2D superconductivity and vortex dynamics in 1T-MoS2

The Berezinskii-Kosterlitz-Thouless phase transition and Bose metal phase, driven by vortices and their dynamics are the hallmark features of a clean two-dimensional superconductor. Materials with a minimal structural disorder and high conductivity are essential for the observation of these features. 1T-MoS2, a relatively unexplored metallic van der Waals material, is a promising candidate in this regard. Here, we report the observation of two-dimensional superconductivity in a few-layer 1T-MoS2 device. The electrical characterisation reveals a transition temperature Tc ~920 mK. Berezinskii-Kosterlitz-Thouless transition and anisotropy in the magneto-transport confirm the dimensionality of the superconductivity. In addition, we observe an enhancement in the parallel upper-critical-field and emergence of the Bose metal state in our sample. The observation of 2D superconductivity in 1T-MoS2 and the capability to scalably engineer this phase on the semiconducting 2H-MoS2 phase opens up a new route for the realisation and study of monolithic hybrid quantum circuits. Omnipresent vortices and their dynamics enrich clean two-dimensional superconducting systems with striking features. The authors experimentally investigate the Berezinskii-Kosterlitz-Thouless transition and the Bose metal phase in 1T-MoS2.

S ystems with reduced dimensions provide an opportunity to test fundamental theorems, formulate new ones and access new technologies. The search for non-Abelian quantum Hall systems in one-dimensional constrictions 1 and superconductivity in two-dimensions (2D) 2,3 are a few examples. Mermin-Wagner theorem forbids ordering and spontaneous symmetry breaking in dimensions D 2 for non-zero temperatures 4 . In contrast, experiments involving 2D systems vary; isolation of a single atomic layer of graphite 5 , 2D metallic state at ultra-low-temperatures 6 and 2D superconductivity 3 invoke discussions on the limit and applicability of the theorem to practical systems. Traditionally, 2D systems are realised on interfaces of bulk materials or thin films which are strongly disordered, while controlled experiments in this direction require systems with a minimal structural disorder. The van der Waals (vW) materials, highly crystalline and inherently two-dimensional in the fewlayer limit, have recently become one of the most sought-after choices for exploring various quantum phenomena [7][8][9][10][11] . Among the vW materials, transition metal dichalcogenides (TMDCs), and in particular MoS 2 , has shown potential as a platform for observing quantum phenomena and for fabricating quantum devices [7][8][9][10][11][12] . Apart from the high degree of electrical tunability, what makes MoS 2 fascinating is the presence of 2D polymorphs displaying a wide spectrum of electrical properties 13; 2H and 3R are large bandgap semiconductors, 1T' and 1T" are narrow bandgap semiconductors and 1T is metallic. The prospect of engineering a 2D superconducting polymorphic phase could make MoS 2 a universal platform for hosting hybrid quantum circuits consisting of semiconducting and superconducting circuit elements [14][15][16] . Recently, superconducting phase transition has been reported in 1T and 1T' phases of MoS 2 [17][18][19] , nevertheless the electrical transport is not explored in detail, and the dimensionality of superconductivity is not addressed. 2D superconductivity is limited to systems having sheet-resistance less than the quantum resistance, R Q ¼ h 4e 2 $ 6:45 kΩ 3,20; the metallic nature and high carrier concentration make 1T-MoS 2 a natural choice.
Vortices and their dynamics are characterising features of superconductivity in the 2D limit. The Berezinskii-Kosterlitz-Thouless (BKT) phase transition 21,22 caused by vortex-antivortex unbinding and the anisotropy in the magneto-transport are considered as the signatures of 2D superconductivity. Systems with a low disorder in the presence of a magnetic field reveal a 2D quantum metallic phase (Bose metal) as a result of dissipation induced by disruption of global phase coherence [23][24][25][26][27] . Until the advent of vW materials 3 2D superconductivity has been confined to disordered material systems 20,28-31 such as the interface of bulk materials and thin films. Low structural disorder and high crystallinity make 2D superconductivity and associated phenomena easier to observe in vW systems compared to disordered systems. A substantial increase in the parallel upper critical-field compared to that predicted by the Pauli limit has been observed in crystalline 2D systems 28,[32][33][34][35] . This has been attributed to spinorbit coupling (SOC) and broken-inversion symmetry, resulting in spin-splitting of conduction electrons making it resilient to the magnetic field induced pair breaking. A similar enhancement in the upper critical field has also been reported in centrosymmetric systems recently 36 .
