Gated Tuned Superconductivity and Phonon Softening in Mono- and Bilayer MoS$_2$

Superconductors at the atomic two-dimensional (2D) limit are the focus of an enduring fascination in the condensed matter community. This is because, with reduced dimensions, the effects of disorders, fluctuations, and correlations in superconductors become particularly prominent at the atomic 2D limit; thus such superconductors provide opportunities to tackle tough theoretical and experimental challenges. Here, based on the observation of ultrathin 2D superconductivity in mono- and bilayer molybdenum disulfide (MoS$_2$) with electric-double-layer (EDL) gating, we found that the critical sheet carrier density required to achieve superconductivity in a monolayer MoS$_2$ flake can be as low as 0.55*10$^{14}$cm$^{-2}$, which is much lower than those values in the bilayer and thicker cases in previous report and also our own observations. Further comparison of the phonon dispersion obtained by ab initio calculations indicated that the phonon softening of the acoustic modes around the M point plays a key role in the gate-induced superconductivity within the Bardeen-Cooper Schrieffer (BCS) theory framework. This result might help enrich the understanding of 2D superconductivity with EDL gating.


Introduction
Discovery of the interfacial superconducting state at the atomic scale has motivated the pursuit of emergent condensed phases in two-dimensional (2D) electronic systems.
Studies of interfacial superconductivity have been generally limited to the regime in which the superconducting order parameter is restricted to 2D. Examples include one-atomic-layer Pb or In films on a Si (111) substrate, [1][2][3][4][5] single-unit-cell FeSe films on a SrTiO3 substrate [6][7][8][9] and few layer NbSe2 crystal. 10,11 Recent advances in electric-double-layer (EDL) gating have enabled the continuous tuning of one of the order parameters-the superfluid density-in 2D superconductors on the surface of bulk crystals with unprecedented control of surface band bending, doping level and vortex interaction, thus opening up new opportunities for understanding 2D superconductivity in the surface accumulation layers. [12][13][14][15][16][17][18] With the reduced dimensionality-especially in a strict 2D monolayer form in layered materials in which the disorder, fluctuation and correlation effects all play particularly important roles-being the key parameter, how low the carrier density can be to realize the interfacial superconductivity approaching the ultimate atomic limit remains elusive. Therefore, to investigate the superconductivity at the monolayer limit starts to be a topic of great interest. 18 As a representative semiconducting layered material, monoand bilayer MoS2 (Figure 1a) was chosen here as an ideal platform to study superconductivity at its 2D limit because of the following advantages: 1. The dramatic change in the band structure once the crystal is thinned down from bilayer to monolayer provides us with the possibility to understand the effect of some specific 4 subband structures and Fermi surfaces on the superconductivity; 2. The unique electric-field-driven Zeeman splitting 16,19,20 in the valleys of conduction bands might potentially provide spin-valley locking or a triplet electron pairing mechanism in the unique band structure of monolayer MoS2; 21  flakes with thickness ranging from mono-to six-layers.
In this work, we also demonstrated gate-induced 2D superconductivity in monoand bilayer MoS2 in an electric-double-layer transistor (EDLT) geometry and we found that the monolayer sample showed a smaller critical sheet carrier density requirement to achieve superconductivity than those of bilayer and bulk samples. The comparison studies of their vortex motions on monolayer and bilayer superconductivity were conducted to gain insight into the differences in the microscopic pictures of the 2D superconductivity. Our further ab initio calculations identified that the superconductivity are mainly induced by the phonon-softening of in-plane acoustic phonon modes and via such a mechanism the monolayer MoS2 can be more easily driven into superconducting phase with the less electron doping compared to the bilayer case. This work might enrich the understanding of gated interfacial superconductivity approaching the ultimate atomic limit and provides a 5 method to achieve new types of superconductors. Figures 1b and 1c show the typical device structure (optical image) and measurement geometry (schematic illustration) of EDLT using an ionic liquid (DEME-TFSI) as the gate medium. Due to the large tunability of the chemical potential using EDLTs, Figure 1d presents an ambipolar operation of bilayer MoS2 EDLT devices at 220 K with an ON-OFF ratio of 10 5 and a maximum attainable sheet carrier density (n) up to ~2 × 10 14 cm -2 by applying sufficiently high gate voltage VLG via the EDL gating medium. As shown in Figure 1e, in a typical case of a bilayer MoS2 device with n = 1.3 × 10 14 cm -2 (measured at 10 K), clear superconducting behavior can be observed.

