In low-dimensional electron systems, charge density waves (CDW) and superconductivity are two of the most fundamental collective quantum phenomena. For all known quasi-two-dimensional superconductors, the origin and exact boundary of the electronic orderings and superconductivity are still attractive problems. Through transport and thermodynamic measurements, we report on the field-temperature phase diagram in 2H-TaS2 single crystals. We show that the superconducting transition temperature (Tc) increases by one order of magnitude from temperatures at 0.98 K up to 9.15 K at 8.7 GPa when the Tc becomes very sharp. Additionally, the effects of 8.7 GPa illustrate a suppression of the CDW ground state, with critically small Fermi surfaces. Below the Tc the lattice of magnetic flux lines melts from a solid-like state to a broad vortex liquid phase region. Our measurements indicate an unconventional s-wave-like picture with two energy gaps evidencing its multi-band nature.
For more than four decades, one of the major subjects in condensed matter physics has been the coexistence of the charge density wave (CDW) order and superconductivity in transition metal dichalcogenides (TMDs)1,2. In CDW materials such a coupling between the electrons and the soft-phonon mode describes the phase transition from the CDW to a normal state3. The superconducting transition temperature (Tc) increases while the CDW lock-in temperature falls by doping4, critical thicknesses5, or by external pressure6,7,8. Recently, Klemm9 has shown that most of the pristine TMDs are highly unconventional in comparison with conventional superconductors. Amongst many TMD materials, 2H-TaS2 (H: hexagonal, see Methods, Extended Data Fig. 1) becomes superconducting at ambient pressure and without doping4. So far, this compound is one of the very few materials where a chiral and polar charge-ordered phase is suggested to exist10,11. Based on scanning tunneling microscopy measurements, the nodal gap structure of a single-layer material has recently been proposed12. Moreover, the lack of agreement on the electronic properties of 2H-TaS2, the information on its magnetic properties and, the Abrikosov vortex dynamics, is also missing up to now13. Therefore, the appearance of superconductivity in 2H-TaS2 in the presence of a CDW is of great interest. This has motivated us to study the low temperature-field dependencies of both transport and thermodynamics in the normal and superconducting states of 2H-TaS2 single crystals to determine their superconducting properties.
The temperature dependencies of the in-plane and out-of-plane zero-field resistivity (ρab and ρc) are shown in Fig. 1(a). Both ρab and ρc exhibit a prominent CDW anomaly at 76 K (see The Methods, Extended Data Fig. 2). A parameter often used to characterize the interlayer coupling, is the anisotropy of the resistivity ρc/ρab. The largest anisotropy ratio found here is ρc/ρab ~ 16 just above the Tc. We noticed in particular that the anisotropy ratio is almost temperature independent. This anisotropy ratio behavior suggests that the in-plane and out-of-plane transport in 2H-TaS2 share the same scattering mechanism. Upon lowering the temperature below the CDW transition, the resistivity displays a drop to zero as shown in the inset of Fig. 1(a). The detailed magnetic field and temperature dependencies of ρab(H) at various temperatures ranging from, 60 mK to 3 K with the field direction parallel to the c-plane of the crystal, are presented in Fig. 2. At low temperatures, the curves are almost parallel to each other in the transition region. With increasing magnetic fields, the onset of superconductivity shifts to lower temperatures gradually. The suppression of superconductivity with a magnetic field applied along the c-direction is more obvious than that in the H||ab configuration, indicating a high anisotropy for a low Tc in 2H-TaS2.