Owing to the limited electrical tunability, attempts to explore superconductivity in Graphene by doping 37 have not met with much success. Many of the transition metal carbides and nitrides exhibit superconductivity in the bulk form 38 , and 2D superconductivity has been identified in Mo 2 C recently 39 . Among the TMDCs, NbSe 2 and TaS 2 have shown superconductivity in the bulk 40,41 and also in the 2D limit 33,42 . Both mono-layer and fewlayer NbSe 2 have shown 2D superconductivity with Ising pairing 33 and emergence of the Bose metallic phase 23 . NbSe 2 and TaS 2 , being metallic in nature, have limited electrical tunability. In addition, they are prone to oxidation, making device fabrication and integration with other materials challenging 23,42,43 . Carrier densities~10 14 cm −2 and an insulating-tosuperconducting transition have been reported in 2H-MoS 2 32,44-46 , 2H-MoSe 2 47 , 2H-WS 2 34,48 and 1T-SnSe 2 49 using the ionic-liquid gating technique. Ising pairing has been observed in ionic-liquid gated 2H-MoS 2 32,45,46 and 2H-WS 2 34,48 devices. A summary of important works on 2D superconductivity and associated features are tabulated in Supplementary Table 1. 1T-MoS 2 , metallic in nature with a high carrier concentration, meets the resistance requirement to exhibit 2D superconductivity. The 1T phase is generally prepared by structural modification of the 2H phase either by chemical intercalation 50 or by physical processes such as high-energy electron irradiation or plasma treatment [51][52][53] . Stability of the 1T phase is a highly debated topic. Chemically converted 1T-MoS 2 is reported to be unstable against time and temperature making it not suitable for the device fabrication, in contrast to those prepared by physical routes [52][53][54][55] . Scalable and controllable engineering of the 1T phase on the 2H phase has been demonstrated recently 53 . The possibility of observing 2D superconductivity on the 1T phase can make MoS 2 a potential choice for hosting 2D monolithic hybrid quantum circuits.
In this manuscript, we report the observation of 2D superconductivity, BKT phase transition and the emergence of Bose metallic state in an~8 nm thick (~12 layers) 1T-MoS 2 sample. The 1T-MoS 2 samples are prepared from the 2H phase using forming-gas microwave plasma assisted phase engineering. We characterise the 1T phase by high-resolution transmission electron microscopy (HR-TEM). Details of the sample preparation and in-depth structural characterisation can be found elsewhere 53 . The electrical characterisations consist of low-noise four-probe transport measurements from 300 K down to 4 K and magneto-transport measurements from 4 K down to 12 mK. Current-voltage (I-V) characteristics show excellent linearity down to 4 K without any dependence on the back-gate voltage, suggesting the presence of a metallic state 13,53 in our sample. The sample resistance shows a sharp drop at a temperature of~1.2 K. The resistance vanishes for lower temperatures signalling a transition to the superconducting state with a T c~9 20 mK. We confirm the 2D nature of the superconductivity from magnetotransport anisotropy and the presence of the BKT phase transition. Magneto-transport measurements reveal a parallel criticalfield manifold of the Pauli limit, possibly due to the presence of high SOC in our system, making it a potential candidate for the realisation of superconducting circuit elements operational at high magnetic fields. The sample also exhibits a transition to Bose metallic state, a signature of a clean 2D superconducting system.

