Introduction

Transition metal dichalcogenides have been the focus of recent research due to their wide range of unique electronic properties in their 2D form with high potential for technological applications1,2,3,4,5,6,7,8. Among them are intrinsic superconductors, such as 2H-NbSe2 and 2H-NbS2. In their 2D form, they have aroused great interest due to the discovery of Ising superconductivity, which allows them to exceed the Pauli limit of superconductivity4,5. While the upper critical field (Hc2) of a spin-singlet type-II superconductor is normally determined by the orbital limit for superconductivity9, there are rare cases where the orbital limit is particularly high. Possible reasons are heavy effective masses of the quasiparticles10,11,12, or a highly anisotropic structure that suppresses the orbital limit due to the open nature of the Fermi surface13,14,15,16,17,18,19,20,21,22,23,24. In this case, superconductivity is in principle abruptly destroyed at the Pauli limit25,26 in the form of a first-order phase transition. There are two ways in which a superconductor can maintain its superconducting (SC) state above the Pauli limit. The first possibility is the formation of a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state27,28, in which the Cooper pairs obtain a finite center-of-mass momentum, resulting in a spatially modulated order parameter in a wide range of their magnetic field vs. temperature phase diagram, which can extend far beyond the Pauli limit. Such spatial modulation of superconductivity has been described as a pair density wave state and is also used to explain the pseudogap phase in cuprates29. Prominent examples of the FFLO state have been found in some layered organic superconductors13,14,15,16,17,18,19,20,21,22,23, but also in form of the prominent Q-phase in the heavy-fermion compound CeCoIn5, where a spatially modulated order parameter coexists with a magnetic spin density wave order10,11,12, and more recently in the iron-based superconductor KFe2As224, and FeSe30. It is generally assumed that the superconductor must be in the clean limit for the FFLO state to be realized, with a mean-free path ℓ longer than the coherence length ξ31. Given the high purity of contemporary dichalcogenide samples and very short ξ values, this criterion is certainly achieved here.

For 2D dichalcogenides, Ising superconductivity is another possibility4,5. Here, the breaking of the mirror symmetry in the plane leads to a very strong pinning of the electron spins out of the plane due to the Ising-Spin-Orbit interaction (ISOI), resulting in opposite spin directions in the adjacent K and K’ electron pockets. This ISOI effectively protects the Cooper pairs when an in-plane magnetic field is applied to the 2D layer, resulting in enormous enhancements of the critical field. This phase is of particular interest due to the recent theoretical prediction of a topological SC state with Majorana zero modes in high parallel applied fields in monolayer NbSe232,33.

While reports on Ising superconductivity usually focus on monolayers, it is also clear that Hc2 in 2D materials with multiple layers still exceeds the Pauli limit5. It is not clear whether ISOI could still have an influence in the bulk form of these layered materials with very weak interlayer coupling. In this letter, we present magnetic torque, specific heat, and thermal expansion measurements on a bulk single crystal of NbS2, where we align the layered structure parallel to the applied magnetic field using a piezo rotary stage with millidegree accuracy. We find that Hc2 significantly exceeds the Pauli limit of 10 T at low temperatures and shows a pronounced characteristic upswing towards low temperatures. This is a clear indication that an unusual SC state is formed14,15,24,34. Thermal expansion measurements as a bulk thermodynamic method, which are closely related to the specific heat, indicate a small transition anomaly near the Pauli limit indicating a phase transition within the SC state. Using theoretical simulations35, we show that Ising superconductivity is not accountable for such a strong Hc2 upturn in bulk NbS2. The existence of the additional phase transition within the SC state points to a formation of the FFLO state.

