Abstract
Recently, the transition metal dichalcogenide (TMD) system 2MWS_{2} has been identified as a Dirac semimetal exhibiting both superconductivity with the highest T_{c} ~ 8.5 K among all the TMD materials and topological surface states. Here we report on muon spin rotation (μSR) and density functional theory studies of microscopic SC properties and the electronic structure in 2MWS_{2} at ambient and under hydrostatic pressures (p_{max} = 1.9 GPa). The SC order parameter in 2MWS_{2} is determined to have singlegap swave symmetry. We further show a strong negative pressure effect on T_{c} and on the SC gap Δ. This may be partly caused by the pressure induced reduction of the size of the electron pocket around the Γpoint. We also find that the superfluid density n_{s} is weakly affected by pressure. The absence of a strong pressure effect on n_{s} and the absence of a correlation between n_{s} and T_{c} in 2MWS_{2}, in contrast to the other SC TMDs T_{d}MoTe_{2} and 2HNbSe_{2}, is explained in terms of its location in the optimal (ambient pressure) and above the optimal (under pressure) superconducting regions of the phase diagram and its large distance to the other possible competing or cooperating orders.
Introduction
The transition metal dichalcogenides (TMDs) are representative layered materials presently attracting very strong interest from research communities working on nanoscience and 2d layered systems.^{1,2,3,4,5,6,7,8,9,10,11,12,13,14} The TMDs share the MX_{2} chemical formula, with M being a transition metal (M = Ti, Zr, Hf, V, Nb, Ta, Mo, W or Re) and X being a chalcogen (X = S, Se, or Te). TMDs are obtained in various crystal structures 1T, 2H, and 1T′ with different stackings of the individual MX_{2} layers. A number of ground states have been observed in these systems including chargedensity waves, superconductivity, semiconductors and topological metals. New physics findings in TMDs include strong excitonic states, novel phenomena from valley splitting, Dirac cone dispersion and topological effects, and more.^{1,4}
Layered materials, with highly anisotropic electronic properties have been found to be potential hosts for unconventional superconductivity. TMDs exhibit many features strikingly similar to those reported in other unconventional superconductors. Superconductivity can be observed under pressure as for the pristine 1T polytype TMD materials which are not superconducting at ambient pressure, but exhibit charge density wave (CDW) states, instead.^{15} The metallic 2H polytypes have two, or more, Fermi surfaces and saddle bands, allowing for dual orderings, which can be coexisting CDW and SC orderings, two superconducting gaps, two CDW gaps, and possibly even pseudogaps above the onset temperature of CDW orderings, as reported in some cuprate superconductors.^{16,17,18} Upon application of pressure, the CDW ground state is suppressed and the superconducting transition temperature is enhanced. A domelike superconducting phase was induced by electrostatic gating in the semiconducting 2HMoSe_{2}, 2HMoS_{2}, 2HMoTe_{2}, and 2HWS_{2} with a T_{c} of up to 10 K.^{19,20,21} It was also shown that using field effect gating, one can induce superconductivity in monolayer semiconducting TMD WS_{2},^{22} and one could access an unprecedented doping range revealing a rich set of competing electronic phases ranging from band insulator, via superconductor, to a reentrant insulator at high doping. Superconductivity with a very low T_{c} ~ 0.1 K was also reported in the Weyl semimetal MoTe_{2} which exhibits a low temperature orthorhombic T_{d} structure.^{5} However, under the application of pressure, the orthorhombic structure transforms into the 1T′ structure and T_{c} is enhanced to about 8 K. Recently, we found a correlation between the superfluid density at T = 0 and the transition temperature T_{c} in the typeII Weyl semimetal T_{d}MoTe_{2}^{23} and the deviation from the linear correlation in 2HNbSe_{2} under pressure.^{24} These observations resemble the ones found earlier in cuprates^{25,26,27,28,29,30} and Febased superconductors.^{31,32,33,34} These examples demonstrate that the TMDs exhibit rather unconventional superconducting and normal state properties.