Results
Characterisation of 1T-MoS 2 samples. Figure 1a shows an HR-TEM image with the selected area electron diffraction (SAED) pattern of a representative 1T-MoS 2 sample prepared from the 2H phase by the plasma treatment 53 . The sample shows extended 1T regions, whose area is primarily limited by the area of the starting 2H flake. The sharp SAED spots are indicative of highly crystalline extended 1T regions in our sample. We also find tiny isolated 2H regions (bounded in green) which we believe are phase transformed back as a result of prolonged exposure to the energetic electron beam. The 1T phase is characterised by octahedral coordination between the Mo and S atoms 55 . Figure 1b shows magnified view of the HR-TEM image of the 1T region while the corresponding atomic arrangement of the Mo and S atoms is shown Fig. 1c. Mo atoms, by virtue of their superior visibility in the TEM images, appear as bright spots in contrast to S atoms 56 . More details on the sample preparation, structural and electrical characterisation can be found elsewhere 53 .
The optical image of a 1T-MoS 2 device on a Si/SiO 2 substrate is shown in Fig. 1d. The electrical contacts are defined by photolithography followed by Cr/Au metallisation. We estimate a thickness of~8 nm (~12 layers) for the 1T-MoS 2 flake using atomic force microscopy (AFM). A height-profile obtained from the AFM along the red-arrowed line is shown in the inset. Figure 1e shows the four-probe I-V characteristics of the sample at 300 K (red) and 4 K (blue). We use probes 7, 8, 9 and 13 for all the transport measurements presented in this manuscript. The I-V characteristics show excellent linearity both at 300 K (red) and 4 K (blue). The traces are offset along the y-axis by 20 μV for better visibility. We observe that the normal state resistance of the sample at 300 K,~70 Ω, undergoes a slight increase, by~2 Ω, as the sample is cooled down to 4 K. In addition, the resistance of the sample shows little variation with the back-gate voltage V g , as shown in the top-left inset to Fig.1e. The linear I-V characteristics without any gate-voltage dependence suggest the metallic nature of the 1T-MoS 2 . From the four-probe resistance (~70 Ω) we calculate a sheet resistance R s~1 08 Ω/□ for the sample. From the $ 239 for our sample, which satisfies the Ioffe-Regel criteria 57 , k F .l ≥ 1, the disorder limit for metallic conduction. k F is the Fermi wave vector, and l is the mean-free-path of electrons in our sample. From the Hall measurements and the sheet resistance, we estimate a carrier concentration of~2.3 × 10 15 cm −2 and a meanfree-path l~20 nm.
The resistance as a function of temperature from 300 K down to 12 mK is shown in the lower-right inset of Fig. 1e. A magnified view of the resistance versus temperature in this range is shown in the Supplementary Fig. 1. The resistance shows negligible temperature dependence down to 4 K. This concurs with the fact that our sample consists of continuous 1T regions as inferred from the TEM analysis 53 . In contrast, transport on 1T-MoS 2 systems prepared by chemical routes has shown strong temperature dependence which has been attributed to hopping or activated behaviour owing to disorder or lower phase purity 13,58 . As the sample is cooled below 4 K, the resistance The traces are offset along the y-axis by 20 μV the for visibility. Top-inset: four-probe resistance versus back-gate voltage at 300 K, bottom inset: Fourprobe resistance versus temperature from 300 K down to 12 mK decreases, drops sharply around 1.2 K and vanishes for lower temperatures signalling the presence of a superconducting state. We note here that the carrier concentration of our sample is twoorders higher in magnitude compared to the minimum carrier concentration reported for the observation of 2D superconductivity in ionic-liquid gated MoS 2 32 . In addition, the sheet resistance of our sample is considerably lower than the quantum resistance, the upper limit to observe 2D superconductivity 3 .