Results

Magnetic torque experiments

Figure 1a shows magnetic torque magnitude (τ) measured at various fixed temperatures with a small 1° misalignment of the field with respect to the basal plane of NbS2. At lower temperatures, when Hc2 reaches higher fields, it is obvious that the transition becomes sharper and more step-like, which may be an indication for Pauli-limited first-order behavior15,24. Only a small tail persists above the step-shaped transition, which indicates a certain persistence of superconductivity. Hc2 reaches a maximum at 1.15 K and then decreases again slightly towards 0.35 K. Here we use the fields where the steepest slope occurs to mark the transition fields that should best represent the abrupt first-order-like transition. Note that other criteria (e.g., the transition onset) only lead to an almost parallel shift of the Hc2 line. To achieve a strictly parallel field orientation, we aligned the layered structure of the NbS2 single crystal by first minimizing τ at a fixed field and temperature by gradually rotating the sample through the parallel orientation24. This allowed us to determine the approximate parallel orientation. Subsequently, we repeated field scans at tiny variations of orientation of 0.1° or less until we found torque data with minimum amplitude and minimal opening of the hysteresis loop. This corresponded to the parallel orientation. A sequence of measurements at small angular variation is shown in Fig. 1b. As the field decreases, the torque builds up much more continuously below Hc2, suggesting that the sharp jumps in Fig. 1a are due to screening currents that build up continuously during the field sweep in the SC state and decay abruptly when approaching the Pauli limit in slightly tilted fields.

Fig. 1: Magnetic torque data in a slightly tilted field of NbS2.
figure 1

a Magnetic torque magnitude τ (H) of NbS2, measured at fixed temperatures as a function of the applied magnetic field H, oriented with a small angle θ = 1° to the basal plane. The data were measured at increasing field. Crosses mark the points where the steepest slope occurs near Hc2. b Magnetic torque measured at T = 0.35 K at various angles near θ = 0° (parallel field). A weak linear normal state contribution was removed from all data and offsets were added for better clarity.

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In Fig. 2a, we show τ data measured at different fixed temperatures during field sweeps for the field applied parallel to the NbS2 basal plane. Here we identify the Hc2 transition from the onset point where the data starts to deviate from zero, as marked by the additional open circles, which is the only sharp feature that occurs. In the same field, a hysteretic opening of the two branches, which was recorded when the field was swept up and down, confirms the onset of superconductivity. At 350 mK Hc2 exceeds our highest field of 15 T, and a hysteresis exists up to 15 T (Fig. 2b).

Fig. 2: Magnetic torque data in parallel fields of NbS2.
figure 2

a Magnetic torque magnitude τ (H) measured at fixed temperatures as a function of the magnetic field applied strictly parallel to the basal plane (θ = 0°) near the onset upper critical field Hc2 (open circles). The solid lines represent data measured at increasing field, while the dotted lines are data measured at decreasing field (overlapping parts of the data for low fields have been omitted and offsets have been added for reasons of clarity). b Magnetic torque data at 0.35 K where Hc2 exceeded the highest field in our magnet cryostat. c Magnetic torque measured in fixed parallel fields as a function of temperature. The data were normalized and a weak linear normal state background was removed for clarity. Open circles mark the onset critical temperatures.

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In Fig. 2c, we present torque data measured in constant, strictly parallel fields as a function of temperature. The SC transition can be identified as a step-like transition. The data were normalized by the jump size at Tc(H) for better clarity. We take the upper onset of the transition to identify Tc(H) as marked by the additional open circles. Note that other criteria to define of Tc(H) or Hc2(T) provide qualitatively similar phase diagrams.