Recently, superconductivity with T_{c} ~ 8.8 K was reported in the new compound 2MWS_{2}, which is constructed from 1T′WS_{2} monolayers.^{35} 2MWS_{2} has the highest T_{c} among all TMD materials at ambient pressure. The system possesses a semimetallic band structure. The crystal structure of 2MWS_{2} has a different stacking of 1T′WS_{2} monolayers along the aaxis, compared to T_{d}MoTe_{2} and 1T′WTe_{2}. In addition to superconductivity, topological surface states with a single Dirac cone were theoretically predicted for 2MWS_{2}. Hence, this system represents one of the few examples of a material with both superconductivity and a topologically nontrivial surface states. The possible occurrence of topological superconductivity (TSC) in 2MWS_{2} after fine tuning of some parameters is also discussed.^{35} TSCs are materials with unique electronic states consisting of a full pairing gap in the bulk and gapless surface states composed of Majorana fermions (MFs).^{36,37,38} Up to now, the only known properties of the superconducting state in 2MWS_{2} are the critical temperatures and fields.^{35} Thus, a thorough exploration of superconductivity in 2MWS_{2} from both experimental and theoretical perspectives is imperative.
To further study superconductivity and its nature in 2MWS_{2}, it is critical to measure the superconducting order parameter at the microscopic level. Thus, we conducted ambient and highpressure^{39,40,41,42} muon spin relaxation/rotation (μSR) measurements of the magnetic penetration depth λ in the polycrystalline sample of 2MWS_{2}. This quantity is one of the fundamental parameters of a superconductor, since it is related to the superfluid density n_{s} via 1/λ^{2} = μ_{0}e^{2}n_{s}/m* [where m* is the effective mass; see also Eq. (2)]. Most importantly, the temperature dependence of λ is particularly sensitive to the topology of the superconducting gap: while in a fully gapped superconductor, Δλ^{−2}(T) ≡ λ^{−2}(0) − λ^{−2}(T) vanishes exponentially upon decreasing the temperature, it vanishes as a power of T in a superconductor containing a gap with nodes. The μSR technique provides a powerful tool to measure λ in the vortex state of type II superconductors in the bulk of the sample,^{43} in contrast to many techniques that probe λ only near the surface. Details are provided in the Methods section.
Here we show that the superconducting order parameter in 2MWS_{2} possesses singlegap swave symmetry, which is consistent with the presence of dominant holetype charge carriers, as deduced from Hall effect measurements. Moreover, we observed a strong negative pressure effect on the critical temperature T_{c} and the superconducting gap Δ as well as on the size of the electron pocket around the Γpoint, at which a band inversion appears up to the highest applied pressure. However, the pressure effect on the superfluid density in 2MWS_{2} is very weak. We find that the ratio T_{c}/n_{s} is located in the Uemura plot close to those of other superconducting TMD systems, as T_{d}MoTe_{2} and 2HNbSe_{2}, and those of other unconventional superconductors. This suggests that 2MWS_{2} exhibits rather unconventional superconducting properties. The absence of a strong pressure effect on the superfluid density and its correlation with T_{c}, which was previously found in T_{d}MoTe_{2} and 2HNbSe_{2}, is explained in terms of the absence of competing or cooperating order to superconductivity, as 2MWS_{2} exhibits optimal superconducting conditions already at ambient pressure. By application of pressure we are pushing the system away from any possible competing/cooperating order. Our findings, therefore, pose a challenge for understanding the underlying physics in these layered TMDs and might ultimately lead to a better understanding of generic aspects of nonBCS behaviors in unconventional superconductors.
