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Evidence for two-dimensional Ising superconductivity in gated MoS2

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Science  11 Dec 2015:
Vol. 350, Issue 6266, pp. 1353-1357
DOI: 10.1126/science.aab2277

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 MoS2 that was surprisingly immune to a magnetic field applied in the plane of the sample (see the Perspective by Suderow). The band structure of MoS2 and its spin-orbit coupling conspired to create an effective magnetic field that reinforced the electron pairing, with spins aligned perpendicular to the sample.

Science, this issue p. 1353; see also p. 1316

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 Bc2 far beyond the Pauli paramagnetic limit, consistent with Zeeman-protected superconductivity. The gating-enhanced Bc2 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 Bc2 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), Bc2 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, Bc2 cannot exceed the Clogston-Chandrasekhar limit (3, 4) or the Pauli paramagnetic limit (in units of tesla), Bp ≈ 1.86 Tc(0). Here, Tc(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 Bp (5). In spin-triplet superconductors, the parallel-aligned spin configuration in Cooper pairs is not affected by Pauli paramagentism, and Bc2 can easily exceed Bp (68). 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 Bc2 (9); however, for an in-plane magnetic field, Bc2 can only be moderately enhanced to Embedded Image Bp (10). Alternatively, electron spins can be randomized by spin-orbit scattering (SOS), which weakens the effect of spin paramagnetism (1115) and hence enhances Bc2.

Superconductivity in thin flakes of MoS2 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 MoS2 flake, forming a capacitor ~1 nm thick (1722). 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 MoS2 monolayer can induce SOC, manifested as a Zeeman-like effective magnetic field Beff (~100 T) oppositely applied at the K and K' points of the Brillouin zone (23). Because electrons of opposite momentum experience opposite Beff, 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 Bc2. 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.

Fig. 1 Inducing superconductivity in thin flakes of MoS2 by gating.

(A) Conduction-band electron pockets near the K and K' points in the hexagonal Brillouin zone of monolayer MoS2. Electrons in opposite K and K' points experience opposite effective magnetic fields Beff and –Beff, respectively (green arrows). The blue and red colored pockets indicate electron spins oriented up and down, respectively. (B) Side view (left) and top view (right) of the four outermost layers in a multilayered MoS2 flake. The vertical dashed lines show the relative positions of Mo and S atoms in 2H-type stacking. In-plane inversion symmetry is broken in each individual layer, but global inversion symmetry is restored in bulk after stacking. (C) Energy-band splitting caused by Beff. Blue and red bands denote spins aligned up and down, respectively. Because of 2H-type stacking, adjacent layers have opposite Beff at the same K points. (D) The red curve (left axis) denotes the theoretical carrier density n2D for the four outermost layers of MoS2 (26) for sample D1, when Tc(0) = 2.37 K. In the phase diagram (right axis), superconducting states with different values of Tc(0) are color-coded; the same color-coding is used across all figures. Here, Tc is determined at the temperature where the resistance drop reaches 90% of RN at 15 K. This criterion is different from the 50% RN criterion used in the rest of the paper; it was chosen to be consistent with that used in the phase diagram of (17). (E) Temperature dependence of Rs, showing different values of Tc corresponding to superconducting states (from samples D1 and D24) denoted in (D).

Because of the alternating stacking order in 2H-type single crystals of transition metal dichalcogenide (TMD) (Fig. 1B), electrons with the same momentum experience Beff with opposite signs for adjacent layers, which weakens the effect of SOC by cancelling out Beff 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 n2D of up to ~1014 cm−2 (17, 25). Theoretical calculations for our devices indicate that the n2D of individual layers decays exponentially from the channel surface (Fig. 1D, left axis), reducing the n2D 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; n2D ~ 6 × 1013 cm−2), the second layer is not even metallic, because metallic transport can be observed only when n2D > 8 × 1012 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 Rs (Fig. 1E). A superconducting state [Tc (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 cm2/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 MoS2, 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 Tc(0) = 7.38 K] with a series of measurements. The temperature dependences of Rs under out-of- and in-plane magnetic fields (Fig. 2, A and B) are highly anisotropic. The angular dependence of Bc2 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 MoS2 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 LaAlO3/SrTiO3 interfaces (31) and ion-gated SrTiO3 surfaces (32). From the voltage-current (V-I) dependence at different temperatures close to Tc(0) (Fig. 2E), we determined that the Berezinskii-Kosterlitz-Thouless temperature TBKT 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 TBKT values effectively reduced by increasing the magnetic field.

Fig. 2 2D superconductivity in gated MoS2 (sample D24).