Specific heat and thermal expansion experiments

In Figs. 3 and 4, we summarize our specific heat and linear thermal expansion data. To achieve the parallel field orientation, we maximized the Tc in a fixed field of a few Tesla by slightly rotating the sample in repeated measurements24. Figure 3a illustrates the specific heat C/T and the linear thermal expansion coefficient \(\alpha =(1/L){\rm{d}}L(T)/{\rm{d}}T\) in a parallel field of 15 T. Both thermodynamic quantities, which are closely related via the thermodynamic Ehrenfest relationship, show a SC transition, which demonstrates that 15 T is not sufficient to suppress superconductivity. In the inset (Fig. 3b), we show specific heat data measured with our ac modulated temperature technique measured during field sweeps at constant temperatures of the thermal bath. A standard BCS superconductor would show the characteristic step-like transition at Hc236. However, the NbS2 data only display a broad bump centered at relatively low fields, while the signal fades very gradually towards higher fields to approach the normal state. There are no additional features to be seen here that could indicate additional phase transitions within the SC state. However, in Fig. 4a, we illustrate the magnetic field derivative of the 300 mK specific heat data together with the linear magnetostriction coefficient \(\lambda \,=\,(1/L){\rm{d}}L(H)/{\rm{d}}H\). For both quantities, a small anomaly indicates a phase transition within the SC state at ~10 T, which occurs at the theoretical Pauli limit. The transition is reproducible when the field is swept up and down. In the following, we will attribute it to the transition that separates the ordinary low-field SC state from a high-field FFLO state. In the magnetic torque τ(Η) in the temperature range of 0.3–2.5 K tiny anomalies are hidden in the large slope near 10 T, but become visible after subtraction of a linear background in the form of a downward step, or as a dip-like structure in the field derivative dτ(H)/dH (Fig. 4b).

Fig. 3: Thermodynamic bulk probes displaying the upper critical field Hc2 transition of NbS2 in high parallel magnetic fields.
figure 3

a Specific heat C/T (circles) and linear thermal expansion coefficient \(\alpha =(1/L){\rm{d}}L(T)/{\rm{d}}T\) (squares) of NbS2 showing the superconducting transition in a magnetic field of 15 T applied strictly parallel to the layer structure. The additional orange star and tilted green triangle mark the transition midpoints to be included in the phase diagram in Fig. 5. b Specific heat of NbS2 measured with an ac technique with a small 1 mK temperature modulation during field sweeps at different fixed temperatures. The triangles mark the fields in which the constant normal state-specific heat value (dotted lines) is reached.

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Fig. 4: Thermodynamic bulk probes displaying the phase transition between the BCS low-field SC phase and the FLLO state in NbS2.
figure 4

a Magnetic field derivative dC(T)/dH of the specific heat, measured at 300 mK during a field sweep (see Fig. 4b for the corresponding C/T data) together with the linear magnetostriction coefficient \(\lambda \,=\,(1/L){\rm{d}}L(H)/{\rm{d}}H\) for magnetic fields applied strictly parallel to the layer structure. Both quantities reproducibly show a small transition anomaly at ~10 T, as marked by the circle and star symbols to be included in the phase diagram in Fig. 5. b Field derivative dτ(H)/dH of the magnetic torque for selected temperatures showing a small step-like anomaly near 10 T, as marked by the circles.

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Discussion

We summarize our results in the H/T phase diagram in Fig. 5. At high temperatures, the Hc2 transition line begins to rise with a non-vertical slope. This is in contrast to the behavior found in 2D samples and shows that there are significant orbital effects on the upper critical transition, which are naturally absent in 2D samples. We also include scaled Hc2 data of the organic superconductor κ-(BEDT-TTF)2Cu(NCS)215, for which numerous studies have demonstrated the existence of an FFLO state13,14,15,18,19,20,21. It is obvious that the initial slope of NbS2 is much weaker than for κ-(BEDT-TTF)2Cu(NCS)2, which proves a smaller Maki parameter \({\alpha }_{m}\,=\,\sqrt{2}\frac{{H}_{{\rm{orb}}}(0)}{{H}_{P}(0)}\). A large Maki parameter in the range between 1.7 and 3.437,38 is considered one of the decisive prerequisites for the formation of an FFLO state. From a fit with the Werthamer–Helfand–Hohenberg model of the initial Hc2 slope of NbS2 we obtain an orbital limit Horb ~23 T, which gives αm = 3.25, which is certainly large enough to support an FFLO state. Moreover, in quasi-2D superconductors with weak intra-layer coupling, the orbital effect is expected to be suppressed in high parallel magnetic fields when the electron wave functions become localized on the layers39. At 3.6 K, the Hc2 line rises above the theoretical BCS Pauli limit at ~10 T and then begins to saturate down to 1.5 K. At lower temperatures it shows a characteristic upswing and raises above 15 T, where it leaves the field region accessible in our experiment. The upswing is strikingly similar to that observed in κ-(BEDT-TTF)2Cu(NCS)2, with both data being almost congruent in the high-field region. Such an upswing indicates a certain change in the SC properties making it more robust against the strong Zeeman fields. It is generally interpreted as an indication of the development of an FFLO state. In fact, the phase diagram looks remarkably similar to other FFLO systems10,11,12,13,14,15,16,17,18,19,20,21,22,23,24, including the field-induced transition at ~10 T, as determined from the data in Fig. 4, which probably indicates the phase change between the ordinary SC state at low fields and the high-field FFLO state. The fact that this transition occurs near the Pauli limit further supports the FFLO scenario, as the ordinary SC state should vanish beyond this field.

