Research ArticleCONDENSED MATTER PHYSICS

Unconventional scaling of the superfluid density with the critical temperature in transition metal dichalcogenides

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Science Advances  29 Nov 2019:
Vol. 5, no. 11, eaav8465
DOI: 10.1126/sciadv.aav8465

Abstract

We report on muon spin rotation experiments probing the magnetic penetration depth λ(T) in the layered superconductors in 2H-NbSe2 and 4H-NbSe2. The current results, along with our earlier findings on 1T′-MoTe2 (Guguchia et al.), demonstrate that the superfluid density scales linearly with Tc in the three transition metal dichalcogenide superconductors. Upon increasing pressure, we observe a substantial increase of the superfluid density in 2H-NbSe2, which we find to correlate with Tc. The correlation deviates from the abovementioned linear trend. A similar deviation from the Uemura line was also observed in previous pressure studies of optimally doped cuprates. This correlation between the superfluid density and Tc is considered a hallmark feature of unconventional superconductivity. Here, we show that this correlation is an intrinsic property of the superconductivity in transition metal dichalcogenides, whereas the ratio Tc/TF is approximately a factor of 20 lower than the ratio observed in hole-doped cuprates. We, furthermore, find that the values of the superconducting gaps are insensitive to the suppression of the charge density wave state.

INTRODUCTION

Transition metal dichalcogenides (TMDs) are a class of layered materials, which presently attract great interest because of their versatile physical, chemical, and mechanical properties (111). The TMDs share the MX2 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). These compounds crystallize in different structural phases resulting from different stacking of the individual MX2 layers, with van der Waals bonding between them.

A variety of previously unknown physical phenomena have been reported in TMD systems. Most recently, topological physics with Dirac-type dispersion, exotic optical, and transport behavior originating from valley splitting were predicted and observed (1, 4). Correlated physics has been studied intensively, especially in 2H-TaS2, 2H-NbSe2, and, more recently, 1T′-MoTe2 (5, 1215). 2H-NbSe2 is a superconductor with a critical temperature Tc ≈7.3 K and hosts a two-dimensional (2D) charge density wave (CDW) with a critical temperature of TCDW ≈ 33 K (1621). The polymorph 4H-NbSe2 undergoes a CDW transition at TCDW ≈ 40 K and becomes superconducting at Tc ≈ 6 K. The superconducting transition temperature of 2H-NbSe2 is remarkably high in comparison with other TMDs, where commonly transition temperatures below 4 K are observed. Recently, it was found that both the superconducting state and the CDW ordering remain intact even for a single layer of this material (8). The superconducting transition temperature is thereby lowered in the 2D limit to Tc ≈ 1.1 K. Previously, the effect of hydrostatic pressure on the CDW ordering and superconductivity was studied in 2H-NbSe2 by means of magnetization and resistivity measurements (22, 23). It was found that upon increasing pressures, CDW ordering is suppressed. The suppression of CDW ordering commonly causes a strong increase of the superconducting transition temperature Tc [see, e.g., (2426)]. The generic occurrence of a “fragile” SC phase in the systems with competing SC and CDW ground states was recently reported (27). In 2H-NbSe2, Tc increases only slightly (ΔTc<1 K) with increasing pressure without any anomaly across the critical pressure at which the CDW state disappears. Moreover, both CDW order parameter amplitude and its static phase coherence length are unaffected when the superconductivity is suppressed in a high magnetic field (18). Recent x-ray experiments also suggest that different phonon branches are responsible for superconductivity and CDW orders (23). These findings suggested that superconductivity is only weakly affected by the suppression of the CDW ordering (1821, 33).

Layered superconductors with highly anisotropic electronic properties have been found to be potential hosts for unconventional superconductivity. Cuprates are the most prominent example, consisting of CuO2 layers and the iron pnictide superconductors, with tetrahedra coordinated FePn4 layers. Other examples for unconventional superconductivity in layered systems are a monolayer of FeSe on strontium titanate substrates (28) and the reported superconductivity in twisted bilayer graphene (29). Recently, we have reported on muon spin rotation (μSR) measurements on the TMD superconductor Td-MoTe2, which is considered to be a type II Weyl semimetal (10, 11, 30). We had found a linear scaling of superfluid density at T = 0 and the transition temperature Tc, similar to the relation found earlier in the cuprates (31) and other Fe-based superconductors (3234).

