## Locking the spins in a superconductor

In Cooper pairs—pairs of electrons responsible for the exotic properties of superconductors—the two electrons' spins typically point in opposite directions. A strong-enough external magnetic field will destroy superconductivity by making the spins point in the same direction. Lu *et al.* observed a two-dimensional superconducting state in the material MoS_{2} that was surprisingly immune to a magnetic field applied in the plane of the sample (see the Perspective by Suderow). The band structure of MoS_{2} and its spin-orbit coupling conspired to create an effective magnetic field that reinforced the electron pairing, with spins aligned perpendicular to the sample.

## Abstract

The Zeeman effect, which is usually detrimental to superconductivity, can be strongly protective when an effective Zeeman field from intrinsic spin-orbit coupling locks the spins of Cooper pairs in a direction orthogonal to an external magnetic field. We performed magnetotransport experiments with ionic-gated molybdenum disulfide transistors, in which gating prepared individual superconducting states with different carrier dopings, and measured an in-plane critical field *B*_{c2} far beyond the Pauli paramagnetic limit, consistent with Zeeman-protected superconductivity. The gating-enhanced *B*_{c2} is more than an order of magnitude larger than it is in the bulk superconducting phases, where the effective Zeeman field is weakened by interlayer coupling. Our study provides experimental evidence of an Ising superconductor, in which spins of the pairing electrons are strongly pinned by an effective Zeeman field.

In conventional superconductors, applying a sufficiently high magnetic field above the upper critical field *B*_{c2} is a direct way to destroy superconductivity by breaking Cooper pairs via the coexisting orbital and Pauli paramagnetic mechanisms. The orbital contribution originates from the coupling between the magnetic field and the electron momentum, whereas the paramagnetic contribution is caused by spin alignment in Cooper pairs by an external magnetic field. When the orbital effect is weakened or eliminated, either by having a large electron mass (*1*) or by reducing dimensionality (*2*), *B*_{c2} is solely determined by the interaction between the magnetic field and the spin degree of freedom of the Cooper pairs. In superconductors where Cooper pairs are formed by electrons with opposite spins, aligning the electron spins by the external magnetic field increases the energy of the system; therefore, *B*_{c2} cannot exceed the Clogston-Chandrasekhar limit (*3*, *4*) or the Pauli paramagnetic limit (in units of tesla), *B*_{p} ≈ 1.86 *T*_{c}(0). Here, *T*_{c}(0) is the zero-field superconducting critical temperature (in units of kelvin) that characterizes the binding energy of a Cooper pair, which competes with the Zeeman splitting energy.

However, in some superconductors, the Pauli limit can be surpassed. For example, forming Fulde-Ferrell-Larkin-Ovchinnikov states with inhomogeneous pairing densities favors the presence of a magnetic field, even above *B*_{p} (*5*). In spin-triplet superconductors, the parallel-aligned spin configuration in Cooper pairs is not affected by Pauli paramagentism, and *B*_{c2} can easily exceed *B*_{p} (*6*–*8*). Spin-orbit interactions have also been shown to align spins to overcome the Pauli limit. Rashba spin-orbit coupling (SOC) in noncentrosymmetric superconductors will lock the spin to the in-plane direction, which can greatly enhance the out-of-plane *B*_{c2} (*9*); however, for an in-plane magnetic field, *B*_{c2} can only be moderately enhanced to *B*_{p} (*10*). Alternatively, electron spins can be randomized by spin-orbit scattering (SOS), which weakens the effect of spin paramagnetism (*11*–*15*) and hence enhances *B*_{c2}.