Results
Magnetic penetration depth and the superconducting gap symmetry
Figure 1a exhibits transversefield (TF) muontime spectra for the polycrystalline sample of 2MWS_{2} measured in an applied magnetic field of μ_{0}H = 60 mT above (T = 10 K) and below (T = 200 mK) the superconducting (SC) transition temperature T_{c}. Above T_{c} the oscillations reflecting the muonspin precession show a small damping due to the random local fields from the nuclear magnetic moments. Below T_{c} the relaxation rate strongly increases upon decreasing the temperature due to the presence of a nonuniform local magnetic field distribution resulting from the formation of the fluxline lattice (FLL) in the Shubnikov phase. By carefully examining the TFμSR data, we find that the sample consists of a superconducting and a nonsuperconducting volume fraction. To visualise these two fractions of the sample, we report on Fig. 1b the Fourier transforms of the μSR time spectra, shown in Fig. 1a. At T = 200 mK the narrow signal around μ_{0}H_{ext} = 60 mT, originates from the nonsuperconducting part of the sample, while the broad signal with a first moment μ_{0}H_{int} < μ_{0}H_{ext}, marked by the solid arrow in Fig. 1b, arises from the superconducting part of the sample. The estimated superconducting volume fraction from the μSR data is ~45%. It is interesting to investigate whether magnetism is present either in the superconducting or nonsuperconducting fractions of the sample. In order to search for magnetism, we have carried out zerofield (ZF) μSR experiments above and below T_{c}, which are shown in Fig. 1c. No sign of static magnetism could be detected in the ZF time spectra down to 1.7 K. Moreover, the ZF relaxation rate is small and does not change between 10 and 1.7 K. ZFμSR spectra are well described by the combination of a Lorentzian and a Gaussian KuboToyabe depolarization function,^{44,45} reflecting the field distribution at the muon site created by the nuclear dipole moments and randomly oriented diluted weak local electronic moments. Returning to the discussion of the TFμSR data, we observe a strong diamagnetic shift of the internal magnetic field μ_{0}H_{int} sensed by the muons below T_{c}. This is evident in Fig. 2a, where we plot the temperature dependence of ΔB_{dia} = μ_{0}(H_{int,SC} − H_{int,NS}), i.e., the difference between the internal field μ_{0}H_{int,SC} measured in the SC fraction and one μ_{0}H_{int,NS} measured in the normal state at T = 10 K. The strong diamagnetic shift along with the negative field dependence of the superconducting depolarization rate σ_{sc} (see Fig. 2b) excludes the occurrence of field induced magnetism in 2MWS_{2}. Note that in some Febased and cuprate high temperature superconductors, where field induced magnetism was detected, a paramagnetic shift and a linear increase of σ_{sc} with magnetic field were observed.^{46,47,48} The absence of magnetism in zerofield or under applied magnetic fields implies that the increase of the TF relaxation rate below T_{c} is solely arising from the FLL in the superconducting fraction. The strong superconducting response from nearly 50% of the sample points to the bulk superconductivity in 2MWS_{2}. We note that the diamagnetic moment, which we deduce from the bulk magnetization measurements suggest a 100% SC volume in the sample, which is consistent with the previous report.^{35} However, bulk magnetisation measures the diamagnetic screening of the whole sample where a 100% screening does not necessarily imply a 100% SC volume. μSR, being a local probe, directly measures the SC volume fraction within the sample. Thus, in this work we provide a more precise estimate for the SC volume in 2MWS_{2}. The μSR evidence for a phase separation between SC and paramagnetic regions in 2MWS_{2} is an interesting finding, which might be an intrinsic feature of this system.