Here, we report on the substantial increase of the superfluid density ns/m* in van der Waals system 2H-NbSe2 (Fig. 1A) under hydrostatic pressure, which appears not to be correlated with the suppression of the CDW ordered state. We find the superfluid density in 2H-NbSe2, 4H-NbSe2, and 1T′-MoTe2 to scale linearly with Tc, which indicates that this linear relation has general validity for TMD superconductors. Such relations are considered to be a hallmark feature of unconventional superconductivity (31, 35, 36) in cuprate and iron-pnictide superconductors. Upon application of pressure on 2H-NbSe2, the ns/m* versus Tc dependence shows the deviation from the abovementioned linear behavior. We find this linear deviation to be very similar to the deviation observed for optimally doped cuprates. Our findings, therefore, pose a challenge for understanding the underlying quantum physics in these layered TMDs and might lead to a better understanding of generic aspects of non–Bardeen-Cooper-Schrieffer (BCS) behaviors in unconventional superconductors.

RESULTS AND DISCUSSION

In Fig. 1B, we show the schematic phase diagram of the pressure dependence of the CDW transition temperature TCDW and the superconducting transition temperature Tc for NbSe2, according to Qi et al. (5). The red arrows mark the pressures at which the T dependence of the penetration depth and the superfluid density was measured. As it can be seen, the CDW transition is strongly reduced within the investigated pressure range between p = 0 and 2.2 GPa. In Fig. 1 (C and D), two representatives of the transverse-field (TF) μSR time spectra for 2H-NbSe2 at ambient pressure and at the maximum applied pressure of p = 2.2 GPa are shown. Both measurements were performed in the pressure cell. For both pressures, the TF muon time spectra in a field of μ0H= 70 mT are shown above (T = 10 K) and below (T = 0.25 K) the superconducting transition temperature Tc. Above Tc, the oscillations show a small relaxation due to the random local fields from the nuclear magnetic moments. Below Tc, the relaxation rate strongly increases with decreasing temperature due to the presence of a nonuniform local magnetic field distribution as a result of the formation of a flux-line lattice (FLL) in the Shubnikov phase.

Fig. 1 Schematic phase diagram and TF μSR time spectra for NbSe2.

(A) Crystal structure of 2H-NbSe2. (B) Pressure dependence of Tc and the CDW temperature TCDW (5). The red arrows mark the pressures at which the T dependence of the penetration depth was measured. The TF spectra, above and below Tc at pressures of (C) p = 0 GPa and (D) p = 2.2 GPa, are shown. The solid lines in (C) and (D) represent fits to the data by means of Eq. 4. The dashed lines are guides to the eyes.

From the obtained TF muon time spectra, we have derived the temperature dependence of the muon spin depolarization rate σsc, which is proportional to the second moment of the field distribution (see Methods). In Fig. 2, we show the temperature dependence of the muon spin depolarization rate σsc of 4H-NbSe2 at ambient pressure and 2H-NbSe2 at pressures of p= 0, 0.4, 0.7, 1.4, and 2.2 GPa. Below Tc, the relaxation rate σsc starts to increase from zero with decreasing temperature owing to the formation of the FLL. The critical temperatures for 2H-NbSe2 and 4H-NbSe2 are Tc ∼ 6 K and 7 K, respectively. We find that the low-temperature value of σsc between 2H-NbSe2 and 4H-NbSe2 increases as much as Tc. Upon application of pressure on 2H-NbSe2, the superconducting transition temperature Tc increases, which is in good agreement with earlier reports (22, 23). Furthermore, we observed a substantial increase of the low-temperature value of the muon spin depolarization rate σsc with increasing pressure. This can be seen most clearly at base temperature, where the muon spin depolarization rate σsc(T = 0.25 K) increases by ∼30% from p = 0 GPa to p = 2.2 GPa. At all pressures, the form of the temperature dependence of σsc, which reflects the topology of the SC gap, shows the saturation at low temperatures, indicating the presence of the isotropic pairing in NbSe2 at all applied pressure.

Fig. 2 Superconducting muon spin depolarization rate σSC for NbSe2.

Temperature dependence of σSC(T) measured in 4H-NbSe2 at ambient pressure and in 2H-NbSe2 at various hydrostatic pressures in an applied magnetic field of μ0H = 70 mT.