Anisotropy in magneto-transport. To understand the nature and dimensionality of superconductivity in our device, we explore magneto-transport in the parallel and perpendicular magnetic field orientations. Figure 2a, b show the resistance of the sample versus temperature for various magnetic field values in the parallel and perpendicular orientations, respectively. Figure 2c, d shows the magneto-resistance of the sample for temperatures between 12 mK and 1.2 K in the parallel and perpendicular magnetic field directions, respectively. The temperature at which the resistance reduces to 50% of the normal state resistance R N (~70 Ω), at zero magnetic fields, is regarded as the critical temperature T c . T c = 920 ± 10 mK for our sample which corresponds to a Bardeen-Cooper-Schrieffer (BCS) energy gap 2Δ BCS = 3.52K B T C~2 80 µeV 59 . The transition moves to lower temperature values with the magnetic field and eventually vanishes above a critical field B k c2 for parallel and B ? c2 for perpendicular orientations. The critical field is estimated as the field at which the resistance of the sample reduces to 50% of R N . We observe a large anisotropy in the transport with respect to the magnetic field orientations and also a broadening of the transition for the parallel configuration. The broadening of the transition may be caused by phase fluctuations induced by the magnetic field 60 . These observations are atypical of a 3D superconducting state and points towards the presence of a 2D superconducting state in our sample.
For a 2D superconductor, the Tinkham's model predicts the behaviour of critical fields for parallel and perpendicular in the parallel orientation and B ? c2 ðTÞ ¼ ϕ 0 2πξ GL ð0Þ 2 1 À T T c in perpendicular orientation 31 , where ϕ 0 is the flux quantum, ξ GL ð0Þ is the Ginzburg-Landau coherence length and d sc is the effective thickness of the superconductor. Figure 2e shows the dependence of the critical magnetic field on the temperature for parallel (green) and perpendicular (blue) configurations. The green and blue dotted lines are fits using Tinkham's model for the parallel and perpendicular orientations, respectively, showing good agreement with the theory. From the fits, we estimate ξ GL 0 ð Þ = 11.9 ± 2.8 nm and d sc = 12.1 ± 3.1 nm. The BCS coherence length, 1:35 ξ GL 0 ð Þ< l implies that our device can be regarded as a clean superconductor. The estimated superconducting thickness obtained from the fit is of the same order but slightly larger than the sample thickness measured by AFM; similar discrepancies have been reported in the past 23,31 . We note here that in the parallel field configuration the critical field is highly sensitive to small angular deviations in the sample orientation which may also contribute to this error 59 .
The Pauli limit 59 described by the BCS theory put an upper bound to B k c2 0 ð Þ ¼ 1:86 T c in 2D superconductors. In our sample, the measured B k c2 at 12 mK is~6.25 T,~four-times the value 1.618 T predicted by the Pauli limit. A similar enhancement in the B k c2 has been observed in systems with high SOC (i) TMDC superconductors in the monolayer limit owing to Ising pairing of electrons [32][33][34] (ii) crystalline thin-film superconductors with high SOC and substrate-induced symmetry breaking 28 (iii) recently in centrosymmetric few layer TMDCs 36 . This has been suggested due to the SOC induced spin splitting of conduction electrons making pair breaking unfavourable under an external magnetic field; further studies are required to understand the detailed mechanism.
The Berezinskii-Kosterlitz-Thouless transition. A superconducting state is characterised by a critical current I c , above which the material turns normal 59 . Figure 3a shows the I-V characteristics from 12 mK (blue) to 4 K (red) at zero magnetic fields. A plot of I c versus temperature is shown in the inset. We observe that as the I c reduces with temperature and finally becomes zero as the normal state is reached. Dynamics of vortices leave a clear signature in the transport characteristics. In a 2D superconductor, in the absence of a magnetic field, the transition to zero resistance as the sample is cooled involves (i) formation of  Figure 3b shows the log-log scale plot of the I-V characteristics in the vicinity of the superconducting transition exhibiting the power law dependence. The inset shows a plot of the scaling exponent α(T) versus temperature as the sample undergoes a transition from the normal to the superconducting state. As the temperature is lowered, the slope α increases from unity corresponding to the Ohmic behaviour and diverges rapidly around α = 3. The grey dotted line in the main panel, and the purple dotted-line in the inset corresponds to α = 3; the corresponding temperature is the BKT transition temperature T BKT~6 75 mK. We observe hysteresis in the I-V traces between forward and reverse current sweeping directions in the superconducting state as shown in Fig. 3c. This is regarded as another sign of the presence of vortex-antivortex pairs in the system 23,62 . Above T BKT the free vortices are stable, and the correlations between distant pairs are destroyed. Below T BKT the free vortices created as a result of the driving current also pair-up, and the voltage developed due to the vortex motion in the unbound state causes the hysteresis in the critical current ΔI c 62,63 . ΔI c as a function of temperature is shown in Fig. 3d. The hysteresis reduces with temperature and disappears around 675 mK which corresponds to the T BKT estimated from the I-V characteristics using the scaling theory.