Superconductivity in thin flakes of MoS_{2} can be induced electrostatically using the electric field effect, mediated by moving ions in a voltage-biased ionic liquid placed on top of the sample [section 1 of (*16*); (*17*)]. Negative carriers (electrons) are induced by accumulating cations above the outermost layer of an MoS_{2} flake, forming a capacitor ~1 nm thick (*17*–*22*). The potential gradient at the surface creates a planar homogenous electronic system with an inhomogeneous vertical doping profile, where conducting electrons are predominantly doped into a few of the outermost layers, forming superconducting states near the *K* and *K'* valleys of the conduction band (Fig. 1A). The in-plane inversion symmetry breaking in a MoS_{2} monolayer can induce SOC, manifested as a Zeeman-like effective magnetic field **B**_{eff} (~100 T) oppositely applied at the *K* and *K'* points of the Brillouin zone (*23*). Because electrons of opposite momentum experience opposite **B**_{eff}, this SOC is then compatible with Cooper pairs also residing at the *K* and *K'* points (*24*). Therefore, spins of electrons in the Cooper pairs are polarized by this large out-of-plane Zeeman field, which is able to protect their orientation from being realigned by an in-plane magnetic field, leading to a large in-plane *B*_{c2}. This alternating spin configuration also provides the essential ingredient for establishing an Ising superconductor, where spins of electrons in the Cooper pairs are strongly pinned by an effective Zeeman field in an Ising-like fashion.

Because of the alternating stacking order in 2*H*-type single crystals of transition metal dichalcogenide (TMD) (Fig. 1B), electrons with the same momentum experience **B**_{eff} with opposite signs for adjacent layers, which weakens the effect of SOC by cancelling out **B**_{eff} mutually in the bulk crystal (Fig. 1C) (a comparison with bulk intercalated TMD is given in section 7 of (*16*)]. However, field-effect doping can strongly confine carriers to the outermost layer, reaching a two-dimensional (2D) carrier density *n*_{2D} of up to ~10^{14} cm^{−2} (*17*, *25*). Theoretical calculations for our devices indicate that the *n*_{2D} of individual layers decays exponentially from the channel surface (Fig. 1D, left axis), reducing the *n*_{2D} of the second-to-outermost layer by almost 90% in comparison with the outermost one (*26*). From the established phase diagram (*17*), if superconductivity is induced close to the quantum critical point (QCP; *n*_{2D} ~ 6 × 10^{13} cm^{−2}), the second layer is not even metallic, because metallic transport can be observed only when *n*_{2D} > 8 × 10^{12} cm^{−2}. Therefore, the outermost layer is well isolated by gating, mimicking a freestanding monolayer (*27*).

We obtained superconducting states across a range of doping concentrations (Fig. 1D, right axis) by varying the gate voltage (*17*); these states have different temperature dependences of sheet resistivity *R*_{s} (Fig. 1E). A superconducting state [*T*_{c} (at *B* = 0) = 2.37 K] at the onset of superconductivity (close to QCP) could be induced without suffering from the inhomogeneity usually encountered at low doping concentrations (Fig. 1E, red curve). Consistently, this well-behaved state also exhibits a high mobility of ~700 cm^{2}/Vs (measured at *T* = 15 K) before reaching zero resistance.

Angle-resolved photoemission spectroscopy (ARPES) measurements (*27*, *28*) and theoretical calculations (*25*, *29*) both showed that electron doping starts near the *K* points of the conduction band. The band structure is modified at higher doping (*25*, *29*), meaning that the simplest superconducting states in MoS_{2}, which are dominated by Cooper pairs at the *K* and *K'* points, should be prepared by minimizing doping [higher doping states are discussed in section 7 of (*16*)].

The charge distribution of our gated system implies that the superconducting state thus formed should exhibit a purely 2D nature. To demonstrate this dimensionality, we have characterized sample D24 [with *T*_{c}(0) = 7.38 K] with a series of measurements. The temperature dependences of *R*_{s} under out-of- and in-plane magnetic fields (Fig. 2, A and B) are highly anisotropic. The angular dependence of *B*_{c2} at *T* = 6.99 K (Fig. 2D) was extracted from Fig. 2C. Curves fitted with the 2D Tinkham formula (red curve) (*30*) and the 3D anisotropic Ginzburg-Landau (GL) model (blue curve) (*2*) show that for θ > ± 1° (where θ is the angle between the *B* field and the MoS_{2} surface), the data are consistent with both models, whereas for θ* <* ± 1° (Fig. 2D, inset), the cusp-shaped dependence can only be explained with a 2D model. These measurements show that our system exhibits 2D superconductivity, similar to LaAlO_{3}/SrTiO_{3} interfaces (*31*) and ion-gated SrTiO_{3} surfaces (*32*). From the voltage-current (*V*-*I*) dependence at different temperatures close to *T*_{c}(0) (Fig. 2E), we determined that the Berezinskii-Kosterlitz-Thouless temperature *T*_{BKT} is 6.3 K for our 2D system (Fig. 2F). *V-I* characteristics in a magnetic field (fig. S3) exhibit similar critical behavior to the zero-field data, with their *T*_{BKT} values effectively reduced by increasing the magnetic field.

