field, the zerobias peak diminishes continuously, corresponding to a shrinking superconducting gap, and becomes very weak at 31.5 T (inset of Fig. 4). This result suggests that H_{c20}^{∥} ≳ 31.5 T, consistent with the result from equation (2).
Finally, we briefly comment on the H_{c2}^{∥}–T_{C} phase diagram for the fewlayer and bulk sample. As N increases, interlayer coupling kicks in, which destroys perfect Ising pairing and introduces orbital effects, both of which decrease the upper critical field H_{c20}^{∥}. However, for layer thickness much smaller than the outofplane penetration depth (∼23 nm; ref. 10), the orbital effect is strongly suppressed^{20}. Similar H_{c2}^{∥}–T_{C} dependences are observed for fewlayer samples with H_{c20}^{∥} > 3H_{P} in both bilayer and trilayer NbSe_{2}. The observation of H_{c20}^{∥} ≫ H_{P} in fewlayer NbSe_{2} samples suggests a weak coupling between the layers. Indeed, as long as the interlayercoupling constant (t_{⊥} ≈ (ℏv_{F⊥}π)/2c ≈ 10 meV, estimated using v_{F⊥} ≈ 10^{4} ms^{−1} for the outofplane Fermi velocity^{45} and c ≈ 0.64 nm for the interlayer distance^{8} in bulk NbSe_{2}) is small compared with the spin splitting energy 2Δ_{SO} ≈ 76 meV, the outofplane spin is still approximately a good quantum number. The electron spin, instead of being locked to the momentum, is now locked to each individual layer, which also protects superconductivity under a parallel magnetic field^{25}. Such a spin–layer locking has been observed in related centrosymmetric groupVI TMDs (refs 6,7) and heavyFermion superconductor–normalmetal superlattices^{46}. On the other hand, in the bulk limit, the orbital effect dominates the phase diagram as for the case of outofplane fields. The estimated inplane upper critical field H_{c20}^{∥} ≈ 17 T (from 70% of the linear extrapolation of the H_{c2}^{∥}–T_{C} dependence at zero temperature^{47}) is about 4 times higher than H_{c20}^{⊥} and is in good agreement with the reported value^{9,10}. We note that the above discussion on the magnetic response of fewlayer samples is very qualitative. More detailed experimental and theoretical studies are warranted for a full understanding of the interplay between spin–layer locking and orbital effects, and of the importance of the complex Fermi surface of NbSe_{2} (refs 13,14). We further note that studies of interplay between SOIs and superconductivity in atomically thin materials may also lead to 2D topological superconductivity^{48}.
Methods
Sample preparation and device fabrication.
Highquality 2HNbSe_{2} single crystals were grown from Nb metal wires of 99.95% purity and Se pellets of 99.999% purity by iodine 99.8% vapour transport in a gradient of 730–700 °C in a sealed quartz tube for 21 days. A very slight excess of Se was introduced (typically 0.2% of the charge) to ensure stoichiometry in the resulting crystals. Thin flakes were mechanically exfoliated from bulk single crystals on silicone elastomer polydimethylsiloxane stamps. Atomically thin samples of good geometry were first identified by optical microscopy and then transferred onto silicon substrates (covered by a 280 nm layer of thermal oxide) with prepatterned Au electrodes. To minimize the environmental effects on the samples, we limited their exposure to air to <1 h. Hexagonal boron nitride (hBN) thin films of 10–20 nm thickness were introduced as a capping layer for further protection. The sample thickness was determined according to their shear mode frequency by Raman spectroscopy^{17}. The crystal quality was characterized by polarized optical secondharmonic generation. (See Supplementary Section 1 for more details.)
Electrical characterization.
Transport and magnetotransport measurements were carried out in a Physical Property Measurement System down to 2.1 K and up to 9 T. For higher magnetic field measurements up to 31.5 T, a Janis He3 cryostat with base temperature of 0.3 K was employed. Unless specifically mentioned, longitudinal electrical resistance was acquired using a fourpoint geometry with excitation current limited to 1 μA to avoid heating and highbias effects. (Dependence on the excitation current was performed to study the fluctuation effects in two dimensions.) The devices were mounted on a rotation stick, which allows alignment of the sample plane with the external magnetic field with high accuracy (<0.5° error). Multiple devices were prepared and measured. All yielded consistent results for samples of the same thickness.
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Acknowledgements
We thank M. H. W. Chan for fruitful discussions. This work was supported by the US Department of Energy, Office of Basic Energy Sciences (contract No. DESC0013883 (K.F.M.) and DESC0012635 (J.S.)). Optical spectroscopy was supported by the National Science Foundation (NSF) under Award No. DMR1410407. The NHMFL is supported by the NSF Cooperative Agreement No. DMR1157490 and the State of Florida. K.T.L. is supported by HKUST3/CRF/13G and the Croucher Innovation Grant. The work in Lausanne was supported by the Swiss National Science Foundation. We also acknowledge support from the NSF MRSEC under Award No. DMR1420451 (Z.W.) and the MRI2D Center at Penn State University (X.X.).
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J.S. and K.F.M. designed the experiments. X.X. and Z.W. performed the experiments with the assistance of W.Z. at the Penn State Physics Low Temperature Laboratory, and of J.H.P. at the National High Magnetic Field Laboratory. X.X., Z.W., J.S. and K.F.M. analysed the data and cowrote the paper. K.T.L. contributed to the interpretation of the results. H.B. and L.F. contributed NbSe_{2} crystals. All authors discussed the results and commented on the manuscript.
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Xi, X., Wang, Z., Zhao, W. et al. Ising pairing in superconducting NbSe_{2} atomic layers. Nature Phys 12, 139–143 (2016). https://doi.org/10.1038/nphys3538
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