Results
The superconducting state was gradually suppressed upon the application of an increasing perpendicular magnetic field and was completely suppressed when the magnetic field reached 1 T (see supplementary information for more details). In addition to the main drop in sheet resistance Rs at 3.3 K, we noted two other small resistance drops at higher temperatures above the superconducting transition, which are likely caused by the slightly inhomogeneous carrier accumulation. These observations suggest that the inhomogeneity-induced fluctuation in the chemical potential of the channel could cause a more notable effect on the electron transport properties while approaching the 2D limit.
To examine the dimensionality nature of such EDL gating-induced superconductivity, we performed 4-probe voltage-current (V-I) measurements and found that the results satisfied the Berezinskii-Kosterlitz-Thouless (BKT) transition . Compared with thicker MoS2 devices (~20 nm), 14 enhancement of back gate modulation due to the atomically thin geometry was also observed (supplementary Figure S5). Superconductivity in a monolayer MoS2 flake is shown in Figure 1f. The magnetic field response (shown in Figure 1f) confirmed that the resistance drop during cooling-down is most likely the superconducting transition. Two remarkable characteristic features must be addressed here. First, before the transition to a superconductor, the resistance of both mono-and bilayer flakes increases with decreasing temperature (more pronounced in the monolayer case), showing a stronger insulating behavior compared with that observed in thicker MoS2 flakes (20 nm). 14 This result indicates that disorder plays a more important role in the 2D phase transition while approaching the thin limit. This disorder might originate from the inhomogeneous charge accumulation or the substrate effect on the channels. Second, the critical carrier density required to achieve the superconducting state in the monolayer case is much lower than those values observed in our bilayer case or reported thicker cases, 14, 16, 20 which we will discuss in detail below. We note that due to the relatively small applied gate voltage to avoid possible electrochemical reactions and the limited temperature accessibility of our equipment, a zero resistance 7 superconducting state was not reached. However, the interface effect between electrodes and sample, 25,26 finite-size effects 24 and slightly inhomogeneous superconductivity 15,27 may also play roles in the non-zero resistance of low dimensional superconductivity. The bilayer superconductivities with zero resistance state under similar conditions suggest that the non-zero resistance case in monolayer MoS2 cannot be explained by finite-size effects or interface effect between electrodes and sample. The small applied gate voltage and slightly inhomogeneous superconductivity may be possible reasons and will require further investigations.
To study the detailed differences in the critical carrier densities required to achieve superconductivity between mono-and bilayer MoS2, we first measured the low-temperature transport properties of bilayer devices at different carrier densities. device, however, a much smaller critical carrier density (~0.55 × 10 14 cm -2 , with the results shown in Figure 2b) was observed than for those observed in bilayer devices.
We summarize the main results in Figure 2c, where Tc [defined as Rs(Tc) = 0.9Rs(10 8 K)] is plotted as a function of n for several typical devices (see a similar plot using number of carrier per primitive cell as the standard of the carrier density in Figure S6, and additional data on homogeneity of carrier densities for multiple pairs of contacts in Figure S7). All the carrier densities were determined by measuring Hall effect at 10 K, where the mono-and bilayer MoS2 maintained normal states. The Hall resistance, Rxy, show antisymmetric and linear characteristics when plot as a function of magnetic field (such as the inset of Figure 2a and 2b). The negative sign of Rxy for positive magnetic field indicate electron-type carriers, which is consistent with the positive gate biases. And the carrier densities were extracted from the Hall coefficient RH for each gate voltage by using the formula = 1 ⁄ . Because disorders or fluctuations are generally believed to play a crucial role in disturbing the superconductivity upon approaching the 2D limit, it is counterintuitive to find that in the ultimate limit case (monolayer), it is "easier" to realize superconductivity than in the bilayer or bulk cases when tuning such a key parameter-carrier density n.