The second moment of the resulting inhomogeneous field distribution is related to the magnetic penetration depth λ as ΔB2σsc2λ4, whereas σsc is the Gaussian relaxation rate due to the formation of FLL (37). To investigate the symmetry of the superconducting gap, we have therefore derived the temperature-dependent London magnetic penetration depth λ(T), which is related to the relaxation rate byσsc(T)γμ=0.06091Φ0λ2(T)(1)

Here, γμ is the gyromagnetic ratio of the muon, and Φ0 is the magnetic-flux quantum. Thus, the flat T dependence of σsc observed at various pressures for low temperatures (see Fig. 2) is consistent with a nodeless superconductor, in which λ−2(T) reaches its zero-temperature value exponentially.

To proceed with a quantitative analysis, we consider the local (London) approximation (λ ≫ ξ, where ξ is the coherence length) and use the empirical α model. The model, widely used in previous investigations of the penetration depth of multiband superconductors (3842), assumes that the gaps occurring in different bands, besides a common Tc, are independent of each other. The superfluid density is calculated for each component separately (38) and added together with a weighting factor. For our purposes, a two-band model suffices, yieldingλ2(T)λ2(0)=ω1λ2(T,Δ0,1)λ2(0,Δ0,1)+ω2λ2(T,Δ0,2)λ2(0,Δ0,2)(2)

Here, λ(0) is the London magnetic penetration depth at zero temperature, Δ0,i is the value of the ith SC gap (i = 1, 2) at T = 0 K, and ωi is the weighting factor, which measures their relative contributions to λ−2 (i.e., ω1 + ω2 = 1).

The results of this analysis are presented in Fig. 3A, where the temperature dependence of λ−2 is plotted for 4H-NbSe2 at ambient pressure and for 2H-NbSe2 at pressures of p = 0, 0.4, 0.7, 1.4, and 2.2 GPa. The dashed and the solid lines for ambient pressure results represent fits for the temperature-dependent London magnetic penetration at ambient pressure using an s-wave and an s + s-wave model, respectively (30). As it can be seen, the s + s-wave provides a much better description of the data, thereby ruling out the simple s-wave model as an adequate description of λ−2(T) for 2H-NbSe2. The two-gap s + s-wave scenario with a small gap Δ1 ≃ 0.55(3) meV and a large gap Δ2 ≃ 1.25(5) meV for p = 0 GPa [with the pressure-independent weighting factor of ω2 = 0.8(1)] describes the experimental data remarkably well.

Fig. 3 Pressure evolution of various quantities.

(A) The temperature dependence of λ−2 measured at ambient pressure for 4H-NbSe2 and at various applied hydrostatic pressures for 2H-NbSe2. The solid lines correspond to a two-gap (s + s)-wave model, and the dashed line represents a fit using a single-gap s-wave model. (B) Pressure dependence of Tc and the zero-temperature value of λ−2(0). (C) Pressure dependence of the zero-temperature values of the small superconducting gap Δ1 and the large superconducting gap Δ2.

The presence of two isotropic gaps in 2H-NbSe2 and their values are in very good agreement with previous results (4347). According to angle-resolved photoemission spectroscopy (ARPES) data, several independent electronic bands (four Nb-derived bands with roughly cylindrical Fermi surfaces centered at the Γ and K points and one Se-derived band with a small ellipsoid pocket around the Γ point) cross the Fermi surface in 2H-NbSe2, and two-gap superconductivity can be understood by assuming that the SC gaps open at two distinct types of bands. We find that two-gap s + s-wave superconductivity is preserved up to the highest applied pressure of p = 2.2 GPa. All the s + s-wave fits for all pressures are shown in Fig. 3C. Furthermore, the pressure dependence of all the parameters extracted from the data analysis within the α model is plotted in Fig. 3 (B and C). The critical temperature Tc increases with pressure only by ∼0.7 K at the maximum applied pressure of p = 2.2 GPa, as shown in Fig. 3B. We, however, observe a substantial increase of the superfluid density λ−2 with increasing pressures, as shown in Fig. 3B. At the maximum applied pressure of p = 2.2 GPa, the increase of λ−2 is Δp ≈ 31.4(8)% compared to the value at ambient pressure. The absolute size of the small gap Δ1 ≃ 0.5(3) meV and that of the large gap Δ2 ≃ 1.25(5) meV remain nearly unchanged by pressure, as shown in Fig. 3C. The two-gap s + s-wave scenario also describes the data for 4H-NbSe2, which was not reported previously. The gap values for 4H-NbSe2 are Δ1 ≃ 0.19(3) and Δ2 ≃ 0.89(1) meV.

