Abstract
Symmetrybreaking has been known to play a key role in noncentrosymmetric superconductors with strong spin–orbit interactions (SOIs; refs 1,2,3,4,5,6). The studies, however, have been so far mainly focused on a particular type of SOI, known as the Rashba SOI (ref. 7), whereby the electron spin is locked to its momentum at a rightangle, thereby leading to an inplane helical spin texture. Here we discuss electricfieldinduced superconductivity in molybdenum disulphide (MoS_{2}), which exhibits a fundamentally different type of intrinsic SOI, manifested by an outofplane Zeemantype spin polarization of energy valleys^{8,9,10}. We find an upper critical field of approximately 52 T at 1.5 K, which indicates an enhancement of the Pauli limit by a factor of four as compared to that in centrosymmetric conventional superconductors. Using realistic tightbinding calculations, we reveal that this unusual behaviour is due to an intervalley pairing that is symmetrically protected by Zeemantype spin–valley locking against external magnetic fields. Our study sheds light on the interplay of inversion asymmetry with SOIs in confined geometries, and its role in superconductivity.
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Main
MoS_{2} is a member of the layered semiconducting transition metal dichalcogenides (TMDs; ref. 11), which have been attracting widespread attention as twodimensional (2D) materials beyond graphene, owing to their multiple functionalities with potential applications such as atomically thin electronics^{12,13,14}, photonics^{15} and valleytronics devices utilizing a coupled spin and valley degree of freedom^{16,17,18}. Also, MoS_{2} is becoming a new platform for investigating quantum physics—for example, with quantum oscillations^{19} and electricfieldinduced superconductivity^{20}. The unit cell of MoS_{2} is composed of two formula units, in each of which one Mo atom is sandwiched between two S atoms, forming a S–Mo–S monolayer stacking along the caxis with D_{3h} symmetry (Fig. 1a). In the isolated monolayer, inplane inversion symmetry is broken, causing outofplane spin polarization together with effective Zeeman fields—namely, Zeemantype spin polarization at zero magnetic field^{8,9,10,21}. This Zeemantype spin splitting reaches 3 meV (ref. 22) and 148 meV (ref. 8) at the bottom of conduction band and the top of the valence band, respectively, both of which are located at the K points, the corner of the hexagonal first Brillouin zone shown in Fig. 1b. Such a zerofield spin splitting is not observed in bulk MoS_{2} with D_{6h}^{4} symmetry^{23,24}. Also, this spin splitting changes its sign at the −K point, because the K and −K points are connected by the timereversal operation. Such a spin splitting unique to monolayer MoS_{2} originates from the fairly strong SOI of transition metal dorbitals, and is commonly observed in the group VI of TMD semiconductors^{8,9}. This valleydependent spin polarization is in marked contrast to the inplane momentumdependent spin polarization caused by the Rashbatype SOI (ref. 7).
A noncentrosymmetric system with considerable SOIs is an ideal platform for exotic superconductivity—in fact, superconductivity occurring in the Rashbatype band structure has been intensively investigated on a variety of systems^{1,2,3,4}, together with the effect of spin–momentum locking. However, the effect of Zeemantype spin polarization on superconductivity has not been discussed previously. Here, we investigate electricfieldinduced superconductivity in MoS_{2} by using an electricdoublelayer transistor (EDLT) configuration (Fig. 1c), which creates a highdensity twodimensional electron system (2DES) on the surface (Fig. 1d) without introducing extrinsic disorder, thereby offering novel opportunities to search for new types of exotic superconductivity^{25,26,27}.