From the TF muon time spectra, we determined the Gaussian superconducting relaxation rate σ_{sc} (after subtracting the nuclear contribution), which is proportional to the second moment of the field distribution (see Methods section). Figure 2b shows the obtained field dependence of σ_{sc} at 1.7 K. Each point was obtained by field cooling the sample from above T_{c} to 1.7 K. As expected from the London model one observes that first σ_{sc} strongly increases with increasing magnetic field until reaching a maximum at 40 mT.^{49} Above 40 mT, σ_{sc} decreases with increasing magnetic field. This appears consistent with a behaviour expected for swave superconductor for an ideal triangular vortex lattice.^{49} Taking into account the critical field of 3 T, the theoretical swave behaviour has been calculated according to the Brandt formula.^{49} It is shown by the dashed line in Fig. 2b, which agrees well with the experimental data. The second moment of the resulting inhomogeneous field distribution is related to the magnetic penetration depth λ by \(\left\langle {\Delta B^2} \right\rangle \propto \sigma _{sc}^2 \propto \lambda ^{  4}\).^{49} In order to investigate the symmetry of the superconducting gap, we have therefore derived the temperaturedependent London magnetic penetration depth λ(T), which is related to the relaxation rate by:
Here, γ_{μ} is the gyromagnetic ratio of the muon, and Φ_{0} is the magneticflux quantum. In order to accurately determine the temperature dependence of λ, it is important to do the temperature dependent measurements of σ_{sc} slightly above the maximum of σ_{sc}(H), where Eq. (1) is valid. Thus, the measurements were done with an applied magnetic field of 60 mT. In Fig. 2c, we show the temperature dependence of the muon spin depolarization rate σ_{sc}. Below T_{c} the relaxation rate σ_{sc} starts to increase from zero with decreasing temperature due to the formation of the FLL and reaches σ_{sc} ≃ 1.43 μs^{−1} at the baseT. The temperature dependence of σ_{sc}, which reflects the topology of the SC gap, exhibits a saturation at low temperatures. This is consistent with a nodeless superconductor, for which the superfluid density n_{s} ∝ σ_{sc} reaches its zerotemperature value exponentially. For a quantitative analysis, we in the local (London) approximation (λ ≫ ξ, where ξ is the coherence length), calculated λ(T). The result of the gap analysis is shown in Fig. 3. The data are perfectly compatible with a single swave gap with a value of Δ = 1.31(1) meV.
The London magnetic penetration depth λ is given as a function of n_{s},^{50} the effective mass m^{*}, ξ and the mean free path l, according to
For systems close to the clean limit, ξ/l → 0, the second factor essentially becomes unity, and the simple relation 1/λ^{2} ∝ n_{s}/m^{*} holds. Considering the upper critical fields H_{c2} of 2MWS_{2}, as reported in detail by Fang et al.,^{35} we can estimate the inplane coherence length to be ξ_{ab} ≃ 8.8 nm at ambient pressure. At ambient pressure, the inplane mean free path l was estimated to be l_{ab} ≃ 200 nm. No estimates are currently available for l under pressure. However, the inplane l is most probably independent of pressure considering the fact that the pressure mostly reduces the interlayer distance due to the unique anisotropy resulting from the stacking of layers with van der Waals type interactions between them. Thus, in view of the short coherence length and relatively large l, we can reliably assume that 2MWS_{2} lies close to the clean limit.^{23,51} With this assumption, we obtain the groundstate value n_{s}/(m*/m_{e}) ≃ 2.8 × 10^{26} m^{−3}, where m_{e} is the free electron mass.
Probing superconductivity and band structure as a function of pressure
To gain further insight into the superconducting properties of 2MWS_{2}, the temperature dependence of the superfluid response was measured under applied pressures up to 1.9 GPa. Moreover, the pressure effect of the electronic structure was also calculated. We note that the SC volume fraction was found to be independent of pressure. In Fig. 4a, b, we show the temperature evolution of σ_{sc}, normalised to the value at ambient pressure, and the diamagnetic shift ΔB_{dia} measured under various hydrostatic pressures. A single gap swave model describes the σ_{sc}(T) data very well up to the highest pressure. Remarkably, both T_{c} and Δ decreases substantially upon application of pressure. T_{c} decreases from 8.5 K at p = 0 GPa to 6 K at 1.9 GPa. Δ decreases from 1.31(1) meV at 0 GPa to 0.82(2) meV at 1.9 GPa. However, the analysis of the temperature dependence of σ_{sc}(T) reflects only a ~5% increase of the zerotemperature value of n_{s}. This means that the absolute value of the superfluid density in 2MWS_{2} is very weakly affected by pressure. The band structures, calculated in the presence of the spinorbit coupling for ambient pressure and under the applied pressures of p = 1.3 GPa and 1.9 GPa are shown in Fig. 4d–f (see also Fig. S1 in the Supplementary Information). The band inversion between the porbital of the S atoms and the dorbital of the W atoms at the Γpoint is found up to the highest investigated pressure. This indicates that the topology of the bands in 2MWS_{2} remains nontrivial under pressure. We also find that the electron pockets at the Γpoint shrink upon application of pressure, which might have some interesting consequences on the physical properties of the system.