Another approach to gain insight into the differences in the microscopic pictures of the 2D superconductivity in mono-and bilayer MoS2 devices is to study their vortex motion. 17 Through analysis of Rs-T plots under various perpendicular magnetic fields on typical bilayer ( Figure 3a) and monolayer ( Figure 3b) devices, we observed the activated behavior of the vortex dynamics at temperatures slightly lower than Tc and at the resistance saturation at even lower temperatures. This behavior is similar to that recently observed in a ZrNCl EDLT superconductor 17 and in disordered metal thin films. 28 At temperatures slightly lower than Tc, the sheet resistance can be described where U(H) is the activation energy, and kB is Boltzmann's constant. The fitting (black dotted lines in Figures 3a and 3b) yields values of U(H). We further plotted the dependence of U(H)/kB on H, as shown in Figure 3c, and found that both mono-and bilayer devices follow the relation of where ~ 256 ⁄ represents the vortex-antivortex binding energy, Φ0 is the flux quantum, d is the interlayer spacing, λ is the London penetration length depth, and H0 ~ Hc2 (defined as Rs(Hc2) = 0.9Rs(10 K)). These results indicate that the vortices in both mono-and bilayer devices exhibit thermally activated flux flow (TAFF). 29 From the fitting results of the bilayer device, we obtained U0/kB = 9.1 K and H0 = 0.37 T (H0 was found be smaller than Hc2 in this device), which are much larger than those of the monolayer device (U0/kB = 0.12 K and H0 = 0.37 T). At lower temperatures, the resistance deviated from the thermally activated behavior, and a magnetic-field-induced metallic ground state emerged. For the bilayer device, we also measured the magnetic field dependence of Rs at 0. The vortex phase diagrams of both mono-and bilayer devices based on our observations are plotted in Figure 3d. Two phases, the TAFF and quantum creep, are 10 confirmed by the measured Hc2 and thermally activated behavior deviation points.
When temperatures drop below Tc, the vortices move through the superconductor by thermal activation, whereas at even lower temperatures in a low magnetic field, vortices move by quantum tunneling. The movement of vortices indicates that even down to atomic thickness, the gate-induced 2D superconductor systems still reside in the weak disorder limit. In gate-induced superconductors in thick flakes of layered materials, 17 the superfluid density is confined in a few layers near the surface, and the 2D superconductors are susceptible only to relatively weak disorder generated by random electric potential from the ions; thus, they exhibit the behavior of a clean 2D superconductor. Herein, the mono-and bilayer MoS2 flakes are exfoliated onto SiO2 wafers, so the 2D superconductor is affected by stronger disorder from the substrate in addition to the ionic liquid. It is worth noting that these systems remain in a regime of a disordered 2D superconductor.
A quantum metallic state was also observed in bilayer NbSe2 crystal covered by Boron nitride (BN). 11 The low temperature resistance of the state fulfills a power-law scaling with magnetic field, which is consistent with the so-called Bose-metal model.  31 the conduction band minima (CBM) of monolayer and bilayer MoS2 are always located at the K (K') point even when electrons are injected into the samples. With increasing doping level, the conduction band edge at the K (K') point is filled, and the relative energy difference between the conduction band edge at the K (K') and Λ (Λ') points becomes smaller. By further increasing the electron doping, the states at the Λ (Λ') points are filled, whose energy splitting induced by the spin orbital coupling is much larger than that at the K (K') point. 32 Therefore, the electronic Fermi surface of the heavily doped MoS2 contains two parts: one is around the K and K' points, and the other is located near the Λ and Λ' points. This result directly 12 suggests that the electronic states around the Λ and Λ' points have a greater chance to be paired to form superconducting states via the phonon modes once the carrier density is sufficiently high to induce superconductivity in this system (as discussed below), which is different from previous reports. 20 In the framework of Bardeen-Cooper-Schrieffer (BCS) theory, the superconducting  In contrast to monolayer MoS2 superconductivity, the accumulated carrier density of a doped bilayer is lower for the same magnitude of Tc, which is observed experimentally in Fig. 2(c). In conclusion, we demonstrate that in the gate-induced 2D superconductivity of both mono-and bilayer MoS2 flakes, the monolayer sample has an apparently smaller critical sheet carrier density to achieve superconductivity than those of bilayer and bulk samples. The ab initio calculation results point to the phonon softening of in-plane acoustic phonon modes as a possible origin of these observations. Our work paves the way for further understanding of gated interfacial superconductivity approaching the ultimate atomic limit and pursuing a new type of superconductor. 14