To extract the anomalous features of electricfieldinduced superconductivity at the highly crystalline multilayer MoS_{2} surface, we fabricated an EDLT structure with a 20nmthick flake, and then performed magnetotransport measurements. The MoS_{2}EDLT underwent a superconducting transition at a gate voltage of V_{G} = 6.5 V and a sheet carrier density of n_{2D} = 1.5 × 10^{14} cm^{−2} measured at 15 K (Fig. 2a). The critical temperature, T_{c}, of this device was 9.7 K, as defined at the midpoint of the transition, with R_{sheet} being 50% of the normal state sheet resistance at 15 K. This carrier density is slightly larger than the optimum value in the domeshaped phase diagram^{20,28,29}. The electrochemical reaction is unlikely even at high gate voltages up to 6.5 V (see Supplementary Section I), according to the reversibility and the absence of hysteresis in the transfer curve (Supplementary Fig. 1). Zoomins to the resistive transition in the lowtemperature region under the application of perpendicular and parallel magnetic fields from 0 to 9 T are shown in Fig. 2b and c, respectively. The superconducting state is completely quenched at 9 T for perpendicular magnetic fields (Fig. 2b), whereas it remains almost unchanged in the inplane magnetic field geometry (Fig. 2c). This behaviour indicates a substantially large anisotropy in the superconductivity. Figure 2d shows the angular dependence of the upper critical field, H_{c2}(θ), at 9.6 K (θ represents the angle between the caxis of the crystal and the direction of the applied magnetic fields). Here, H_{c2}(θ) is also determined by the midpoint of the resistive transition. A cusplike peak in the inset of Fig. 2d is described well by the 2D Tinkham model^{30} (and not by the 3D anisotropic mass model) as frequently observed in interfacial superconductivity^{31,32,33}. In addition, the dependence of H_{c2} on the temperature T for both the outofplane and inplane magnetic fields (Fig. 2e) are fitted well by the phenomenological 2D Ginzburg–Landau (GL) model,
and
where ϕ_{0}, ξ_{GL}(0) and d_{SC} denote a flux quantum, the inplane GL coherence length at T = 0 K, and the effective thickness of superconductivity, respectively. We find ξ_{GL}(0) = 8.0 nm and d_{SC} = 1.5 nm. Note that the extremely sharp rise of H_{c2}^{‖}(T) near T_{c} shows a marked contrast to that in conventional bulk layered superconductors such as Csdoped MoS_{2} (ref. 34), demonstrating that the present system is extremely 2D in nature. In fact, H_{c2}^{‖}(T) can seemingly go far beyond the Pauli limit, H_{P}^{BCS}, for weak coupling Bardeen–Cooper–Schrieffer (BCS) superconductors, T, where k_{B} and Δ_{0} are the Boltzmann constant and the BCStheorybased superconducting gap at T = 0 K, respectively.
To investigate H_{c2}^{‖} at much lower temperatures, we measured the magnetoresistance of another MoS_{2}EDLT by applying pulsed magnetic fields up to 55 T (see Supplementary Section II and Supplementary Fig. 2). A clear resistance drop at a T_{c} of 6.5 K was observed, which was defined as the temperature where R_{sheet} reached 75% of the normal state sheet resistance, indicating a superconducting signature, although the MoS_{2}EDLT used for the measurements in high magnetic fields (n_{2D} = 8.5 × 10^{13} cm^{−2} at V_{G} = 5.5 V and T = 15 K) did not exhibit zero resistance. The magnetoresistance of the MoS_{2}EDLT is shown in Fig. 3a and b for outofplane and inplane magnetic fields, respectively, at several temperatures between 1.5 and 8.0 K. In the outofplane magnetic field geometry, the superconducting state is completely destroyed by the application of magnetic fields stronger than 5 T. On the other hand, for the inplane magnetic fields, the superconductivity is not completely suppressed, nor does it revert to the normal state even on applying a 55 T magnetic field at 1.5 K. We summarize both H_{c2}^{‖}(T) and H_{c2}^{⊥}(T) in Fig. 3c. We note that H_{c2}^{‖}(T) increases with decreasing temperature and eventually saturates at approximately 52 T at 1.5 K, which is more than four times larger than μ_{0}H_{P}^{BCS} = 12 T. Because the orbital limit is supposed to be large owing to confinement of the geometry by the EDLT, the saturating behaviour of H_{c2}^{‖}(T) at low temperatures is suggestive of the Pauli limit, as seen in the Paulilimited superconductor^{35}.