The London magnetic penetration depth λ is given as a function of ns, m*, ξ, and the mean free path l, according to1λ2=4πnse2m*c2×11+ξ/l(3)

For systems close to the clean limit, ξ/l → 0, the second term essentially becomes unity, and the simple relation 1/λ2ns/m* holds. Considering the upper critical fields Hc2 of 2H-NbSe2, as reported in detail by Soto et al. (48), we can estimate the in-plane coherence length to be ξab ≃ 7.9 nm at ambient pressure p = 0 GPa. At ambient pressure, the in-plane mean free path l was estimated to be lab ≃ 183 nm (48). No estimates are currently available for l under pressure. However, the in-plane l is most probably independent of pressure, considering the fact that the effect of compression is mostly interlayered; i.e., the intralayer Nb-Se bond length remains nearly unchanged (especially in the here investigated pressure region) (22). This very small effect of compression can be attributed 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 2H-NbSe2 lies close to the clean limit (30, 49). With this assumption, we obtain the ground-state value ns/(m*/me) ≃ 5.7 × 1027 m−3 and 7.5 × 1027 m−3 for p = 0 and 2.2 GPa, respectively.

The strong enhancement of the superfluid density λ−2(0) ∝ ns/(m*/me) in 2H-NbSe2 under pressure, as discussed above, is an essential finding of this paper. Note that the impairment of the CDW ordering and the associated lowering of the CDW ordering transition TCDW under pressure may cause the restoring of some electronic density of states at the Fermi surface. However, the electron and hole states that condense into the CDW ordered state comprise only ≈1% of the total density of states at the Fermi surface in 2H-NbSe2 (5052). The expected maximal increase of the total density of states caused by a complete suppression of the CDW would be ΔD(EF) ≈ 1%, which cannot solely attribute for the observed ∼30% enhancement of the superfluid density in 2H-NbSe2. Thus, the large pressure effect on ns/(m*/me) has a more complex origin. We also observed that both superconducting gaps Δ1 and Δ2 are nearly pressure independent up to a pressure of p = 2.2 GPa (Fig. 3C), while the CDW transition temperature is largely reduced in the pressure range p = 0 to 2.2 GPa (22, 23). This implies that the gap values are insensitive to the suppression of the CDW state. This observation strongly supports the idea that superconducting and CDW orders are somewhat isolated from each other and that the CDW pairing has only a minimal effect on the superconductivity in 2H-NbSe2 (23). Furthermore, we find the superfluid density in three TMD superconductors—2H-NbSe2, 4H-NbSe2, and 1T′-MoTe2—to scale linearly with Tc, as shown in Fig. 4 (A and B), which is not expected within BCS theory. This means that the ratio between the superfluid density and the critical temperature Tc in 2H-NbSe2 and 4H-NbSe2 is nearly the same as that for the TMD superconductor 1T′-MoTe2 (30), indicating a common mechanism and related electronic origin for the superconductivity. We observed that upon application of pressure on 2H-NbSe2, the dependence between ns/m* and Tc deviates from the linear correlation, as shown in Fig. 4B. Note that such a deviation was previously found in optimally doped cuprate and Fe-based superconductors under pressure. As an example, the inset of Fig. 4B demonstrates the deviation from the Uemura line in the optimally doped La2−xBaxCuO4 (x = 0.155) under pressure (73). Note that in the case of 1T′-MoTe2, the linear relation with the right slope holds even under pressure, up to the highest investigated pressure of 1.3 GPa, but within this pressure range, the maximum Tc of 1T′-MoTe2 is 2.7 K, which is far below from the optimal superconducting region, where Tc is about 8 K. In the case of 2H-NbSe2, the system has a Tc = 7 K already at ambient pressure, which is still below but quite close to the optimal superconducting region of the phase diagram. Application of pressure pushes the system toward the optimal superconductivity, and as can be seen from our data, the pressure of 1.7 GPa is enough to reach the maximum Tc ∼ 8 K in 2H-NbSe2. Since, historically, the linear increase of Tc with ns/(m*/me) is observed only in the underdoped region of the phase diagram of unconventional superconductors, the deviation from the linear relationship for 2H-NbSe2 under pressure can be explained by locating the system 2H-NbSe2 within or close to the optimal superconducting region under pressure. The fact that TMDs studied in this work exhibit markedly similar features in the relation between the superfluid density and the critical temperature to those reported in other unconventional superconductors implies that TMDs exhibit unconventional superconducting properties. We also show in 2H-NbSe2 that the extracted gap sizes do not depend on Tc, which is an additional evidence of unconventional behavior.