Discussion
We can now discuss the main results of our measurements on 2MWS_{2}:

(i)
The temperature dependence of the superfluid density, reflected by the one of the relaxation rate σ_{sc}, is well described at all pressures by a single gap swave model. This is surprising since a multiband bulk electronic structure with holelike bands around the N point and electronlike bands around the Γ point was found for 2MWS_{2}. On the other hand, Hall effect measurements showed that the holetype carriers are dominant with a very high density of n_{e,h} = 8.58 × 10^{26} m^{−3} at T = 10 K. It is interesting that the superfluid density n_{s}/(m*/m_{e}) ≃ 2.8 × 10^{26} m^{−3}, which we estimated for 2MWS_{2} is very close to the normal state hole density n_{e,h}, indicating that the superconductivity stems mostly from hole carriers. Thus, single gap superconductivity in this multiband system may be explained by the fact that the superconducting gap occurs only on the holelike Fermi surface and that the carrier density on the electron Fermi surface is not sufficient to induce superconductivity. This conclusion is also substantiated by the fact that superfluid density is insensitive to the shrinkage of the electron pocket, caused by pressure. If superconductivity would exist on the electron pocket, the pressure induced reduction of the size of the electron pocket would result to the reduction of the associated electron carrier concentration and thus the reduction of the superfluid density, which is not the case in 2MWS_{2}. Single gap swave superconductivity in 2MWS_{2} is also different from the twogap s + swave superconductivity, observed in the superconducting TMDs T_{d}MoTe_{2} and 2HNbSe_{2}. This again might be related to the fact that in both these systems both hole and electron bands contribute to superconductivity, while in 2MWS_{2} the superconductivity most likely arises predominantly from the hole like band. Regarding the surface states, a single Dirac cone at the Γ point was reported. When the Fermi level is exactly at the Dirac point, only a single Fermi surface was obtained.^{35} A single Fermi surface at the surface and a dominant holelike Fermi surface in the bulk is consistent with single gap superconductivity, observed in our study. It was also suggested that if the superconductivity is induced on this single Fermi surface, topological superconductivity is expected. As the muons penetrate deep into the material, we cannot obtain information on the superconductivity on the surface and about its topological nature. However, because of the semimetallic bulk electronic structure, the topological surface states are partially mixed with bulk states. Thus, the superconducting properties in the bulk could, to some extent, reflect the properties also on the surface.