The enhancement of H_{c2} in a dirtylimit superconductor with strong SOI has been discussed in terms of the spin–orbit scattering caused by disorder. This is expected to cause randomization of electron spins, and thus result in suppression of the effect of spin paramagnetism^{36,37,38}. To evaluate the contribution of this effect, we fitted our H_{c2}^{‖}(T) data by using the microscopic Klemm–Luther–Beasely (KLB) theory^{38}, which is applicable to dirtylimit layered superconductors with strong SOIs (l ≪ ξ_{Pippard} and τ ≪ τ_{SO}, where l, ξ_{Pippard}, τ and τ_{SO} are the mean free length, the Pippard coherence length, the total scattering time and the spin–orbit scattering time, respectively). Our H_{c2}^{‖}(T) data are fitted well by the KLB theory (Supplementary Fig. 3); however, we found that in all cases the values of τ are larger than those of τ_{SO}(τ > τ_{SO}) (Supplementary Table 1). This is an unphysical situation which contradicts with the initial assumption required for this theory (τ ≪ τ_{SO}). Thus, the model with the effect of spin–orbit scattering does not explain the enhancement of H_{c2}^{‖} consistently.
To find a more plausible origin of the enhancement of the Pauli limit in the present system, we first performed a set of abintiobased tightbinding supercell calculations on bulk MoS_{2}, incorporating the nearsurface band bending effect via an electrostatic potential term obtained by selfconsistently solving the Poisson equation (details in Supplementary Section IV). Our calculations suggest that, under the application of a strong electric field, a highdensity 2DES is created at the surface of MoS_{2}. As schematically shown in Fig. 1d, this results in the formation of an accumulation layer, which is effectively confined within the topmost MoS_{2} layer^{39,40}, indicating that noncentrosymmetric quasisinglelayer superconductivity is realized in our system. Such a “quasisinglelayer” 2DES, therefore, ought to have an effective D_{3h} symmetry, leading to many interesting features in the momentum space. For example, once a positive gate voltage is switched on, the conduction band minimum shifts to the ±K points^{40}. This situation is in stark contrast to the case of bulk MoS_{2}, where the conduction band minima are located at six symmetrically equivalent k points along Γ–K directions, also known as the T (or Q) points^{23,24}. Accordingly, the electricfieldinduced 2D superconductivity in MoS_{2} is expected to be solely mediated by the ±K valleys, and thus the most likely ground state of the Cooper pair should be the intervalley pairing between the +K and −K valleys to maintain zero momentum for the centreofmass of the Cooper pairs. Note that the intravalley spinsinglet Cooper pairs are not stabilized in the presence of the Zeemantype SOI, which requires nonzero momentum.
At a sheet carrier density of n_{2D} = 8.7 × 10^{13} cm^{−2}, which is nearly the same as the value in the highfield measurement, the bands are spinsplit by ∼3 meV at the ±K points, at zero magnetic field. Slightly away from the K point, these spinsplit bands cross each other such that the splitting becomes ∼13 meV at the Fermi level. The corresponding band dispersion and spin texture at the Fermi surface are shown in Fig. 4a and b, respectively. All these features of the band structure are qualitatively equivalent to those in the monolayer MoS_{2} derived from the tightbinding method^{41} and the k ⋅ p model^{42}. This agreement indicates that bulk or multilayer TMDs under a strong electric field can effectively behave as monolayers. Such a monolayerlike behaviour has already been experimentally demonstrated in bilayer systems, exhibiting circularly polarized photoluminescence under an electric field^{43}, and bulk systems showing gateinduced weak antilocalization behaviour in magnetoconductance^{21}. In addition to these works, recent optical measurements on WSe_{2} multilayers have shown that these systems can emit an electrically switchable circularly polarized electroluminescence^{16}. The circularly polarized luminescence is believed to be a unique feature of the monolayer. Hence, the observation of the same phenomenon in a gated multilayer system provides strong evidence that TMDs such as MoS_{2} can behave like a monolayer under an electric field.