Fig. 4 Superfluid density versus Tc.

(A) Logarithmic plot of Tc against the λ−2(0) obtained from our μSR experiments in 2H-NbSe2, 4H-NbSe2, and MoTe2. The dashed red line represents the linear fit to the MoTe2 data. Uemura relation for hole- and electron-doped cuprates are shown as solid (31, 5355) and dashed lines (57), respectively. The points for various conventional BCS superconductors are also shown. (B) Linear plot of Tc against the λ−2(0). Inset shows the Tc against the λ−2(0) as a function of pressure for optimally doped cuprate superconductor La2−xBaxCuO4 (73).

The nearly linear relationship between Tc and the superfluid density was originally observed in hole-doped cuprates (31, 35), where the ratio between Tc and their effective Fermi temperature TF is about Tc/TF ∼ 0.05, which means about four to five times reduction of Tc from the ideal Bose condensation temperature for a noninteracting Bose gas. These results were discussed in terms of the crossover from Bose-Einstein condensation (BEC) to BCS-like condensation (5355). Within the picture of BEC to BCS crossover, systems exhibiting small Tc/TF (large TF) are considered to be in the BCS-like side, while the linear relationship between Tc and TF is expected only in the BEC-like side. This relationship has been used in the past for the characterization of BCS-like, so-called conventional superconductors and BEC-like, so-called unconventional superconductors. The present results on 2H-NbSe2 and 4H-NbSe2, together with our previously reported results on 1T′-MoTe2, demonstrate that a linear relation between Tc and the superfluid density holds for these TMD systems. However, we find the ratio Tc/TF to be reduced further by a factor of ∼20. These systems fall into the clean limit, and therefore, the linear relation is unrelated to pair breaking and can be regarded to hold between Tc and ns/m*. This implies that the BEC-like linear relationship may exist in systems with Tc/TF reduced even by a factor of 20 from the ratio in hole-doped cuprates.

In (5456), one of the present authors pointed out that there seem to exist at least two factors that determine Tc in unconventional superconductors: One is the superfluid density, and the other is the closeness to the competing state. The second factor can be seen in the energy of the magnetic resonance mode, which represents the difference in free energy between the superconducting state and the competing magnetically ordered state. In the case of hole-doped cuprates, the competing state is characterized by antiferromagnetic order but frustrated by the introduction of doped holes. In the case of electron-doped cuprates, the competing state develops in an antiferromagnetic network diluted by the doped carriers. In the case of present TMD systems, the competing state comes from CDW or structural orders. These systematic differences of competing states might be related to the three different ratios of Tc/TF seen in the three different families of superconductors shown in Fig. 4.

Note that the similar relation between the superfluid density and the critical temperature, observed in layered dichalcogenides and the cuprates, extends a long list of analogies between this different class of materials: pressure and doping phase diagrams (58), Nernst effect (59), optics (60), Hall effect (61), “kinks” in dispersion (62), pseudogap, and Fermi surface “arcs” (6365). Moreover, the presence of two s-wave superconducting gaps in TMDs, observed by μSR, STM, and ARPES, is analogous to the two-gap (s + s-wave) superconducting gap symmetry of the unconventional Fe-based superconductors (3234). Thus, this report contributes to an extensive list of analogies between known unconventional superconductors (5867) (i.e., the cuprates and the pnictides) and layered TMDs. Our findings are all the more the clearest, systematic observation of unconventional superconducting properties in these compounds to date.