(ii)
We observed, for the first time, a strong negative pressure effect on the critical temperature T_{c} and the superconducting gap Δ. Both quantities decrease by ~30% under 1.9 GPa. Note that the superconducting transition temperature, estimated by the Allen and Dynes modified McMillan formula,^{35} was found to be ~2 K, which is a factor of 4 smaller than the experimental value T_{c} ≃ 8.5 K (see Fig. S2 in the Supplementary Information). This indicates that some dynamical and anharmonic effects on the electronphonon coupling as well as interband scattering between electron and hole pockets are most likely the reasons for highT_{c} in 2MWS_{2}. According to our DFT calculations, the electron pocket around the Γ point shrinks by pressure, which would lead to the suppression of interband processes and this might have some effect on T_{c}. Pressure could also weaken the anharmonic electronlattice interactions, giving rise to the suppression of T_{c}. Some additional experiments are certainly needed to understand the negative pressure effect on the critical temperature. The superfluid density n_{s}/(m^{*}/m_{e}) is essentially pressure independent, displaying at the most a slight increase of about 5%. The ratio between the superfluid density and the critical temperature is located in the Uemura plot close to those of T_{d}MoTe_{2} and 2HNbSe_{2} and other unconventional superconductors, as shown in Fig. 5. This suggests that 2MWS_{2} exhibits rather unconventional SC properties and might suggest a common mechanism and related electronic origin for the superconductivity in these superconducting TMDs. On the other hand, the insensitivity of the superfluid density to the strong change in T_{c} is in contrast to the strong correlation between these two quantities in T_{d}MoTe_{2} and 2HNbSe_{2}. Let us try to understand this apparent discrepancy. The nearly linear relationship between T_{c} and the superfluid density was originally observed in holedoped cuprates in the underdoped region of the phase diagram,^{25,26} where the ratio between T_{c} and their effective Fermi temperature T_{F} is about T_{c}/T_{F} ~ 0.05 corresponding to a 4–5 times reduction of T_{c} from the ideal Bose condensation temperature for a noninteracting Bose gas. These results were discussed in terms of a crossover from BoseEinstein condensation to BCSlike condensation.^{28} It is important to note that in cuprates the relationship between T_{c} and n_{s}/(m*/m_{e}) was shown to depend on which part of the SC dome the samples are located. Thus, the leftside of the dome (underdoped), which has close proximity with the competing/cooperating antiferromagnetic phase, the center of the dome (optimally doped), which is away from the AFM phase or the right side of the dome (overdoped) region, which is clearly seperated from the AFM phase. The linear increase of T_{c} with n_{s}/(m*/m_{e}) is observed only in the underdoped region of the phase diagram. Within the optimal doping region T_{c} does not change despite the fact that n_{s}/(m^{*}/m_{e}) increases. In the overdoped region, T_{c} decreases, but n_{s}/(m*/m_{e}) shows a small increase only. A similar behaviour was also observed in Febased superconductors.^{52} With this in mind, we compare the temperaturepressure phase diagrams for the systems 2MWS_{2}, T_{d}MoTe_{2}, 2HNbSe_{2}. The phase diagrams are shown in Fig. 5b–d. There are two obvious differences between the phase diagrams. In the case of 2HNbSe_{2} and T_{d}MoTe_{2} (Fig. 5c, d), the phase diagram consists of two different orders at ambient pressure: superconductivity and either charge density wave order or noncentrosymmetric structural order. In addition, T_{c} of both systems increases with increasing pressure within the investigated pressure region (0–2 GPa), indicating that the samples are located in the lefthand side of the dome with close proximity to another phase. In the case of 2MWS_{2} (Fig. 5b), the system has solely superconducting order and there is no additional competing/cooperating order. In addition, the system 2MWS_{2} has the highest T_{c} at ambient pressure, suggesting that the system is already in its optimal conditions in terms of superconductivity. Upon application of pressure, T_{c} decreases substantially, pushing the system towards the righthand side of the dome, i.e. pushing it even further away from any possible competing phase. So, the location of 2MWS_{2} in the optimal (ambient pressure) and above the optimal (under pressure) superconducting regions of the phase diagram might be a reason for the absence of the correlation between the superfluid density and the critical temperature in this system.
In conclusion, using DFT, we show that the topology of the bands in 2MWS_{2} remains nontrivial up to the highest applied pressure. We further provide the first microscopic investigation of the superconductivity in the layered superconductor 2MWS_{2} at ambient pressure as well as under hydrostatic pressures by means of muonspin rotation. Specifically, the zerotemperature magnetic penetration depth λ(0) and the temperature dependence of the superfluid density were studied in 2MWS_{2} by means of μSR experiments with pressures up to p ≃ 1.9 GPa. The temperature dependence of the superfluid density λ^{−2} is well described by a single gap swave scenario with a gap value of Δ ≃ 1.31(1) meV at ambient pressure. Considering the previous observations of the dominant holelike Fermi surface in the bulk and single Fermi surface on the surface we suggest that the well resolvable SC gap occurs only on the holelike Fermi surface. We also find that the ratio T_{c}/n_{s} is located in the Uemura plot close to those of T_{d}MoTe_{2} and 2HNbSe_{2} and other unconventional superconductors, suggesting that 2MWS_{2} exhibits nonstandard SC properties. Moreover, we observed the strong negative pressure effect on T_{c} and Δ, which may partly be due to the reduction of the size of the electron pocket at the Γ point, at which a band inversion appears up to the highest applied pressure. The superfluid density shows only a very weak positive pressure effect and thus the correlation between the superfluid density and the critical temperature is absent in 2MWS_{2}. This is in contrast to the strong pressure effects on λ^{−2} and its correlation with T_{c}, observed in the superconducting TMDs T_{d}MoTe_{2} and 2HNbSe_{2}. This discrepancy is discussed in terms of the absence of competing or cooperating order to superconductivity in 2MWS_{2}. These results hint towards the complex nature of the superconductivity in TMDs, despite its swave nature, which might have far reaching consequences for the future development of devices based on these materials.