Fig. 5: Magnetic field vs. temperature phase diagram of NbS2 in magnetic field applied parallel to the layer structure.
figure 5

a Filled black circles represent the upper critical field Hc2(T) as measured by torque magnetometry during field scans at constant temperatures (B-sweep, Fig. 2a, b), red open circles represent the critical temperature Tc(H) measured by torque during temperature scans (T-sweep, Fig. 2c). Blue triangles are Hc2(T) data obtained from ac calorimetry measured during field scans (Cp B-sweep, Fig. 3b). The orange star and green triangle mark the critical temperature in Cp(T) (Cp T-sweep) and thermal expansion α(T) data measured in 15 T (Fig. 3a). The violet star marks a small additional transition anomaly (FFLO transition) in the magnetostriction and filled magenta circle marks a similar anomaly visible in the field derivative of the specific heat near the Pauli limit due to the transition separating the ordinary low-field superconducting phase from the high-field FFLO state (Fig. 4a). In addition, the dark cyan circles correspond to a small anomaly in the magnetic torque attributed to the same phase transition within the superconducting state (Fig. 4b). The orange and dark cyan lines are theoretical predictions of the low-temperature Hc2 line for superconductivity with the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state34 and for Ising superconductivity, respectively. The FFLO phase is marked by the violet-shaded background. We have also added a scaled Hc2 curve of the organic superconductor κ-(BEDT-TTF)2Cu(NCS)2, which shows a very similar upturn of the Hc2 line due to the formation of an FFLO state15. Note that the characteristic temperatures/fields have been marked in the data shown in Figs. 1–4. b Hc2 data obtained from torque magnetometry in field scans measured with a small misalignment angle (θ = 1°, Fig. 1), for which no FFLO state is found, and for a field applied perpendicular to the layer structure (θ = 90°).

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The inset of Fig. 5 shows Hc2 lines for magnetic fields with two different out-of-plane orientations (θ =1° and 90°). It is obvious that the characteristic upturn is suppressed when the NbS2 sample is misaligned by only 1° and remains absent for the field oriented perpendicular to the layers. This illustrates how the stronger orbital effects suppress the FFLO state even at small misalignment angles.

In most other FFLO systems, the Hc2 transition typically sharpens and becomes of first order as one approaches the Pauli limit11,14,24. In NbS2, however, the Hc2 transition in high parallel fields remains very continuous, indicating a very gradual decay of superconductivity towards the normal state in the form of a second-order transition. In ref. 40, it was shown that the first-order nature of the Hc2 transition in high fields occurs only in the absence of the orbital effect, while in its presence it remains of second order anywhere. With orbital effect, it is expected that only the transition between the FFLO state and the BCS state should be of first-order nature. This is fully consistent with our observation: The finite Hc2 slope in the high-temperature range is clearly due to the orbital effect, and the additional transition at 10 T, observed as a spike in the magnetostriction coefficient, indicates a first-order nature of the FFLO to BCS transition. However, a small misalignment of only 1° sharpens the transition observed in torque to a step-like more first-order-like transition and indicates that superconductivity becomes Pauli limited. The fact that the Hc2 transition line then goes through a maximum is indeed expected for the Pauli-limited case in tilted fields40.