In summary, we provide the first microscopic investigation of the superconductivity under hydrostatic pressure in the layered superconductors 2H-NbSe2 and 4H-NbSe2. Specifically, the zero-temperature magnetic penetration depth λeff(0) and the temperature dependence of λeff2 were studied by means of μSR experiments in 2H-NbSe2 as a function of pressure up to p ≃ 2.2 GPa and in 4H-NbSe2 at ambient pressure. The superfluid densities in both samples and at all pressures are best described by a two-gap s + s-wave scenario. Considering the current data on 2H-NbSe2 and 4H-NbSe2 at ambient pressure and our previous observations on 1T′-MoTe2 (30) at ambient as well as at low pressures, we conclude that the superfluid density ns/m* ∝ 1/λ2 scales linearly with Tc in the three TMD superconductors 1T′-MoTe2, 2H-NbSe2, and 4H-NbSe2. We also find that the application of pressure on 2H-NbSe2 causes a substantial increase of the superfluid density ns/m*, which correlates with Tc. However, the ns/m* versus Tc dependence shows a linear deviation from the Uemura relation. Such a deviation was also previously found in optimally doped cuprate and Fe-based superconductors under pressure. We explain this deviation by considering the fact that the system 1T′-MoTe2 at low pressures and 4H-NbSe2 and 2H-NbSe2 at ambient pressure are located below the optimal SC region of the phase diagram and thus show excellent linear scaling between ns/m* and Tc, while 2H-NbSe2 is pushed into the optimal SC region by pressure, causing the deviation from the Uemura relation between ns/m* and Tc; the same is observed for cuprates. Our results demonstrate that a linear relation holds for the above-studied TMD superconductors located below the optimal superconducting region of the phase diagram. A ratio Tc/TF for TMDs is reduced by a factor of 20 from the ratio in hole-doped cuprates. This implies that the superfluid density of TMDs is about an order of magnitude higher than that in the cuprates with respect to their critical temperatures Tc. The fact that TMDs exhibit markedly similar features in the relation between the superfluid density and the critical temperature to those reported in other unconventional superconductor implies that TMDs exhibit rather unconventional superconducting properties. We also find that the values of the superconducting gaps are insensitive to the suppression of the CDW ordered state, indicating that CDW pairing has only a minimal effect on the superconductivity in 2H-NbSe2. These results hint toward a common mechanism and electronic origin for superconductivity in TMDs, which might have far-reaching consequences for the future development of devices based on these materials.

METHODS

Sample preparation

Single-phase polycrystalline samples of 2H-NbSe2 and 4H-NbSe2 were prepared by means of high-temperature solid-state synthesis. Stoichiometric amounts of niobium powder (99.99%) and selenium shots (99.999%) were mixed (for 4H-NbSe2, an excess of selenium was used) and heated in a sealed quartz tube under an inert atmosphere at 750°C (2H-NbSe2) and at 900°C (4H-NbSe2) for 3 days.

Pressure cell

Pressures up to 2.2 GPa were generated in a double-wall piston cylinder type of cell made of CuBe material, especially designed to perform μSR experiments under pressure (6871). As a pressure-transmitting medium, Daphne oil was used. The pressure was measured by tracking the SC transition of a very small indium plate by AC susceptibility. The filling factor of the pressure cell was maximized. The fraction of the muons stopping in the sample was approximately 40%.

μSR experiment

In a μSR experiment (72), nearly 100% spin-polarized muons μ+ were implanted into the sample one at a time. The positively charged μ+ thermalize at interstitial lattice sites, where they act as magnetic microprobes. In a magnetic material, the muon spin precesses in the local field Bμ at the muon site with the Larmor frequency νμ = γμ/(2π)Bμ [muon gyromagnetic ratio γμ/(2π) = 135.5 MHz T−1]. Using the μSR technique, important length scales of superconductors can be measured, namely, the magnetic penetration depth λ and the coherence length ξ. If a type II superconductor is cooled below Tc in an applied magnetic field ranged between the lower (Hc1) and the upper (Hc2) critical fields, a vortex lattice is formed, which, in general, is incommensurate with the crystal lattice with vortex cores separated by much larger distances than those of the unit cell. Because the implanted muons stop at given crystallographic sites, they will randomly probe the field distribution of the vortex lattice. These measurements need to be performed in a field applied perpendicular to the initial muon spin polarization (so-called TF configuration).