It is worth mentioning that the Tc in resistivity at ambient pressure is anomalously wide. It is about 0.62 K from the onset Tc value to the zero resistivity value of the Tc at 0.98 K, i.e. about 50% of the Tc. This anomalous could have several sources: chemical or electronic inhomogeneity, fluctuations, or vortex effects. Inhomogeneity is indeed expected to widen the transition of this compound because studies show that even a small concentration of dopants enhance the Tc dramatically4,14. This is why superconductivity above 1 K in nominally pure 2H-TaS2 is explained by a small Ta excess or by the presence of sub 1% quantities of impurity atoms4. An intrinsic electronic inhomogeneity related to inhomogeneous CDW is quite possible as the chiral CDW reported for this compound supposes a domain structure4 which is inline with the observed narrowing of the Tc with CDW destruction, for example with doping by Ni14. On the other hand, both kinds of inhomogeneity could also affect the width of the transition in heat capacity; however that is only about 0.2 K, much narrower than the . This suggests that the effects of these inhomogeneities are limited by 0.2 K, while the rest of the is related to fluctuations and/or vortices. The dissipative vortex motion could either be due to the flux flow through low pinning centers or due to the free motion of individual vortices in the vortex liquid state. Since in our resistivity experiments we used the lowest current 0.1μA and the was not sensitive to its small enhancement, we may consider the vortex liquid state as the most probable mechanism of widening the transition, similarly to CuxTiSe215. The vortex liquid can be considered a result of fluctuations in the vortex lattice below the Tc, while fluctuations in the superconducting order parameter lead to the appearance of preformed pairs above the Tc. The measurement of both fluctuation regions is the Ginzburg number Gi = δT/Tc, which is usually extremely small for the low-temperature superconductors Gi ~ (Tc/EF)4 ~ 10−12–10−14, even for two-dimensional ones, for which Gi ~ Tc/EF or τ−1/EF for the clean and dirty limits respectively16. Here δT is the range of temperatures in which fluctuation corrections are relevant and τ−1 is the quasiparticle scattering rate at the Fermi energy (EF). However, in the CDW state, the reconstructed FS may have small and very shallow pockets for which the EF could not be much larger than Tc or 1/τ17. Therefore, the broadening of the Tc due to the interplay with CDW is further supported by the sharp Tc after the suppression of the CDW upon compression.
Enhancement of Superconductivity Upon Compression
In low-dimensional electron systems, CDW and superconductivity are two of the most fundamental collective quantum phenomena1,2. Unconventional superconductivity is nearly always found in the vicinity of another ordered state, such as antiferromagnetism, CDW, or stripe order. This suggests a fundamental connection between superconductivity and fluctuations in some other order parameter18. To better understand this connection, we used high-pressure resistivity to directly study the CDW order in 2H-TaS2. The effect of pressure on 2H-TaS2 is presented in Fig. 1(b). Upon 3.1 GPa, the CDW slightly shifts to 69 K. The effects of 8.7 GPa illustrate a suppression of the TCDW. In addition, a very sharp drop in resistivity indicates the onset of superconductivity and dramatically enhances the modest Tc to ~9.15 K upon 8.7 GPa. Similarly to recently reported data19, our resistance measurements show that the Tc increases from temperatures below 1 K up to 8.5 K at 9.5 GPa. Additionally, the authors observed a kink in the pressure dependence of TCDW at about 4 GPa that they attributed to the lock-in transition from an incommensurate CDW to a commensurate CDW. Above this pressure, the commensurate TCDW slowly decreases, coexisting with superconductivity within our full pressure range. These observations show that the enhancement in superconductivity is due to the consequent changes of Fermi surface (FS) upon compression. However, this is not direct evidence that confirms where such features act on superconductivity independently of the CDW. In the CDW state, a gap opens up over part of the FS in the direction of the q vectors of the CDW8. This reduces the average density of states at the FS. Upon compression, TCDW is suppressed. The amplitude of the CDW lattice distortion also suppresses, thus gradually restoring the FS and increasing the Tc. Therefore, one can see that both superconductivity and the CDW involve widely different parts of the FS associated with the absence of or small interband correlations. It is worth noting that superconductivity in 2H-NbSe2 is only moderately affected by pressure20,21 and the CDW already disappears at 5 GPa20,22. The weak pressure dependence of the TCDW at higher pressures indicates that the CDW in this pressure range is remarkably robust to a reduction in the lattice parameters19. Very recently23, in 2H-NbSe2 the rapid destruction of the CDW under pressure was found to be related to the quantum fluctuations of the lattice renormalized by the anharmonic part of the lattice potential. In addition, the connection between CDWs and superconductivity arises from the fact that high-energy optical phonon modes have a strong contribution to the Eliashberg function, whereas the low-energy longitudinal acoustic mode that drives the CDW transition barely contributes to superconductivity
