Symmetry-breaking has been known to play a key role in non-centrosymmetric 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 right-angle, thereby leading to an in-plane helical spin texture. Here we discuss electric-field-induced superconductivity in molybdenum disulphide (MoS2), which exhibits a fundamentally different type of intrinsic SOI, manifested by an out-of-plane Zeeman-type spin polarization of energy valleys8,9,10. 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 tight-binding calculations, we reveal that this unusual behaviour is due to an inter-valley pairing that is symmetrically protected by Zeeman-type 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.
MoS2 is a member of the layered semiconducting transition metal dichalcogenides (TMDs; ref. 11), which have been attracting widespread attention as two-dimensional (2D) materials beyond graphene, owing to their multiple functionalities with potential applications such as atomically thin electronics12,13,14, photonics15 and valleytronics devices utilizing a coupled spin and valley degree of freedom16,17,18. Also, MoS2 is becoming a new platform for investigating quantum physics—for example, with quantum oscillations19 and electric-field-induced superconductivity20. The unit cell of MoS2 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 c-axis with D3h symmetry (Fig. 1a). In the isolated monolayer, in-plane inversion symmetry is broken, causing out-of-plane spin polarization together with effective Zeeman fields—namely, Zeeman-type spin polarization at zero magnetic field8,9,10,21. This Zeeman-type 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 zero-field spin splitting is not observed in bulk MoS2 with D6h4 symmetry23,24. Also, this spin splitting changes its sign at the −K point, because the K and −K points are connected by the time-reversal operation. Such a spin splitting unique to monolayer MoS2 originates from the fairly strong SOI of transition metal d-orbitals, and is commonly observed in the group VI of TMD semiconductors8,9. This valley-dependent spin polarization is in marked contrast to the in-plane momentum-dependent spin polarization caused by the Rashba-type SOI (ref. 7).
A non-centrosymmetric system with considerable SOIs is an ideal platform for exotic superconductivity—in fact, superconductivity occurring in the Rashba-type band structure has been intensively investigated on a variety of systems1,2,3,4, together with the effect of spin–momentum locking. However, the effect of Zeeman-type spin polarization on superconductivity has not been discussed previously. Here, we investigate electric-field-induced superconductivity in MoS2 by using an electric-double-layer transistor (EDLT) configuration (Fig. 1c), which creates a high-density two-dimensional electron system (2DES) on the surface (Fig. 1d) without introducing extrinsic disorder, thereby offering novel opportunities to search for new types of exotic superconductivity25,26,27.
To extract the anomalous features of electric-field-induced superconductivity at the highly crystalline multilayer MoS2 surface, we fabricated an EDLT structure with a 20-nm-thick flake, and then performed magneto-transport measurements. The MoS2-EDLT underwent a superconducting transition at a gate voltage of VG = 6.5 V and a sheet carrier density of n2D = 1.5 × 1014 cm−2 measured at 15 K (Fig. 2a). The critical temperature, Tc, of this device was 9.7 K, as defined at the midpoint of the transition, with Rsheet being 50% of the normal state sheet resistance at 15 K. This carrier density is slightly larger than the optimum value in the dome-shaped phase diagram20,28,29. 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). Zoom-ins to the resistive transition in the low-temperature 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 in-plane 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, Hc2(θ), at 9.6 K (θ represents the angle between the c-axis of the crystal and the direction of the applied magnetic fields). Here, Hc2(θ) is also determined by the midpoint of the resistive transition. A cusp-like peak in the inset of Fig. 2d is described well by the 2D Tinkham model30 (and not by the 3D anisotropic mass model) as frequently observed in interfacial superconductivity31,32,33. In addition, the dependence of Hc2 on the temperature T for both the out-of-plane and in-plane magnetic fields (Fig. 2e) are fitted well by the phenomenological 2D Ginzburg–Landau (GL) model, and where ϕ0, ξGL(0) and dSC denote a flux quantum, the in-plane GL coherence length at T = 0 K, and the effective thickness of superconductivity, respectively. We find ξGL(0) = 8.0 nm and dSC = 1.5 nm. Note that the extremely sharp rise of Hc2(T) near Tc shows a marked contrast to that in conventional bulk layered superconductors such as Cs-doped MoS2 (ref. 34), demonstrating that the present system is extremely 2D in nature. In fact, Hc2(T) can seemingly go far beyond the Pauli limit, HPBCS, for weak coupling Bardeen–Cooper–Schrieffer (BCS) superconductors, T, where kB and Δ0 are the Boltzmann constant and the BCS-theory-based superconducting gap at T = 0 K, respectively.
