Unusual evolution of B_{c2} and T_c with inclined fields in restacked TaS_2 nanosheets

Recently we reported an enhanced superconductivity in restacked monolayer TaS_2 nanosheets compared with the bulk TaS_2, pointing to the exotic physical properties of low dimensional systems. Here we tune the superconducting properties of this system with magnetic field along different directions, where a strong Pauli paramagnetic spin-splitting effect is found in this system. Importantly, an unusual enhancement as high as 3.8 times of the upper critical field B_{c2}, as compered with the Ginzburg-Landau (GL) model and Tinkham model, is observed under the inclined external magnetic field. Moreover, with the out-of-plane field fixed, we find that the superconducting transition temperature T_c can be enhanced by increasing the in-plane field and forms a dome-shaped phase diagram. An extended GL model considering the special microstructure with wrinkles was proposed to describe the results. The restacked crystal structure without inversion center along with the strong spin-orbit coupling may also play an important role for our observations.


Introduction
Superconductivity in low-dimensional systems was investigated extensively recently, due to the fertile physical phenomenon and exotic properties [1][2][3][4] . At present, the gate of this research field has just been opened and more interesting phenomena are waiting to be explored. Because of the strong spin-orbit coupling, superconducting transition metal dichalcogenides (TMDs) are investigated intensively in the two-dimensional (2D) limit in recent years [5][6][7][8] . A clear enhancement of the in-plane upper critical field was frequently reported in these materials, which was interpreted by the Zeeman-protected Ising superconductivity mechanism. Using a chemical exfoliation method, we have obtained the monolayer TaS 2 nanosheets, which were assembled layer-by-layer by vacuum filtration 9,10 . Such a restacked material shows superconductivity with T c (∼ 3.2 K) several times higher than the pristine bulk 2H-TaS 2 , which supplies a significant platform for studying the intrinsic physical properties of unconventional superconductivity in TMDs. Such an enhancement of T c is consistent with a previous work by E. N. Moratalla et al 11 , although they didn't reach the one-layer limit, and finally confirmed by other two groups by mechanically exfoliating TaS 2 to monolayer 12,13 . The enhancement of T c was believed to originate from the suppression of the charge-density wave and the increase of the density of states by the process of thickness reduction. 9,12 However, an in-depth investigation on the physical behaviors of the restacked TaS 2 is lacking and more experiments are required at present. Magnetic field is one of the fundamental tuning parameters to affect the behaviors of a superconductor. In the type-II superconductors, the magnetic field can penetrate into the bulk in the form of quantized vortex lines when it exceeds the lower critical field B c1 14 Fig. 1c for the two orientations. Instead of the square root behavior for the in-plane upper critical field (B ab c2 ∼ 1 − T /T c ) expected for the 2D superconductors [5][6][7][8]12 , an opposite tendency with a positive curvature is observed. This has been found to be a universal feature of anisotropic three-dimensional (3D) layered superconductors 15,16 . This reflects the influence of inter-layer coupling on the in-plane upper critical field of our samples, although such an inter-layer stacking manner doesn't affect T c . The value of B c2 at zero temperature can be estimated using the Werthamer-Helfand-Hohenberg relation 17 B c2 = −0.693 × dB c2 (T )/dT | Tc × T c after the slope dB c2 (T )/dT | Tc is obtained from Fig. 1c. In addition, the paramagnetic limiting field B P has a simple relation with T c , B P = 1.84 × T c based on the conventional BCS theory 18 . The resultant values for the three characteristic fields B ab c2 (in-plane B c2 ), B c c2 (out-of-plane B c2 ) and B P are denoted by arrows in Figure 1c and summarized in Table 1, from which the anisotropy of upper critical field Γ = B ab c2 /B c c2 = 11 is obtained. This value is larger than most of the iron-based superconductors and the copper-based superconductor YBCO 16,19 . Moreover, a clear relative relation B c c2 < B P < B ab c2 can be deduced.
Field-angle resolved experiments were performed by measuring the field (B) and angle (θ) dependence of resistivity at a fixed temperature 2.