As shown in Fig. 4a, each band is almost fully outofplane spin polarized. The inplane Rashbatype component, which originates from the asymmetric potential along the caxis produced by the strong electric field (∼50 MV cm^{−1}) (Fig. 1d), is calculated to be very small, with less than 2% of the total spin polarization. This is indeed expected by group theory, ruling that no inplane component is allowed at the K points owing to their threefold rotational (C_{3}) symmetry^{44,45}. In the presence of the finite Rashbatype SOI, a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state^{4,5,6} (a helical state^{46}), where Cooper pairs have nonzero momentum, with s + fwave symmetry^{47}, is likely to be realized. However, we confirmed by a numerical calculation that the enhancement of the upper critical field due to the FFLO state, or induced spintriplet components derived from Zeemantype SOI, is negligible (see ref. 48 and Supplementary Fig. 6). Note that this FFLO state, where Cooper pairs have a finite centre of mass momentum which is much smaller than K, should be distinguished from the intravalley pairing. Also, as other possibilities for the enhancement of H_{c2}, Rashbatype SOI (refs 1,2,3), quantum critical point^{49} and modified electron gfactor^{1,30} can be ruled out in the present system (details in Supplementary Section III). Therefore, spin–valley locking due to intrinsic Zeemantype SOI is considered to be the most promising origin for the enhancement of H_{c2}^{‖}.
We theoretically estimated the realistic Pauli limit of the present system by considering both the Zeemantype and the small Rashbatype SOIs. For this purpose, we constructed a simpler tightbinding model reproducing the 2DES subband structure shown in Fig. 4b (see Supplementary Section IV and Supplementary Fig. 4). Assuming isotropic swave superconductivity, we then calculated the Pauli limit in this model by solving the linearized BCS gap equation using a diagrammatic technique based on the 2DES subband structure^{50} (see Supplementary Section V). Figure 4c shows the theoretical curves of the Pauli limit in this system. Considering only the Zeemantype SOI, the Pauli limit is considerably enhanced, as it is larger than 70 T at T = 1 K (see also Supplementary Fig. 5). This result indicates that the moderately large valleydependent Zeemantype spin splitting in the vicinity of the K points (∼13 meV) protects singlet Cooper pairing between the K and −K valleys (Fig. 4d)—namely, the Cooper pairing locked by outofplane spin polarization to the two opposite directions, referred to as intervalley Ising pairing, enhances H_{c2}^{‖} much more than the H_{P}^{BCS}.
By contrast, once the small Rashbatype SOI is included, the enhanced Pauli limit is considerably suppressed, indicating that the symmetrical protection by spin–valley locking is weakened (Fig. 4c). This is because the inplane polarized spin components due to the Rashbatype SOI are much more susceptible to an external inplane magnetic field in comparison to the outofplane polarized spins due to the intrinsic Zeemantype SOI. The best agreement with the experimental data is obtained for a moderate Rashbatype SOI of 10% of the Zeemantype SOI, although such a Rashbatype SOI is unlikely according to the firstprinciplesbased band calculations, as mentioned above. We discuss three possible origins for this discrepancy between the theoretical results based on a singlelayer tightbinding model and the experimental results in Supplementary Section VI. In addition, according to our numerical calculations, which include the dependence on both the carrier density and T_{c}, the Pauli limit is predominantly controlled by both the Zeemantype SOI and T_{c}, and the contribution of the Rashbatype SOI is negligibly small, in the range of carrier density where superconductivity is realized in this system (Supplementary Fig. 7). These results demonstrate that, by the application of a strong electric field, MoS_{2}, which is believed to be a conventional superconductor in the intercalated bulk form, becomes an unconventional 2D Ising superconductor in which Cooper pairs are protected by Zeemantype spin–valley locking, and are thereby very robust against external magnetic fields, which results in the marked enhancement of the Pauli limit. Our findings therefore indicate that, combined with highly crystalline materials, the exotic properties of superconductivity are now accessible through geometrical confinement using strong electric fields, which suggests that electricfieldinduced superconductivity offers an ideal platform for unveiling the intrinsic nature of matter.