Materials and devices
The mono-and bilayer MoS2 flakes on silicon wafers (covered by 300-nm-thick SiO2) were fabricated by standard mechanical exfoliation of bulk MoS2 (SPI supplies). The

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Competing financial interests
The authors declare no competing financial interests.  Supplementary Note 2: Rs-T curves and magnetoresistance data Fig. S2a shows an Rs-T curve of a bilayer MoS2 device with a larger temperature range (the same device as shown in Fig. 1e) with n = 1.3 × 10 14 cm -2 (measured at T = 10 K). A metal-insulator transition was observed at ~15 K, and a superconducting state was observed when T < 1.7 K. As shown in Fig. S2b, by applying a perpendicular magnetic field at a base temperature (300 mK), the zero resistance state could be suppressed starting with μ0H ~ 0.15 T and recovered to the normal state with μ0H = 1 T. Fig. S2c shows the magnetoresistance measurements of a monolayer device (the same device as shown in Fig. 1f) at different temperatures. Figure S2 | Rs-T curves and magnetoresistance data. a, Larger temperature range Rs-T curve of a bilayer MoS2 device (the same device as shown in Fig. 1e). b, Magnetoresistance measurement performed at 0.3 K. c, Magnetoresistance measurements of a monolayer device (the same device as shown in Fig. 1f) at different temperatures.

Supplementary Note 3: T-and B-dependent V-I characteristics of bilayer MoS2
We measured the 4-probe voltage-current (V-I) characteristics using a standard four-terminal dc technique. As shown by the results obtained from a typical bilayer MoS2 device (Fig. S3a and S3b), a well-defined critical current Ic ~11.09 μA was observed at a base temperature of 280 mK. The zero resistance state was suppressed gradually with either increased temperature (Fig. S3a) or the application of perpendicular magnetic field B (Fig. S3b). A complete suppression (showing linear V-I curves) was observed when T ≥ 3 K or μ0H ≥ 8 T. Fig. S3c shows the color plot of V-I curves with continual sweeping of B (ranging from -1 T to 1 T).

Supplementary Note 4: BKT transition of bilayer MoS2
As shown in Fig. S4a, a clear V~I α power law dependence was observed in a bilayer MoS2 device (n = 1.3 × 10 14 cm -2 ) at various temperatures ranging from 0.28 to 5 K, suggesting the occurrence of the Berezinskii-Kosterlitz-Thouless (BKT) transition 5-7 for 2D superconductivity. As shown in Fig. S4b, the fitted exponent α decreased with increasing temperature and approached 3 at T ~2 K, indicating a BKT transition temperature TBKT ~2 K. An additional test for the BKT transition is to check whether [d(lnR)/dT] −2/3 varies linearly with T when above TBKT, 5-7 which was also observed (shown in Fig. S4c). TBKT was determined to be ~2.17 K from extrapolation, consistent with the previous estimation (~2 K) derived in Fig. S4b.  In MoS2 EDLT devices, n2D of individual layer decays exponentially from the channel surface, reducing n2D of the second-to-outermost layer by almost 90% in comparison with the outermost one. 8,9 In our bilayer MoS2 case, the 2D carrier density showed in the main text were ~10 14 cm −2 in total, and we consider that most of the carriers were accumulated in the top monolayer as well. Therefore, it is reasonable to use the sheet carrier density as the standard of the critical carrier density for inducing superconductivity.
The number of carriers per primitive cell can be used as another meaningful standard of the critical carrier density. Such plot is shown in Fig. S6. Same conclusion can be reached that the critical carrier density for inducing superconductivity in