To investigate Hc2 at much lower temperatures, we measured the magnetoresistance of another MoS2-EDLT by applying pulsed magnetic fields up to 55 T (see Supplementary Section II and Supplementary Fig. 2). A clear resistance drop at a Tc of 6.5 K was observed, which was defined as the temperature where Rsheet reached 75% of the normal state sheet resistance, indicating a superconducting signature, although the MoS2-EDLT used for the measurements in high magnetic fields (n2D = 8.5 × 1013 cm−2 at VG = 5.5 V and T = 15 K) did not exhibit zero resistance. The magnetoresistance of the MoS2-EDLT is shown in Fig. 3a and b for out-of-plane and in-plane magnetic fields, respectively, at several temperatures between 1.5 and 8.0 K. In the out-of-plane 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 in-plane 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 Hc2(T) and Hc2⊥(T) in Fig. 3c. We note that Hc2(T) increases with decreasing temperature and eventually saturates at approximately 52 T at 1.5 K, which is more than four times larger than μ0HPBCS = 12 T. Because the orbital limit is supposed to be large owing to confinement of the geometry by the EDLT, the saturating behaviour of Hc2(T) at low temperatures is suggestive of the Pauli limit, as seen in the Pauli-limited superconductor35.
The enhancement of Hc2 in a dirty-limit 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 paramagnetism36,37,38. To evaluate the contribution of this effect, we fitted our Hc2(T) data by using the microscopic Klemm–Luther–Beasely (KLB) theory38, which is applicable to dirty-limit layered superconductors with strong SOIs (l ≪ ξPippard and τ ≪ τSO, where l, ξPippard, τ and τSO are the mean free length, the Pippard coherence length, the total scattering time and the spin–orbit scattering time, respectively). Our Hc2(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 τSO(τ > τSO) (Supplementary Table 1). This is an unphysical situation which contradicts with the initial assumption required for this theory (τ ≪ τSO). Thus, the model with the effect of spin–orbit scattering does not explain the enhancement of Hc2 consistently.
To find a more plausible origin of the enhancement of the Pauli limit in the present system, we first performed a set of ab-intio-based tight-binding supercell calculations on bulk MoS2, incorporating the near-surface band bending effect via an electrostatic potential term obtained by self-consistently solving the Poisson equation (details in Supplementary Section IV). Our calculations suggest that, under the application of a strong electric field, a high-density 2DES is created at the surface of MoS2. As schematically shown in Fig. 1d, this results in the formation of an accumulation layer, which is effectively confined within the topmost MoS2 layer39,40, indicating that non-centrosymmetric quasi-single-layer superconductivity is realized in our system. Such a “quasi-single-layer” 2DES, therefore, ought to have an effective D3h 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 points40. This situation is in stark contrast to the case of bulk MoS2, where the conduction band minima are located at six symmetrically equivalent k points along Γ–K directions, also known as the T (or Q) points23,24. Accordingly, the electric-field-induced 2D superconductivity in MoS2 is expected to be solely mediated by the ±K valleys, and thus the most likely ground state of the Cooper pair should be the inter-valley pairing between the +K and −K valleys to maintain zero momentum for the centre-of-mass of the Cooper pairs. Note that the intra-valley spin-singlet Cooper pairs are not stabilized in the presence of the Zeeman-type SOI, which requires non-zero momentum.