Methods
Sample preparation
Topochemical synthesis procedure of the metastable, monoclinic tungsten diselenide (2MWS_{2}) can be found elsewhere.^{35,54} In a first step, a hexagonal phase of 2MWS_{2} (SG P6_{3}/mmc  No 194), was synthetized by solid state reaction method at 850 °C for 35 h from a stoichiometric mixture of elemental sulfur (5N, Alfa Aesar) and tungsten (5N, Alfa Aesar). Than potassium intercalation and growth of KWS_{2} was conducted by annealing the stoichiometric mixture of K (3N5, Aldrich) and 2MWS_{2} at 850 °C. After 35 h a specimen was cooled down to 550 °C with a rate of 6 °C/h. All syntheses reactions and sample growth processes described above where conducted in a double wall, evacuated and sealed quartz ampules. In a third step K_{x}(H_{2}O)yWS_{2} compound was created by washing 3 g of KWS_{2} with deionized water. Finally K_{x}(H_{2}O)yWS_{2} was oxidized with 200 ml of 0.1N K_{2}Cr_{2}O_{7}/0.1N H_{2}SO_{4} aqueous solution in order to get polycrystalline samples of 2MWS_{2}. Powders from the same batch were pressed in the form of pellets and were measured with various instruments. For the low background GPS instrument and lowtemperature HAL9500 instrument, equipped with BlueFors vacuumloaded cryogenfree dilution refrigerator (DR), we used one pellet with 10 mm in diameter. For high pressure GPD instrument, several pellets with 5.9 mm in diameter were prepared and placed inside the pressure cell.
Pressure cell
Double wall pistoncylinder type of cell made of CuBe/MP35N material was used in μSR experiments to generate pressures up to 1.9 GPa.^{39,40,41,42} Small indium plate was placed together with the sample in the pressure cell filled with the Daphne oil. The pressure was estimated by tracking the SC transition of a indium plate by AC susceptibility. The filling factor of the pressure cell was maximised. The fraction of the muons stopping in the sample was ~45%. The background of the pressure cell is well known and characterised.^{39}
μSR experiment
Spinpolarized muons μ^{+} are extremely sensitive magnetic probes. Muonspin experiences the Larmor precession either in the local field or in an applied magnetic field. By probing the field distribution in the vortex state of a TypeII superconductor using transverse field μSR technique, fundamental parameters such as the magnetic penetration depth λ and the coherence length ξ can be measured in the bulk of a superconductor.