According to the Halperin-Nelson equation 64,65 , the resistance in a 2D superconductor in the vicinity of the transition varies as exp Àb T À T BKT ð Þ À 1 2 . A plot of resistance versus temperature with the fit to the equation (blue line) is shown in Fig. 3e. We extract a value of 689 ± 7 mK for T BKT from the fit. This value is consistent with the T BKT,~6 75 mK, estimated from the I-V curves (using both the scaling law and the hysteresis data shown in Fig. 3d).
The emergence of the Bose metal phase. A strongly disordered 2D superconductor makes a transition to an insulating state under the application of a perpendicular magnetic field 25 . In contrast, systems with a low disorder have shown a quantum phase transition to an intermediate 2D metallic state, Bose metal, with a resistance much lower than R N 23,66 . This state is characterised by a saturation of the resistance as T→0. Besides, the resistance obeys a power law with the magnetic field 23,66,67 . These observations have been theoretically addressed using various Bosonic models [25][26][27] . Disruption of the global phase coherence is considered as the key to this dissipation. Das and Doniach, using the Bose-Hubbard model attribute this to the dynamical gauge field fluctuations 24,25 . Phillips and Dalidovich model the system as a glassy phase-landscape making the motion of Bosonic charge carriers dissipative 26,27 .
Arrhenius plot of resistance for perpendicular magnetic fields below B ? c2 shown in Fig. 4a display the existence of Bose metal phase in our sample. Though a sharp drop in the resistance is observed at the superconducting transition point, the system does not exhibit a zero-resistance state for finite field values and saturates to small values compared to R N as the temperature is lowered. The resistance exhibits an activated behaviour for temperatures close to T c due to temperature-driven depairing of vortex-antivortex pairs. The pair dissociation energy U(B) is obtained from the fit to Fig. 4a using R / exp À U B c2 . The crossover temperature from the activated transport behaviour to the resistance-saturation and the activation energy reduces with the magnetic field ruling out the possibility of the state being an artefact of sample heating 68 . The sample is protected from the electrical noise and radiation using extensive filtering and radiation shields installed at various temperature stages of the measurement system, details of which can be found in the methods section. The superconductor to the Bose metal transition is characterised by a resistance scaling R / B À B c0 ð Þ 2ν where B c (0) is the critical field for the superconductor to Bose metal transition and ν is the exponent by which the superfluid correlation length diverges 23,25,26 . Figure 4c is a log-log scale plot of resistance versus perpendicular magnetic field for temperatures below T BKT with a fit to the above equation. A plot of the scaling exponent 2ν extracted from the fit versus temperature is shown in the inset. The exponent assumes a value of unity near T BKT corresponding to a linear behaviour and grows nonlinear as the temperature is reduced with 2ν = 2.37 ± 0.16 for lower temperatures. We note that the exponent has been shown to vary with sample thickness and the value of the exponent we obtained is comparable to those observed elsewhere for similar systems 23 . Figure 4d shows the B-T phase diagram displaying different regimes of transport arising due to vortex dynamics. The B ? c2 versus T data shown in red-dots extracted from the magnetotransport measurements separate these phases from the normal state. As the temperature is lowered below T c , the system goes into a thermally activated flux flow (TAFF) regime 20,23 , shaded in blue. The 2D superconducting phase (2DS) with vanishing resistance, shown in green, exists only at sufficiently low temperatures and magnetic fields. The Bose metal (BM) state characterised by finite dissipation at higher magnetic field values is shown in purple. The cross-over from the thermally activated transport to a resistance saturation, shown in blue squares, defines the boundary between the TAFF and the BM regimes. The 2DS state shows a hysteresis in the I-V characteristics and the vanishing of this hysteretic behaviour defines the boundary 23 .