ISOI is expected to exist predominantly in monolayers with their broken inversion symmetry leading to spin splitting of the Fermi surface, while in bilayers and the bulk it is spin degenerate5. However, at least bilayers still exhibit Ising superconductivity with Hc2 values exceeding the Pauli limit by a factor of ~45, which is a consequence of the very weak inter-layer coupling in TMD materials. It remains unclear at what thickness the Ising superconductivity disappears and whether Ising superconductivity can still have some influence in the bulk. Therefore, one concern regarding the FFLO scenario as an explanation of the phase diagram is that the ISOI could still have a considerable influence in the bulk. ISOI is not compatible with a finite-momentum pair density wave state in the FFLO phase because it suppresses the effect of the in-plane Zeeman field on the Fermi surface. However, NbS2 has multiple Fermi pockets at the Γ and K/K’ points. While the K pockets are protected by ISOI, the spin-orbit interaction at the Γ pocket is weak and therefore compatible with the FFLO state. To test whether the phase diagram could alternatively be explained by Ising superconductivity, we calculated the critical field numerically by solving the linearized gap equation, using a three-band tight-binding model41 with a spin-orbit coupling strength of βSO = 87 meV (see “Methods” section for details). In order to simulate a bulk sample, we used a thickness of ten layers, which proved to be sufficient since we found that increasing the thickness further does not affect the critical field much. The obtained low-temperature Hc2 line is included in Fig. 5. While for samples in the 2D limit it could be shown that Ising superconductivity can cause very pronounced Hc2 upturns at low temperature similar to FFLO states6,35,42, it is obvious that for reasonable values of the spin-orbit coupling strength the upturn of bulk NbS2 is much too weak to explain our phase diagram.

To conclude, with the additional evidence of a field-driven phase transition within the SC state near the Pauli limit and the good agreement with theoretical simulations, it can be concluded that the phase diagram we observed for a bulk NbS2 single crystal in magnetic fields strictly parallel to its layer structure, with its pronounced upswing of the Hc2 line far above the Pauli limit for superconductivity, is most plausibly explained by an FFLO state that forms in magnetic fields above 10 T. This high-field phase requires further systematic experiments including nuclear magnetic resonance to directly monitor the nature of the FFLO pair density wave state.

Methods

Sample preparation

The growth of the high-quality 2H-NbS2 single crystal was conducted in a quartz glass ampule, which was placed in a furnace with a temperature gradient, which serves to achieve oversaturation. The hot temperature was regulated in the range of 850–400 °C, while the cold side was held at 100–70 °C lower temperature. Upon gradually cooling the multicomponent flux through this temperature range the solubility of the components was reduced so that crystallization occurred. The feed of the chalcogenide was placed in the hot part of the reaction vessel, gradually dissolved into a growth medium in form of a CsCl–KCl–NaCl salt melt mixture of eutectic composition, and migrated to the cold end of the ampule, where it formed the crystals. The crystal growth lasted about 3 weeks. Detailed characterization of the process is found in ref. 43. The thin platelets of high-quality single crystals of NbS2 with optically flat surfaces on both sides were cut into a bar shape. Tablet-like crystals of NbS2 grew up to 1 mm in size when using an evaporation method44. The sample under investigation represented a small square-shaped platelet, which was completely flat on the macroscopic scale. The zero-field specific heat (Fig. 6) displays a sharp SC transition jump centered at Tc = 5.5 K indicative for a large SC volume fraction and good quality of the single crystal.

Fig. 6: Zero-field specific heat C/T of the NbS2 single crystal used in this study.
figure 6

A sharp superconducting transition is centered at Tc = 5.5 K.

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Experimental techniques

All our experimental sensors fit on the same piezo rotary stage mounted on our 3He probe in a 15 T magnet cryostat. The rotator allows the sample to be aligned in relation to the field direction with millidegree accuracy. Field sweeps were conducted at a rate of 0.1–0.5 T/min. Temperature-dependent measurements at different fixed fields were performed during sweeps at 0.04 K/min.