Note added in proof: We became aware of recent published similar experimental works on NbSe_{2} (ref. 51) and iongated MoS_{2} (ref. 52).
Methods
Device fabrication.
Bulk 2Hpolytype MoS_{2} single crystals were cleaved into thin flakes with tens of nanometres in thickness using the Scotchtape method. The flakes were then transferred onto Si/SiO_{2} substrates or Nbdoped SrTiO_{3}/HfO_{2} substrates. Au (90 nm)/Cr (5 nm) electrodes were patterned onto an isolated thin flake in a Hall bar configuration, and a side gate electrode was patterned onto the substrate. We covered the device with ZEP 520A (used as the resist for electron beam lithography), except for the channel surface, to avoid chemical intercalation from the edge of the flake, allowing us to focus on the field effect. A droplet of ionic liquid covered both the channel area and the gate electrode. The ionic liquid N, NdiethylN(2methoxyethyl)Nmethylammonium bis (trifluoromethylsulphonyl) imide (DEMETFSI) was selected as a gate medium.
Transport measurements.
The temperaturedependent resistance, under magnetic fields, of the MoS_{2}EDLT (shown in Fig. 2) was measured with a standard fourprobe geometry in a Quantum Design Physical Property Measurement System (PPMS) with a Horizontal Rotator Probe with an error below 0.01°, combined with two kinds of a.c. lockin amplifier (Stanford Research Systems Model SR830 DSP lockin amplifier and Signal Recovery Model 5210 lockin amplifier). The gate voltage was supplied by a Keithley 2400 sourcemeter. We applied gate voltages to the device at 220 K, which is just above the glass transition temperature of DEMETFSI, under high vacuum (less than 10^{−4} torr), and cooled down to low temperatures. The excitation source–drain current used in the PPMS setup was limited to 1 μA to avoid heating and largecurrent effects on the superconductivity.
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Acknowledgements
We thank T. Gokuden for technical support, and M. Yoshida and K. Kikutake for fruitful discussions. Y.S. and Y. Nakamura were supported by the Japan Society for the Promotion of Science (JSPS) through a Research Fellowship for Young Scientists. This work was supported by GrantinAid for Specially Promoted Research (no. 25000003) from JSPS and Grantin Aid for Scientific Research on Innovative Areas (no. 22103004) from MEXT of Japan.
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Y.S. and Y.I. conceived the idea and designed the experiments. Y.S., Y. Nakagawa and M.O. fabricated MoS_{2}EDLT devices. Y.S. conducted cryogenic transport measurements with the PPMS setup, and analysed the data. M.S.B. carried out ab initiobased tightbinding supercell calculations. Y. Nakamura performed numerical calculations of the upper critical field. Y.S., Y. Kasahara and Y. Kohama carried out highfield measurements in the Institute for Solid State Physics. J.Y. took leadership of the initial highfield experiment when he was in the University of Tokyo and RIKEN. M.T., Y. Kasahara and T.N. led physical discussions. Y.S., M.S.B., T.N., Y.Y. and Y.I. wrote the manuscript.
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Saito, Y., Nakamura, Y., Bahramy, M. et al. Superconductivity protected by spin–valley locking in iongated MoS_{2}. Nature Phys 12, 144–149 (2016). https://doi.org/10.1038/nphys3580
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DOI: https://doi.org/10.1038/nphys3580
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