A moderate in-plane *B* field of up to 12 T shows little effect on the superconducting transition temperature [where *T*_{c}(0) = 7.38 K and the Pauli limit *B*_{P} = 13.7 T (Fig. 2B)]; thus, the *B*_{c2} of the system must be far above *B*_{P}. To confirm this, we performed a high field measurement up to 37 T [section 2 of (*16*)] on sample D1 after observing a steep increase in *B*_{c2} near *T*_{c}(0) = 5.5 K (Fig. 3C, green dots). By controlling the gating strength, superconducting states with *T*_{c}(0) = 2.37 and 7.64 K were induced in sample D1. For *T*_{c}(0) = 2.37 K, we obtained *B*_{c2} as the magnetic field required to reach 50% of the normal state resistivity (*R*_{N}) (Fig. 3A). *B*_{c2} is above 20 T at 1.46 K (Fig. 3C, red circles), which is more than four times the *B*_{P}. The data from the second gating [*T*_{c}(0) = 7.64 K (Fig. 3B)] show only a weak reduction of *T*_{c} by ~1 K at even the highest magnetic field, 32.5 T (~ 2.3 × *B*_{p}).

The temperature dependences of in-plane *B*_{c2} for sample D1 in three different states (Fig. 3C) are fitted using a phenomenological GL theory in the 2D limit (*2*) and the microscopic Klemm-Luther-Beasley (KLB) theory (*12*, *15*, *33*). The extrapolated zero-temperature in-plane *B*_{c2} is far beyond *B*_{p} for all three superconducting states. The zero-temperature *B*_{c2} predicted by 2D GL theory, without taking spin into account, is larger than that estimated by the KLB theory, which considers both the limiting effect from spin paramagnetism and the enhancing effect by the SOS from disorder. To fit the data using the KLB theory (dashed curves in Fig. 3C), the interlayer coupling has to be set to zero. This strongly suggests that the induced superconductivity is 2D, which is consistent with the conclusion drawn from (Fig. 2 and previous theoretical calculations (*17*, *26*) and ARPES measurements (*27*, *28*) regarding predominant doping in the outermost layer. Curves fitted with the KLB theory yield a very short SOS time of ~24 fs (fig. S5), which is less than the total scattering time of 185 fs estimated from resistivity measurements at 15 K (table S2) and much shorter than the estimation of nanoseconds calculated for MoS_{2} at the carrier density range accessed by this work (*34*). Short spin-orbit scattering times of ~40 to 50 fs have also been observed in organic molecule–intercalated TaS_{2} (*35*–*37*), (LaSe)_{1.14}(NbSe_{2}) (*38*, *39*), and the organic superconductor κ-(ET)_{4}Hg_{2.89}Br_{8} [ET, bis(ethylenedithio)tetrathiafulvalene] (*40*).

The temperature dependence of *B*_{c2} in bulk superconducting MoS_{2} intercalated by alkali metals (*41*) near *T*_{c}(0) is linear instead of square root (Fig. 3C). The slight upturn of *B*_{c2} toward lower temperatures away from *T*_{c}(0) is the evidence of crossover from 3D to 2D superconducting states (*12*, *33*, *36*–*38*) caused by the layered nature of the bulk crystal. In these bulk phases, the measured *B*_{c2} values are much smaller than or comparable (when Cs dopants are intercalated) to *B*_{p} (*41*).