At a sheet carrier density of n2D = 8.7 × 1013 cm−2, which is nearly the same as the value in the high-field measurement, the bands are spin-split by ∼3 meV at the ±K points, at zero magnetic field. Slightly away from the K point, these spin-split 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 MoS2 derived from the tight-binding method41 and the k ⋅ p model42. This agreement indicates that bulk or multilayer TMDs under a strong electric field can effectively behave as monolayers. Such a monolayer-like behaviour has already been experimentally demonstrated in bilayer systems, exhibiting circularly polarized photoluminescence under an electric field43, and bulk systems showing gate-induced weak anti-localization behaviour in magnetoconductance21. In addition to these works, recent optical measurements on WSe2 multilayers have shown that these systems can emit an electrically switchable circularly polarized electroluminescence16. 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 MoS2 can behave like a monolayer under an electric field.
As shown in Fig. 4a, each band is almost fully out-of-plane spin polarized. The in-plane Rashba-type component, which originates from the asymmetric potential along the c-axis 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 in-plane component is allowed at the K points owing to their three-fold rotational (C3) symmetry44,45. In the presence of the finite Rashba-type SOI, a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state4,5,6 (a helical state46), where Cooper pairs have non-zero momentum, with s + f-wave symmetry47, 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 spin-triplet components derived from Zeeman-type 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 intra-valley pairing. Also, as other possibilities for the enhancement of Hc2, Rashba-type SOI (refs 1,2,3), quantum critical point49 and modified electron g-factor1,30 can be ruled out in the present system (details in Supplementary Section III). Therefore, spin–valley locking due to intrinsic Zeeman-type SOI is considered to be the most promising origin for the enhancement of Hc2.
We theoretically estimated the realistic Pauli limit of the present system by considering both the Zeeman-type and the small Rashba-type SOIs. For this purpose, we constructed a simpler tight-binding model reproducing the 2DES subband structure shown in Fig. 4b (see Supplementary Section IV and Supplementary Fig. 4). Assuming isotropic s-wave 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 structure50 (see Supplementary Section V). Figure 4c shows the theoretical curves of the Pauli limit in this system. Considering only the Zeeman-type 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 valley-dependent Zeeman-type 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 out-of-plane spin polarization to the two opposite directions, referred to as inter-valley Ising pairing, enhances Hc2 much more than the HPBCS.
By contrast, once the small Rashba-type 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 in-plane polarized spin components due to the Rashba-type SOI are much more susceptible to an external in-plane magnetic field in comparison to the out-of-plane polarized spins due to the intrinsic Zeeman-type SOI. The best agreement with the experimental data is obtained for a moderate Rashba-type SOI of 10% of the Zeeman-type SOI, although such a Rashba-type SOI is unlikely according to the first-principles-based band calculations, as mentioned above. We discuss three possible origins for this discrepancy between the theoretical results based on a single-layer tight-binding 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 Tc, the Pauli limit is predominantly controlled by both the Zeeman-type SOI and Tc, and the contribution of the Rashba-type 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, MoS2, 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 Zeeman-type 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 electric-field-induced superconductivity offers an ideal platform for unveiling the intrinsic nature of matter.
Bulk 2H-polytype MoS2 single crystals were cleaved into thin flakes with tens of nanometres in thickness using the Scotch-tape method. The flakes were then transferred onto Si/SiO2 substrates or Nb-doped SrTiO3/HfO2 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, N-diethyl-N-(2-methoxyethyl)-N-methylammonium bis (trifluoromethylsulphonyl) imide (DEME-TFSI) was selected as a gate medium.
The temperature-dependent resistance, under magnetic fields, of the MoS2-EDLT (shown in Fig. 2) was measured with a standard four-probe 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. lock-in amplifier (Stanford Research Systems Model SR830 DSP lock-in amplifier and Signal Recovery Model 5210 lock-in 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 DEME-TFSI, under high vacuum (less than 10−4 torr), and cooled down to low temperatures. The excitation source–drain current used in the PPMS set-up was limited to 1 μA to avoid heating and large-current effects on the superconductivity.
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 Grant-in-Aid for Specially Promoted Research (no. 25000003) from JSPS and Grant-in Aid for Scientific Research on Innovative Areas (no. 22103004) from MEXT of Japan.