Discussion
In this manuscript, we explored 2D superconductivity in a large area few-layer 1T-MoS 2 sample. We investigated the transport properties of the sample from 300 K down to 4 K in a closed cycle cryostat and from 4 K down to 12 mK in a dilution refrigerator. The carrier concentration of~2.3 × 10 15 cm −2 and a sheet resistivity of~108 Ω/□ put the sample in the clean metallic regime. In addition, the linear I-V characteristics with no dependence on the back-gate voltage underline metallic conduction in our sample. Below a temperature of~1.2 K, the sample undergoes a superconducting transition with a T c~9 20 mK. We observe a clear anisotropy in the magneto-transport obeying Tinkham's model. The superconductivity in our sample is accompanied by the BKT phase transition, enhanced parallel critical-field and a transition to the Bose metal state, which are characteristic features of a clean 2D superconducting system. The BKT transition is identified from the exponential scaling of the I-V characteristics, hysteresis in the critical current and also from resistance versus temperature characteristics obeying the Halperin-Nelson equation. The sample undergoes a transition from the superconducting to a 2D quantum metallic state, the Bose metal, under the application of a perpendicular magnetic field. Such behaviour is generally observed in systems with low disorder density. In addition, we observed a manifold increase in the parallel upper critical field beyond the Pauli limit possibly due to high SOC in our sample. The metallic 1T-MoS 2 is a widely studied system from the materials point of view, though, lacks in-depth electrical characterisations for device applications. We believe this work will add depth to the current understanding of the nature of transport in the system. The enhanced upper critical-field, even in the multilayer regime, makes 1T-MoS 2 a potential system for on-chip superconducting circuit elements and cavity QED applications requiring operation at higher external magnetic fields. The possibility of scalably and controllably engineering the 2D superconducting 1T phase on the semiconducting 2H phase opens up a new landscape for hybrid quantum device circuits.

Methods
Device fabrication. MoS 2 samples commercially procured from SPI supplies are exfoliated and transferred onto a clean Si wafer hosting 300 nm SiO 2 layer. The 2H to 1T phase conversion is achieved by exposing the exfoliated samples to forminggas (90% Ar + 10% H 2 ) microwave plasma. The details of the plasma system and the 1T conversion process can be found elsewhere 53 . Contacts on the 1T samples are made by standard photolithography followed by Cr/Au metallisation.
Low-noise transport measurements. Low-noise transport measurements are performed in a cryogen-free dilution refrigerator equipped with a superconducting magnet and also in a closed cycle 4 K cryo-cooler. The measurement setup consists of various filtering circuits and radiation shields at different temperature stages to minimise the electrical noise on the sample. The sample is shielded from external radiation using metallic shields attached successively on various temperature stages; 10 mK (mixing chamber), 700 mK (Still), 3.5 K (superconducting magnet), 50 K and 300 K (room temperature flange). Outside the cryostat, the measurements wires are Faraday shielded from any ambient radiation using metallic enclosures and shielded cables. The circuit consists of low-pass π-filters and RC filters at room temperature, low-pass RC filter at 10 mK (mixing chamber) and a high-frequency copper-powder filter 69 at 10 mK (mixing chamber). The cumulative cut-off frequency (−3 dB) for all the measurement lines is~1.4 kHz. The attenuation for all signals around 10 kHz is~40 dB and above 1 GHz exceeds~60 dB. In addition, a dedicated low-noise earthing arrangement is used to define the ground points for the entire measurement system.

Data availability:
All data generated or analysed during this study are included in this published article and its supplementary information files.