The magnetic torque τ was measured using a capacitive cantilever technique24. The cantilever leg is insulated from the counterplate of the capacitor by a thin sapphire sheet. This allows reversible measurements of the torque both as a function of the field or temperature. The capacitance is measured with a General Radio 1615-A capacitance bridge in combination with a SR830 lock-in amplifier. The vector quantity torque τ is directly related to the anisotropic DC magnetization by the relation \({\boldsymbol{\tau }}\,=\,{\bf{M}}\times {\bf{H}}\), where H is the applied magnetic field. Since for such layered superconductors the DC magnetization is expected to be greatest in the out-of-plane direction, this relationship suggests that τ vanishes in parallel fields. In reality, it reaches a minimum, but does not disappear completely during a complete field sweep due to higher-order quadrupole components15,24. The ultra-high resolution allows us to detect the tiny signal with a very good signal-to-noise ratio. The data are presented as the magnitude of the magnetic torque τ, where the direction is perpendicular to the applied field.

The thermal expansion was measured with a miniature capacitance dilatometer45, in which the sample is pressed with a screw mechanism against one of the plates of a parallel-plate capacitor suspended by a firm spring mechanism. A change in the sample length, induced either by a change in temperature (thermal expansion) or by the field (magnetostriction), changes the distance between the plates and thus causes small changes in the capacitance, which is measured in the same way as described for the torque. Thermal expansion is a bulk thermodynamic quantity closely related to specific heat and allows us to detect small changes in the sample length that occur during phase transitions.

The specific heat Cp was measured using an alternating temperature (ac) technique with a mK temperature modulation amplitude24. To account for the relatively flat temperature dependence of the Hc2 line in the low-temperature/high magnetic field (H/T) regime, we performed the experiments at fixed base temperature during field sweeps. Note that the data measured in this somewhat unusual way still represent the thermal response of the sample with respect to a small temperature change, although here we probe the variation of the specific heat with respect to the field. The reason for this is that near the Pauli limit the slope of the Hc2 line in the magnetic field vs. temperature (H-T) phase diagram becomes very small and measurements during field sweep sampling reveal much more details in the important part of the phase diagram. The calorimeter platform was supported by thin nylon wires that become stiff at low temperature and serve as a support to prevent magnetic torque effects.

Numerical simulation of the critical field to test for the influence of Ising superconductivity

To test whether Ising superconductivity could account for the pronounced upturn of the Hc2 line in Fig. 5 at low temperature, we calculated the upper critical field numerically by solving the linearized gap equation, which can be written as

$$\frac{1}{g}\,=\,T{\int }_{BZ}{d}^{2}{\bf{k}}{\sum }_{n,l,m}\frac{{|\langle l,{\bf{k}}|\delta {{\bf{H}}}_{\Delta }|m,-{\bf{k}}\rangle |}^{2}}{[i{\omega }_{n}-{\xi }_{l}({\bf{k}})][i{\omega }_{n}\,+\,{\xi }_{m}(-{\bf{k}})]}$$
(1)

where \({\omega }_{n}\,=\,\pi T(2n+1)\) is the Matsubara frequency, \({\xi }_{m}({\bf{k}})\) is the m-th eigenenergy of the normal Hamiltonian (including the Zeeman field) of wave vector \({\bf{k}}\), \({|m},{\bf{k}}\rangle\) is the corresponding eigenstate, and \({{\bf{H}}}_{\Delta }\) is the matrix form of the pairing terms in the Hamiltonian

$$\delta {\hat{{\bf{H}}}}_{\Delta }\,=\,{\sum }_{j}{\psi }_{j\uparrow }^{\dagger }({\bf{k}}){\psi }_{j\downarrow }^{\dagger }(-{\bf{k}})$$
(2)

The index j includes the layer index and the band index. With the number of layers being ten and the number of bands in each layer being 3, the dimension of the matrix \(\delta {{\bf{H}}}_{\Delta }\) is 60 by 60 (with a factor of two from the spin). For a certain interaction strength \(g\), Eq. (1) provides the relation between the temperature \(T\) and the critical field \({H}_{c2}\). The actual \(g\) is determined by the measured critical temperature \({T}_{c}\) at zero-field.