Specific Heat Measurements
To further elucidate the bulk superconductivity in 2H-TaSe2, we performed heat capacity studies down to 70 mK. Figure 3 summarizes the T-dependence of the specific heat data in various magnetic fields applied parallel and perpendicular to the ab plane. We observed a clear sharp anomaly at Tc = 1.4 K, close to that determined by our resistivity measurements. The specific heat jump systematically shifted to lower temperatures upon the application of magnetic fields. Our data of small fields close to the Tc shows the evolution of a small fluctuation, peak, overlapped with the specific-heat jump [see inset of Fig. 3(b)]. On the other hand, both kinds of chemical or electronic inhomogeneity should also affect the width of the transition in heat capacity, however that is only about 0.2 K, much narrower than the . This suggests that the effects of inhomogeneities are limited by 0.2 K, while the rest of could be related to fluctuations and/or vortices. A clear maximum of specific heat data at 76 K, typically found in 2H-TaS2 which is weakly first-order, is an indication of the CDW transition [see the inset of Fig. 3(a)]. Note that there is no upturn (Schottky nuclear contribution) in the specific heat data measured to temperatures as low as 70 mK, thus, the zero-field specific heat above Tc can be well fitted to Cp/T = γn + βT2, where γn and β are the electronic and lattice coefficients, respectively [see the dashed line in Fig. 3(b)]. The γn value is found to be around 8.8 mJ/mol K2, indicating that 2H-TaS2 in the CDW state is characterized by a modest density of states. This value agrees with the γn value found by refs 4 and 24 in which Cp was just measured between 1.8 and 10 K. The phononic coefficient β is found to be 0.35 mJ/mol K4. Using the relation θD = (12π4RN/5β)1/3, we obtained the Debye temperature θD = 249(2) K, which is comparable with values reported by DiSalvo et al.2. From the determined γn value, we found that ΔCel/γnTc = 0.72. This value is smaller than the prediction of the weak coupling BCS theory (ΔCel/γnTc = 1.43) and comparable to that in the intercalated compound24. This indicates that the specific-heat data cannot be described by a simple BCS gap (see Methods, Extended Data Fig. 3). However, in a clean situation with negligible pair-breaking effects, the reduced jump in the specific heat compared to that of a single-band s-wave superconductor might be related to unconventional superconductivity with nodes and/or a pronounced multiband character with rather different partial densities of states and gap values25. In addition, evidence of coupling effects arises from the normalized discontinuity value of the specific-heat slopes at the Tc, (Tc/ΔC)(dC/dT. In the single-band weak coupling BCS theory this ratio is 2.64, whereas a value of 3.35 can be deduced in the two-band superconductor MgB226. From our data, we obtained a value of (Tc/ΔC)(dC/dT ~ 3.54, which is very close to MgB2.
H - T Phase Diagram
The Hc2 provides a valuable insight into the nature of the interaction responsible for the formation of Cooper pairs25,27,28. The temperature dependencies of Hc2 and Hirr obtained from C(T, H) and ρ(T, H) with both H||c and H||ab are plotted in Fig. 4 for both orientations. Specific heat Tc(H) values were deduced from the classical entropy conservation construction. The criteria of the normal state in resistivity was used to extract the Tc at each magnetic field. The irreversible magnetic field Hirr was obtained from the zero value of Tc in ρab curves. However, the width of the resistive transition is shown in the inset of Fig. 4(b) and is proportional to μ0H2/3. This is inline with Tinkham’s theoretical prediction29 of the ΔT ∝ μ0H2/3. The large area between the Hc2 and Hirr curves suggests that the vortex dissipation level is still low in this region. Moreover, the possible existence of a distinct Hirr(T) far below Hc2 is due to the fact that the vortex lattices are soft and easily melted into vortex liquid by the magnetic field or thermal fluctuations30. The zero-temperature values for and are estimated to be approximately 0.31 and 1.38 T, respectively. From those we estimated the anisotropic coherence length = 32.6 nm and ξc = 7.3 nm. One can also estimate the coherence length from the uncertainty principle and BCS model. From the Faber-Pippard ratio, ξ = 0.18ħvF/kBTc = 260 nm, for Tc = 1.4 K and an average Fermi velocity vF ≈ 1.5 eVÅ17, which is considered to be similar for 2H-TaSe2, 2H-NbSe2 and 2H-TaS2. This shows that both anisotropy and CDW effects on electronic structure should be taken into account. Furthermore, it has been reported31 that the field-induced antiferromagnetism can extend outside the effective vortex core region where the superconducting order parameter is finite. Such an extended magnetic order is expected to suppress the superconducting order parameter around vortices. This effect will enlarge the vortex core size, which in turn will suppress the Hc2. The effective core size has been found to be around three times that of the coherence length in 2H-NbSe232. From the behavior of Hc2 vs. T for the different field orientations, we have calculated the anisotropy as = ξab/ξc. The anisotropy Γ increases upon approaching the Tc and reaches about 4(1). This indicates that the orbital pair breaking also accounts for the suppression of superconductivity close to Tc in 2H-TaS2.