μSR experiments under pressure were performed at the μE1 beamline of the Paul Scherrer Institute (Villigen, Switzerland), where an intense high-energy (pμ = 100 MeV/c) beam of muons was implanted in the sample through the pressure cell. The low-background Dolly instrument was used to study the polycrystalline samples of 2H-NbSe2 and 4H-NbSe2 at ambient pressure.

Analysis of TF-μSR data

The TF μSR data were analyzed by using the following functional form (39)P(t)=Asexp [(σsc2+σnm2)t22]cos (γμBint,st+φ)+Apcexp [σpc2t22]cos (γμBint,pct+φ)(4)

Here, As and Apc denote the initial assymmetries of the sample and the pressure cell, respectively. γ/(2π) ≃ 135.5 MHz/T is the muon gyromagnetic ratio, φ is the initial phase of the muon-spin ensemble, and Bint represents the internal magnetic field at the muon site. The relaxation rates σsc and σnm characterize the damping due to the formation of the FLL in the SC state and of the nuclear magnetic dipolar contribution, respectively. In the analysis, σnm was assumed to be constant over the entire temperature range and was fixed to the value obtained above Tc, where only nuclear magnetic moments contribute to the muon depolarization rate σ. The Gaussian relaxation rate, σpc, reflects the depolarization owing to the nuclear moments of the pressure cell. The width of the pressure cell signal increases below Tc. As shown previously (70), this is due to the influence of the diamagnetic moment of the SC sample on the pressure cell, leading to the temperature-dependent σpc below Tc. To consider this influence, we assumed the linear coupling between σpc and the field shift of the internal magnetic field in the SC state, σpc(T) = σpc(T > Tc) + C(T)(μ0Hint, NS ─ μ0Hint, SC), where apc(T > Tc) = 0.25 μs−1 is the temperature-independent Gaussian relaxation rate. μ0Hint, NS and μ0Hint, SC are the internal magnetic fields measured in the normal and in the SC state, respectively. As indicated by the solid lines in Fig. 1 (C and D), the μSR data were well described by Eq. 4. The good agreement between the fits and the data demonstrates that the model used describes the data rather well.

Analysis of λ(T)

λ(T) was calculated within the local (London) approximation (λ ≫ ξ) by the following expression (39, 40)λ2(T,Δ0,i)λ2(0,Δ0,i)=1+1π02πΔ(T,φ)(fE)EdEdφE2Δi(T,φ)2(5)where f = [1 + exp (E/kBT)]−1 is the Fermi function, φ is the angle along the Fermi surface, and Δi(T, φ) = Δ0,iΓ(T/Tc)g(φ) (Δ0, i is the maximum gap value at T = 0). The temperature dependence of the gap was approximated by the expression Γ(T/Tc) = tanh {1.82[1.018(Tc/T − 1)]0.51} (41), while g(φ) describes the angular dependence of the gap and it is replaced by 1 for both an s-wave and an s + s-wave gap and ∣cos (2φ)∣ for a d-wave gap.

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REFERENCES AND NOTES

Acknowledgments: The μSR experiments were carried out at the Swiss Muon Source (SμS) Paul Scherrer Insitute, Villigen, Switzerland. Funding: Z.G. acknowledges the financial support by the Swiss National Science Foundation (SNF fellowship P300P2-177832). The work at the University of Zurich was supported by the Swiss National Science Foundation under grant no. PZ00P2_174015. Work at Department of Physics of Columbia University was supported by U.S. NSF DMR-1436095 (DMREF) and NSF DMR-1610633. M.Z.H. was supported by U.S. DOE/BES grant no. DE-FG-02-05ER46200. R.K. acknowledges the Swiss National Science Foundation (grants 200021_149486 and 200021_175935). A.N. acknowledges funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement no. 701647. Author contributions: Project planning: Z.G. and F.O.v.R. Sample growth: F.O.v.R. and C.W. Experiments and corresponding discussions: Z.G., J.-C.O., R.K., Z.S., A.N., A.R.W., A.N.P., J.C., M.Z.H., A.A., H.L., and Y.J.U. μSR data analysis: Z.G. Data interpretation and draft writing: Z.G. and F.O.v.R. with contributions and/or comments from all authors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper. Additional data related to this paper may be requested from the authors.
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