Temperature dependence of Rs under a constant out-of-plane (A) and in-plane (B) magnetic field, up to 12 T. In (B), the left inset shows a close-up view of the data near RN/2 within 1 K. In the right inset, θ is the angle between the B field and the MoS2 surface. (C) Angular dependence of Rs, where the dashed line denotes Rs = RN/2. In the inset, the data are shown in detail within ±1° of the in-plane field configuration (θ = 0°). (D) Angular dependence of Bc2, which is fitted by both the 2D Tinkham model (red) and the 3D anisotropic GL model (blue). In the inset, the angular dependence of Bc2 is shown in detail within ±1° of the in-plane field configuration (θ = 0°). (E) The V-I relationship at different temperatures close to Tc, plotted on a logarithmic scale. The black lines are fits close to metal-superconductor transitions. The long black line denotes V I3, which gives TBKT. (F) Temperature dependence of α from fitting the power law dependence of V Iα from the black lines in (E). TBKT = 6.3 K is obtained for α = 3.

A moderate in-plane B field of up to 12 T shows little effect on the superconducting transition temperature [where Tc(0) = 7.38 K and the Pauli limit BP = 13.7 T (Fig. 2B)]; thus, the Bc2 of the system must be far above BP. 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 Bc2 near Tc(0) = 5.5 K (Fig. 3C, green dots). By controlling the gating strength, superconducting states with Tc(0) = 2.37 and 7.64 K were induced in sample D1. For Tc(0) = 2.37 K, we obtained Bc2 as the magnetic field required to reach 50% of the normal state resistivity (RN) (Fig. 3A). Bc2 is above 20 T at 1.46 K (Fig. 3C, red circles), which is more than four times the BP. The data from the second gating [Tc(0) = 7.64 K (Fig. 3B)] show only a weak reduction of Tc by ~1 K at even the highest magnetic field, 32.5 T (~ 2.3 × Bp).

Fig. 3 Determining the in-plane upper critical field Bc2 at different Tc (samples D1 and D24).

(A) Magnetoresistance of sample D1 [with Tc(0) = 2.37 K near the onset of the superconducting phase] as a function of an in-plane magnetic field up to 37 T, at various temperatures. (B) Temperature dependence of Rs for sample D1 [with Tc(0) = 7.64 K] under different in-plane magnetic fields up to 32.5 T. The dashed lines in (A) and (B) indicate RN/2. Bc2 is determined as the intercept between dashed lines and Rs curves. (C) Temperature dependence of Bc2 for superconducting states induced in sample D1 with different Tc [solid circles; colors follow (D)]. The Bc2 for alkali metal–intercalated bulk MoS2 compounds is from (41) and is shown for comparison. The Bc2 for gate-induced states is fitted as a function of temperature using the 2D GL (solid line) and KLB (dashed line) models. (D) Bc2 normalized by Bp, as a function of reduced temperature T/Tc, including superconducting states from alkali-doped bulk phases and gated-induced phases (samples D1 and D24). The dashed line denotes Bp and sets the boundary of the Pauli limited regime (shaded).

The temperature dependences of in-plane Bc2 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 Bc2 is far beyond Bp for all three superconducting states. The zero-temperature Bc2 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 MoS2 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 TaS2 (3537), (LaSe)1.14(NbSe2) (38, 39), and the organic superconductor κ-(ET)4Hg2.89Br8 [ET, bis(ethylenedithio)tetrathiafulvalene] (40).

The temperature dependence of Bc2 in bulk superconducting MoS2 intercalated by alkali metals (41) near Tc(0) is linear instead of square root (Fig. 3C). The slight upturn of Bc2 toward lower temperatures away from Tc(0) is the evidence of crossover from 3D to 2D superconducting states (12, 33, 3638) caused by the layered nature of the bulk crystal. In these bulk phases, the measured Bc2 values are much smaller than or comparable (when Cs dopants are intercalated) to Bp (41).

This behavior is visualized in Fig. 3D, where the in-plane Bc2 normalized by Bp 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 Bp (dashed line) and bulk-phase Bc2. The D1 with Tc(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 Bc2 in gate-induced phases, we would expect it to also enhance Bc2 in the bulk phases. The difference shown in Fig. 3D indicates that SOS is unlikely to be the origin of the enhancement of Bc2 in the gated phases.

Excluding SOS as the principal mechanism for the strong enhancement of the in-plane Bc2, and taking into account recent developments in understanding the band structures of monolayer MoS2 (42, 43), we propose that this Bc2 enhancement is mainly caused by the intrinsic spin-orbit coupling in MoS2. Near the K points of the Brillouin zone (Fig. 1A) and on the basis of spin-up and -down electrons Embedded Image, the normal-state Hamiltonian of monolayer MoS2 in the presence of an external field can be described by (24)

Embedded Image(1)

Here, Embedded Image denotes the kinetic energy with chemical potential μ; k = (kx, ky, 0) is the kinetic momentum of electrons in the K and K′ valleys; K is the kinetic momentum of the K valley; m is the effective mass of the electrons; σ = (σx, σy, σz) are the Pauli matrices; gF = (ky, –kx, 0) denotes the Rashba vector (lying in-plane); αR and βSO 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 = μBB is the external Zeeman field (where μB is the Bohr magneton). The intrinsic SOC term ϵβSOσz, due to in-plane inversion symmetry breaking, induces an effective magnetic field pointing out of the plane (z direction), which has opposite signs at opposite valleys (green arrows in Fig. 1A). This Zeeman-like effective magnetic field Beff = ϵβSOEmbedded Image/gμB (g, gyromagnetic ratio; Embedded Image, 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 BRa = αRgF/gμB.