Discussion
The detailed investigations on the thickness dependence of superconducting behaviors of 2H-TaS 2 12 supply a good coordinate to make a comparison with our results. It is found that the critical transition temperature T c increases with the decrease of thickness and reaches 3.4 K for monolayer TaS 2 . This value is very close to our sample, confirming the monolayer-features of our sample and suggesting that the restacking process imposed on the monolayer TaS 2 doesn't affect T c of this system. Moreover, the normal state resistivity displays a T 2.45 behavior in low temperature in our sample (see Fig. S4), corresponding to the situation between 3-layer (∼ T 2 ) and 7-layer (∼ T 3 ) for the ordered stacked TaS 2 12 . This implies that the inter-layer coupling in our samples shows a certain degree of influence on the electrical transport behavior. A more careful examination shows that the out-of-plane upper critical field B c c2 is similar to the bilayer TaS 2 , while the in-plane B ab c2 and the anisotropy are only one third of the bilayer TaS 2 . Nevertheless, B ab c2 of our samples is clearly larger than that of the bulk samples since the latter doesn't exceed the Pauli limit. All in all, the present samples are different from both the monolayered and the bulk TaS 2 . The restacked TaS 2 monolayers maintain the enhanced T c (compared with the bulk material), while lose the 2D characters and show an anisotropic 3D features. This is the basis of the following discussions.
Field-induced superconductivity has been theoretically proposed for ferromagnetic materials and is known as Jaccarino-Peter compensation effect. 24 In this effect, the internal magnetic field created by the magnetic moments through the exchange interaction can be compensated by the external magnetic field and superconductivity will occur. This mechanism has been realized in Eu-Sn molybdenum chalcogenides experimentally. 25 Moreover, another theory proposed by Kharitonov and Feigelman considered the polarization of magnetic impurity spins induced by the in-plane field and predicted an enhancement of superconductivity, especially in disordered films. 26 Evaluating the performances of our samples, we found that neither of the two theories is applicable. First of all, our magnetization measurement shows a paramagnetic behavior in our samples (see Fig. S3(c)), which excludes the presence of long-range magnetic moments and magnetic impurities. Secondly, the enhancement of T c by in-plane field in our samples can only be observed in the presence of out-of-plane field, which is rather different from the case of the above-mentioned scenarios.
Intuitively, the theoretical proposal 27-29 predicting a field-induced triplet component in the order parameter of the singlet superconductors due to the Pauli paramagnetic spin-splitting effect is very consistent with our observation shown in Fig. 2(d). According to their arguments, such an enhancement of B c2 (θ) should be conspicuous in anisotropic superconductors with B c c2 ≪ B P ≪ B ab c2 . However, we note that such an exotic behavior is absent in so many low-dimensional TMDs with even stronger Pauli paramagnetic spin-splitting effect 6,7 , which weakens the persuasion of this interpretation. Other important factors should be considered to interpret our experiments.
One important clue for exploring the physical origination is that such a rare behavior was also observed in restacked 1T'-MoS 2 (see Fig. S7) prepared with the similar process 30 to that used in restacked TaS 2 . This implies that the unique and common features of such restacked monolayered materials are key factors for our observations. One most conspicuous feature for such restacked monolayer materials is the noncentrosymmetric crystal structure as mentioned in the beginning of the Results section. It has been discussed a lot both theoretically and experimentally that the noncentrosymmetric crystal structure along with strong spin-orbit coupling (SOC), which also exists in the present compound with 5d metal, is very favorable to incur the spin-triplet component in the superconducting order parameter 31-37 . This may be one possible origin of our observations.
One positive evidence for this scenario is that the restacked 1T'-MoS 2 , which have a weaker SOC because of the lighter 4d Mo element, shows an inconspicuous enhancement of B c2 (θ) compared with restacked TaS 2 (see Fig. S7).
One possible extrinsic origination to explain our results comes from the possible orientation mismatch or wrinkles (see Fig. S2) of the monolayer TaS 2 sheets during the restacking process, which affects the c-axis orientation of the restacked samples. In section 9 of SI, we carefully analyzed the influences of different orientations on the estimation of B c2 and found that B c2 is mainly determined by the high-T c portion where the included angle between the surface and field is the smallest. The presence of the low-T c portion with a larger included angle can lower the onset transition temperature slightly (see Fig. S8). The simple GL model 20 is not applicable any more in this case. Considering the misaligned angle ∆θ resulting from the wrinkles as schematically displayed in the inset of Fig. 4, we proposed an extended GL (EGL) model: where Θ = min |θ − ∆θ| is the smallest included angle between the surface and field, and A(θ) is an adjusting parameter taking into account the effect of the low-T c portion with other orientations.
Here we simply assume that A(θ) is proportional to the GL formula, because A(θ) will be larger when T c is more sensitive with the orientation, which can be reflected by the GL formula. Typically the misaligned angle ∆θ has a distribution range with a maximum ∆θ M ax . Then we have By tuning the value of parameters, it is found that when ∆θ M ax = 27 o , this EGL model can well describe the experimental data in the range θ > ∆θ M ax , as shown in Fig. 4a. As for the case of θ ≤ ∆θ M ax , B c2 (θ) is only determined by the adjusting parameter A(θ). However, the influence factors on A(θ) are rather complicated in this region, so we could not get a good description at present. At this stage, we could not exclude the possibility that the wrinkles have an important influence on our observations. Even in this scenario, our results supplied an interesting prototypical system showing an unambiguous relationship between the microstructure and the physical (superconducting) performance. Moreover, this is also valuable in designing devices for applications.
It is notable that the enhancement of T c by B ab at the present of B c is simply a natural consequence of the unusual behavior in B c2 (θ). As shown in Fig. 4b and 4c, supposing a linear suppression of B c2 with the temperature approaching T c : B c2 (T ) ∼ 1−T /T c , angle dependence of B c2 (θ) at other temperatures can be derived from the data at 2.2 K for both the experimental data and GL model, respectively. The total values of field in the experiment of Fig. 3