This behavior is visualized in Fig. 3D, where the in-plane *B*_{c2} normalized by *B*_{p} for bulk superconducting phases falls within the shaded area bounded by the Pauli limit. In contrast, all gate-induced phases (from samples D1 and D24) are far above both *B*_{p} (dashed line) and bulk-phase *B*_{c2}. The D1 with *T*_{c}(0) = 2.37 K, which is separated from the other gate-induced states, exhibits the largest enhancement. If the large SOS rate extracted from the KLB fitting (Fig. 3C) were the reason for the enhancement of *B*_{c2} in gate-induced phases, we would expect it to also enhance *B*_{c2} in the bulk phases. The difference shown in Fig. 3D indicates that SOS is unlikely to be the origin of the enhancement of *B*_{c2} in the gated phases.

Excluding SOS as the principal mechanism for the strong enhancement of the in-plane *B*_{c2}, and taking into account recent developments in understanding the band structures of monolayer MoS_{2} (*42*, *43*), we propose that this *B*_{c2} enhancement is mainly caused by the intrinsic spin-orbit coupling in MoS_{2}. Near the *K* points of the Brillouin zone (Fig. 1A) and on the basis of spin-up and -down electrons , the normal-state Hamiltonian of monolayer MoS_{2} in the presence of an external field can be described by (*24*)

Here, denotes the kinetic energy with chemical potential μ; **k** = (*k _{x}*,

*k*, 0) is the kinetic momentum of electrons in the

_{y}*K*and

*K*′ valleys;

**K**is the kinetic momentum of the

*K*valley;

*m*is the effective mass of the electrons; σ = (σ

*, σ*

_{x}*, σ*

_{y}*) are the Pauli matrices;*

_{z}**g**

_{F}= (

*k*, –

_{y}*k*, 0) denotes the Rashba vector (lying in-plane); α

_{x}_{R}and β

_{SO}are the strength of Rashba and intrinsic SOC, respectively; ϵ = ±1 is the valley index (1 at the

*K*valley and –1 at the

*K'*valley); and

**b**= μ

_{B}

**B**is the external Zeeman ﬁeld (where μ

_{B}is the Bohr magneton). The intrinsic SOC term ϵβ

_{SO}σ

*, due to in-plane inversion symmetry breaking, induces an effective magnetic field pointing out of the plane (*

_{z}*z*direction), which has opposite signs at opposite valleys (green arrows in Fig. 1A). This Zeeman-like effective magnetic field

**B**

_{eff}= ϵβ

_{SO}/

*g*μ

_{B}(

*g*, gyromagnetic ratio; , unit vector in the out-of-plane direction) will only appear in our multilayered system after applying a strong electric field, which isolates the outermost layers from the other layers (

*17*,

*44*), thus mimicking a monolayer system. The large electric field generated by gating reaches ~50 million volts/cm (

*17*) in our system, causing additional out-of-plane inversion symmetry breaking and creating a Rashba-type effective magnetic field

**B**

_{Ra}= α

_{R}

**g**

_{F}/

*g*μ

_{B}.

The total energy in a magnetic field is schematically shown in Fig. 4, A to D. If the electron spin aligned by **B**_{eff} (**B**_{Ra}) stays parallel to the external magnetic field **B**_{ex} (Fig. 4, A and C), the system gains energy through coupling between spin and external fields as μ_{B}**B**_{ex}. Therefore, *B*_{c2} is limited by *B*_{p} (Fig. 4A), or it can reach *B*_{p} (Fig. 4C) when coupling is reduced in a Rashba-type spin configuration (*10*). When **B**_{eff} and **B**_{Ra} are perpendicular to **B**_{ex}, as respectively shown in Fig. 4, B and D, the spin aligned by both effective fields is orthogonal to **B**_{ex}. Hence, the coupling between spin and **B**_{ex} is minimized, and *B*_{c2} can easily surpass *B*_{p} in these two cases.