In the case of multi-band superconductivity33,34,35,36 the low-temperature Hc2-curve may exceed the single-band Werthamer-Helfand-Hohenberg predictions37. However, a noticeable upward curvature in the Hc2(T) observed in some compounds has been attributed to multiband effects38. Using typical renormalized Fermi velocities derived from preliminary ARPES-data17 and Tc = 1.4 K, one also estimates, that in principle by a two-band approach adopting s-symmetry38, the slope-value is: Hc2,c = , where c1 → c2 → 1/2 and in the case of a dominant interband pairing results in - = 0.14 T/K near the Tc which is already very close to our experimentally determined value. By fitting it using the two-band theory33,39,40, one can obtain the band diffusivities D1, D2 and the intraband and interband coupling constants λ11, λ12 and λ21. The exact relations can be found in ref. 38. Using the band diffusivity ratio η = D2/D1 = 800, λ11 = 0.5 and λ12 = λ21 = 0.25, we fitted our data for 2H-TaS2. The obtained two-band fitting agrees well with the experimental data. To add more insight to the pairing symmetry for the 2H-TaS2 superconductor, we investigated the temperature dependence of the specific heat. The detailed electronic specific heat data and analysis are given in the Extended Data Fig. 3.
Summarizing, we have reported the first superconducting fluctuations investigation across the effect of pressure on the CDW state in 2H-TaS2. From an extensive thermodynamic study, we found a considerable broadening of the Tc at ambient pressure and its sharp transition at high pressures together with an unexpectedly broad region of vortex liquid phase in the vortex phase diagram. These results suggest the presence of the the superconducting fluctuations in the CDW state. Besides of a clear fundamental interest in our system, this finding can be used to control the fluctuations in quantum devices.
Low-temperature transport (down to 60 mK) and specific heat (down to 70 mK) measurements were performed using a dilution refrigerator. The conductance anisotropy in layered material single crystals is large therefore using traditional four-terminal methods to determine the resistivity along the c axis, ρc and in the ab plane, ρab, may be unreliable41. We used six terminals to determine each principal component of resistivity. In the latter method, the current was injected through the outermost contacts on one surface. Voltages were measured across the innermost contacts of each surface. The Laplace equation was then solved and inverted to find ρc and ρab42. In addition, this method allowed testing the sample homogeneity by permuting the electrodes which were used for the current and voltage41,42. Four contacts were used to measure the high-pressure in-plane resistivity. The investigated 2H-TaS2 single crystals were synthesized at hq graphene and were of high purity (>99.995%). The resistivity and specific heat measurements down to 0.4 K were measured in a Physical Property Measurement System (Quantum Design) with an adiabatic thermal relaxation technique.
Any additional Methods, Extended Data display items and Source Data are available in the online version of the paper; references unique to these sections appear only in the online paper.
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We acknowledge Goran Karapetrov for discussions. Z.H. and J.Z. acknowledge support from the SPSP (No. 13PJ1401100).
The authors declare no competing financial interests.
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Abdel-Hafiez, M., Zhao, XM., Kordyuk, A. et al. Enhancement of superconductivity under pressure and the magnetic phase diagram of tantalum disulfide single crystals. Sci Rep 6, 31824 (2016). https://doi.org/10.1038/srep31824
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