Analysis of TFμSR data
The following function is used to analyse the TF μSR data:^{55,56}
Here A_{s} and A_{pc} are the initial asymmetries of the μSR signal for the sample and the pressure cell, respectively. γ/(2π) ≃ 135.5 MHz/T is the muon gyromagnetic ratio and φ is the initial phase of the muonspin ensemble. B_{int} is the internal magnetic field at the muon site. The depolarization rate σ_{sc} is due to the formation of the flux line lattice in the SC state and σ_{nm} describes the damping due to the nuclear dipolar contribution. σ_{nm} was obtained above T_{c} and was assumed to be constant over the entire temperature range. The relaxation rate σ_{pc} reflects the depolarization due to the nuclear moments of the pressure cell. It exhibits the temperature dependence below T_{c}^{41} due to the influence of the diamagnetic response of the SC sample on the pressure cell. In order to consider this influence we assume a linear coupling between σ_{pc} and the field shift of the internal magnetic field in the SC state: σ_{pc}(T) = σ_{pc}(T > T_{c}) + C(T)(μ_{0}H_{int,NS} − μ_{0}H_{int,SC}), where σ_{pc}(T > T_{c}) = 0.25 μs^{−1} is the temperature independent relaxation rate. μ_{0}H_{int,NS} and μ_{0}H_{int,SC} are the internal magnetic fields measured in the normal and in the SC state, respectively.
Analysis of λ(T)
λ(T) was calculated within the London approximation (λ ≫ ξ) by the following function:^{50,55}
where f = [1 + exp(E/k_{B}T)]^{−1} is the Fermi function, φ is the angle along the Fermi surface, and Δ_{i}(T, φ) = Δ_{0,i}Γ(T/T_{c})g(φ) (Δ_{0,i} is the maximum gap value at T = 0). The temperature dependence of the gap is given by the expression Γ(T/T_{c}) = tanh{1.82[1.018(T_{c}/T − 1)]^{0.51}},^{57} while g(φ) describes the angular dependence of the gap. g(φ) = 1 for both an swave and an s + swave gap, and g(φ) = cos(2φ) for a dwave gap.
DFT calculations
2MWS_{2} crystal has a monoclinic structure with the space group symmetry C_{2/m} (number 12) and the following lattice parameters a = 12.856 Å, b = 3.2257 Å, c = 5.7109 Å, α = γ = 90° and β = 112.942° (crystallographic data CCDC 1853656). Fullrelativistic electronic structure calculations were performed in Vienna ab initio simulation package (VASP)^{58,59} using Perdew–Burke–Ernzerhof (PBE)^{60,61} generalized gradient approximation (GGA) exchangecorrelation functional and projected augmented wave (PAW) potentials.^{62,63} The van der Waals interactions were taken into account within the DFTD2 approach by Grimme.^{64} The plane wave basis cutoff was set to 600 eV and 9 × 9 × 9 kpoints grids were generated within Monkhorst–Pack method.
Data availability
All relevant data are available from the authors. The data can also be found at the following link http://musruser.psi.ch/cgibin/SearchDB.cgi using the following details: Area: HAL and GPD. Year: 2018. Run numbers: 10091042 for HAL and 31993281 for GPD.
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Acknowledgements
The μSR experiments were carried out at the Swiss Muon Source (SμS), Paul Scherrer Insitute, Villigen, Switzerland using the HAL9500 μSR spectrometer (πE3 beamline), GPS instrument (πM3 beamline) and high pressure GPD instrument (μE1 beamline). The work at the University of Zürich was supported by the Swiss National Science Foundation under Grant No. PZ00P2_174015. R.K. acknowledges the Swiss National Science Foundation (grants 200021_149486 and 200021_175935). T.N. and S.S.T. acknowledge the funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programm (ERCStGNeupert757867PARATOP). M.Z.H. is supported by US DOE/BES grant no. DEFG0205ER46200.
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Project planning and coordination: Z.G. Sample growth and XRD experiment: D.G. and E.P. μSR experiments: Z.G., R.K., J.V., H.L., A.A. μSR data analysis: Z.G. Magnetization measurements: F.v.R. DFT calculations: M.B., S.S.T. and T.N. Data interpretation and draft writing: Z.G. with contributions and comments from all authors.
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Correspondence to Zurab Guguchia or Dariusz J. Gawryluk.
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Guguchia, Z., Gawryluk, D.J., Brzezinska, M. et al. Nodeless superconductivity and its evolution with pressure in the layered dirac semimetal 2MWS_{2}. npj Quantum Mater. 4, 50 (2019). https://doi.org/10.1038/s4153501901895
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