To theoretically describe our system when subjected to an in-plane external magnetic field (combining the cases shown in Fig. 4, B and C), we introduced the pairing potential terms into *H*(**k**) and solved the self-consistent mean field gap equation [section 6 of (*16*); *h.c.*, hermitian conjugate]. The in-plane *B*_{c2} for a sample with a given *T*_{c} can then be determined by including the intrinsic SOC term β_{SO} and the Rashba energy α_{R}*k*_{F}, where *k*_{F} is the Fermi momentum.

For the most extensive data set from sample D1 [*T*_{c}(0) = 2.37 K], the relationship between *B*_{c2}/*B*_{p} and the reduced temperature *T*/*T*_{c}, shown in Fig. 4E, can be fitted well with β_{SO} = 6.2 meV and α_{R}*k*_{F} = 0.88 meV. The value obtained for β_{SO} corresponds to an out-of-plane field of ~114 T, which is comparable to the value expected from theoretical calculation at the *K* point (3 meV) (*23*). The Rashba energy obtained can be regarded as an upper bound, because the present model does not include impurity scattering, which can also reduce *B*_{c2} (*45*).

The scale of *B*_{c2} enhancement is determined by a destructive interplay between intrinsic β_{SO} and α_{R}*k*_{F}. Reaching higher *T*_{c}(0) requires stronger doping under higher electric fields, with a concomitant increase of **B**_{Ra}. As a result of this competition, the in-plane *B*_{c2} protection should be weakened with the increase of *T*_{c}(0). To support this argument, we chose two other superconducting samples that showed consecutively higher *T*_{c}(0) (from D1 and D24). By assuming identical β_{SO} (6.2 meV), *B*_{c2} from D1 with *T*_{c}(0) = 5.5 K and* B*_{c2} from D24 with *T*_{c}(0) = 7.38 K can be well fitted using α_{R}*k*_{F} = 1.94 and 3.02 meV, respectively; these values are consistent with the expected increase of α_{R}*k*_{F} with *T*_{c}(0) (Fig. 4E).

The effective Zeeman field and its orthogonal protection in individual layers can also be induced by reducing the interlayer coupling in bulk superconducting TMDs (*33*, *35*, *38*, *46*, *47*). Therefore, a large in-plane *B*_{c2} was also observed in bulk when lattice symmetry was lowered by intercalating organic molecules and alkali elements with large radii (Cs-intercalated MoS_{2} shows the highest *B*_{c2} among bulk phases in Fig. 3D) or by forming a charge density wave (*46*).

We compared our *B*_{c2} results with those obtained from other superconductors with enhanced *B*_{c2} under their maximum spin protection along the labeled crystal axis (Fig. 4F); we found that the Zeeman field–protected states in our samples are among the states that are most robust against external magnetic fields. Given the very similar band structures found in 2*H*-type TMDs with universal Zeeman-type spin splitting and the recent successes in inducing more TMD superconductors using the field effect (*17*, *48*, *49*), we would expect a family of Ising superconductors in 2*H*-type TMDs. The concept of the Ising superconductor is also applicable to other layered systems, where similar intrinsic SOC could be induced by symmetry breaking.

## Supplementary Materials

www.sciencemag.org/content/350/6266/1353/suppl/DC1

Materials and Methods

Figs. S1 to S5

Tables S1 and S2

## References and Notes

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**Acknowledgments:**We acknowledge support from the High Field Magnet Laboratory Nijmegen (HFML-RU/FOM), a member of the European Magnetic Field Laboratory. J.T.Y. acknowledges funding from the European Research Council (consolidator grant no. 648855 Ig-QPD). U.Z. was supported by the DESCO program (2-Dimensional Electron Systems in Complex Oxides, program no. 149) of the Foundation for Fundamental Research on Matter, which is part of the Netherlands Organization for Scientific Research. K.T.L. and N.F.Q.Y. were supported by the Hong Kong Research Grants Council and the Croucher Foundation through grants HKUST3/CRF/13G, 602813, 605512, and 16303014 and an Innovation Grant.