The total energy in a magnetic field is schematically shown in Fig. 4, A to D. If the electron spin aligned by Beff (BRa) stays parallel to the external magnetic field Bex (Fig. 4, A and C), the system gains energy through coupling between spin and external fields as μBBex. Therefore, Bc2 is limited by Bp (Fig. 4A), or it can reach Embedded Image Bp (Fig. 4C) when coupling is reduced in a Rashba-type spin configuration (10). When Beff and BRa are perpendicular to Bex, as respectively shown in Fig. 4, B and D, the spin aligned by both effective fields is orthogonal to Bex. Hence, the coupling between spin and Bex is minimized, and Bc2 can easily surpass Bp in these two cases.

Fig. 4 Interplay between an external magnetic field and the spins of Cooper pairs aligned by Zeeman and Rashba-type effective magnetic fields.

(A to D) Illustration of the acquisition of Zeeman energy through coupling between an external magnetic field and the spins of Cooper pairs formed near the K and K' points of the Brillouin zone (not to scale). When Rashba or Zeeman SOC aligns the spins of Cooper pairs parallel to the external field, the increase in Zeeman energy due to parallel coupling between the field and the spin eventually can cause the pair to break [(A) and (C)]. In (B) and (D), the acquired Zeeman energy is minimized as a result of the orthogonal coupling between the field and the aligned spins, which effectively protects the Cooper pairs from depairing. (E) Theoretical fitting of the relationship between Bc2/Bp and T/Tc for samples D1 [Tc(0) = 2.37 K and 5.5 K] and D24 [Tc(0) = 7.38 K], using a fixed effective Zeeman field (βSO = 6.2 meV) and an increasing Rashba field (αRkF ranges from 10 to ~50% of βSO) [section 6 of (16)]. Two dashed lines show the special cases calculated by equation S3, when only the Rashba field (αRkF = 30 meV; βSO = 0) is considered (red), and when both the Zeeman and Rashba fields are zero (black). In the former case, a large αRkF causes a moderate increase of Bc2 to ~Embedded ImageBp (10). In the latter case, the conventional Pauli limit at zero temperature is recovered. (F) Plot of Bc2 versus Tc for different superconductors [a magnetic field was applied along crystal axes a, b, or c or to a polycrystalline (poly)]. The data shown are from well-known systems including noncentrosymmetric (pink circles), triplet (purple squares) (6, 8, 9), low-dimensional organic (green triangles) (40, 5052), and bulk TMD superconductors (blue triangles) (3538, 47). The robustness of the spin protection can be measured by the vertical distance between Bc2 and the red dashed line denoting Bp. Gate-induced superconductivity from samples D1 and D24 are among the states with the highest Bc2/Bp ratio. In (LaSe)1.14(NbSe2), Tc was determined at 95% of RN; Tc in organic molecule–intercalated TMDs was obtained by extrapolating to zero resistance; and all other systems use the standard of 50% of RN.

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 Embedded Image into H(k) and solved the self-consistent mean field gap equation [section 6 of (16); h.c., hermitian conjugate]. The in-plane Bc2 for a sample with a given Tc can then be determined by including the intrinsic SOC term βSO and the Rashba energy αRkF, where kF is the Fermi momentum.

For the most extensive data set from sample D1 [Tc(0) = 2.37 K], the relationship between Bc2/Bp and the reduced temperature T/Tc, shown in Fig. 4E, can be fitted well with βSO = 6.2 meV and αRkF = 0.88 meV. The value obtained for βSO 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 Bc2 (45).

The scale of Bc2 enhancement is determined by a destructive interplay between intrinsic βSO and αRkF. Reaching higher Tc(0) requires stronger doping under higher electric fields, with a concomitant increase of BRa. As a result of this competition, the in-plane Bc2 protection should be weakened with the increase of Tc(0). To support this argument, we chose two other superconducting samples that showed consecutively higher Tc(0) (from D1 and D24). By assuming identical βSO (6.2 meV), Bc2 from D1 with Tc(0) = 5.5 K and Bc2 from D24 with Tc(0) = 7.38 K can be well fitted using αRkF = 1.94 and 3.02 meV, respectively; these values are consistent with the expected increase of αRkF with Tc(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 Bc2 was also observed in bulk when lattice symmetry was lowered by intercalating organic molecules and alkali elements with large radii (Cs-intercalated MoS2 shows the highest Bc2 among bulk phases in Fig. 3D) or by forming a charge density wave (46).

We compared our Bc2 results with those obtained from other superconductors with enhanced Bc2 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 2H-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 2H-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 (5365)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. 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.
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