Quantum and classical ratchet motions of vortices in a 2D trigonal superconductor

Dynamical behavior of vortices plays central roles in the quantum phenomena of two-dimensional (2D) superconductors. Quantum metallic state, for example, showing an anomalous temperature-independent resistive state down to low-temperatures, has been a common subject in recently developed 2D crystalline superconductors, whose microscopic origin is still under debate. Here, we unveil a new aspect of the vortex dynamics in a noncentrosymmetric 2D crystalline superconductor of MoS$_{2}$ through the nonreciprocal transport measurement. The second harmonic resistance $R^{2w}$ at low temperature with high current indicates the classical vortex flow accompanying the ratchet motion. Furthermore, we found that $R^{2w}$ is substantially suppressed in the quantum metallic state with low current region, allowing identification of the quantum and classical ratchet motions of vortices by the magnitude of the second harmonic generation. This suggests that nonreciprocal transport measurement can be a powerful tool to probe the vortex dynamics in noncentrosymmetric 2D superconductors.


3
As a new probe of vortex states, we introduce, in this study, nonreciprocal phenomena, which is sensitively probed by the second harmonic magnetoresistance. It is generally known that, in a noncentrosymmetric system, the electrical resistance becomes dependent on the current direction, when the time reversal symmetry is broken 17 . Although nonreciprocal transport was originally studied in artificial helical structures or interfaces 17,18 , it is nowadays applied to various noncentrosymmetric crystalline systems 19,20 , including superconductors without inversion symmetry 21,22 . Among them, ion-gated MoS2, considered to be a trigonal 2D superconductor (Fig. 1a), can be an ideal platform for investigating the vortex dynamics through nonreciprocal transport.
Here, we report a study on nonreciprocal charge transport, which sensitively probes the vortex ratchet motion reflecting lattice symmetry, in ion-gated MoS2 single crystals. We found that the anisotropic behavior in the second harmonic resistance that satisfies the selection rules for threefold symmetry of single layer MoS2, indicating that the gate-induced superconductivity in multilayer MoS2 obeys the symmetry of monolayer. More importantly, the nonreciprocal magnetoresistance observed down to the lowest temperature at a relatively high current density can be understood in terms of not the paraconductivity scenario near Tc 22 but the ratchet motion of vortices in the plastic flow regime, which is caused by the asymmetric restraining force with threefold symmetry. Furthermore, we found that the second harmonic resistance signal at a low current is substantially suppressed below a certain magnetic field, being consistent with the quantum creep (tunneling) picture of vortices in noncentrosymmetric media. The present result indicates that nonreciprocal transport serves as an effective probe to differentiate the quantum and classical nature of vortex dynamics in noncentrosymmetric superconductors.

Sample fabrications and transport properties in MoS2-EDLTs
We prepared three EDLT samples (Fig. 1b, see Methods) with different configurations using two different current directions: parallel to the zigzag direction (configuration A; sample 1 and 3) and parallel to the armchair direction (configuration B; sample 2), as shown in Fig. 2b. Carrier densities of sample 1, 2 and 3 are 1.2×10 14 cm -2 , 1.8×10 14 cm -2 and 1.3×10 14 cm -2 , respectively, which were estimated by the Hall resistance measurements at 15 K. The crystal orientations of the MoS2 flakes were determined from the shape of the edge according to a previous study 23 .
We then performed the AC transport measurements (see Methods). First, we measured the first (R  ) harmonic signals of longitudinal resistance (Rxx) to show typical superconducting properties in the present system. Figure 1c shows Rxx as a function of temperature T for sample 1 (black) and sample 2 (orange) in zero magnetic field at a gate voltage of VG = 5 V. As T decreases, Rxx continuously decreases and suddenly goes to zero below 10 K. The Tc values of samples 1 and 2, defined as the midpoints of the resistive transitions, are 8.8 and 6.8 K, respectively. Figure 1d shows Rxx-T in sample 3 with Tc = 8.3 K under various magnetic fields B (0-9 T) at a source-drain current (I) of 0.5 A. The drop of sheet resistance becomes broadened with increasing B, and the transition disappears near 9 T. Figure 1e shows an Arrhenius plot of Rxx in sample 3 (the same data as those in Fig. 1d). At high temperatures near Tc, the data shows activated behavior indicating the thermal creep region 6,24,25 , while at lower temperatures, the resistance deviates from the thermally activated behavior and shows the almost temperature-independent behavior. Figure 1f shows a vortex phase diagram based on the analyses in Fig. 1e for sample 3, where the upper critical field curve Bc2(T) is derived from B and T with 90% of normal state resistance, which is confirmed to be close to the mean field value 24 and consistent with the Wertharmer-Helfand-Hohenberg (WHH) theory 26 at low temperature. Below Tcross, which is determined as temperatures at which Rxx deviates from the 5 activated behavior (white circles in Fig. 1e), the phase diagram is dominated by the quantum metallic state, in agreement with the previous study 24 .
Second harmonic measurements in 2D superconducting MoS2 Next, we focus on the second  A. We found that the amplitude of 2ω xx R at the peak position ( 2ω peak R ) increases with decreasing 6 temperature ( Fig. 3c) and is indiscernible in T > Tc. This increase implies that not the paraconductivity but the vortex dynamics plays a crucial role in the enhancement of nonreciprocal transport because the transport far below Tc and Bc2 is governed by vortex motion.
In order to elucidate the origin of nonreciprocal transport at lower temperatures far below of the rectification effect at high B is due to the effective weakening of the pinning potentials by increasing the AC current. In addition, according to a theoretical paper, nonreciprocal transport occurs even in the plastic vortex flow regime, where the vortices move among the pinning potentials as classical particles, in two-dimensional noncentrosymmetric superconductors 32 . Therefore, in the present case of MoS2, we propose that asymmetric pinning potentials, which might be caused by defects of the 2D crystals such as sulfur vacancies, is the origin of an inequivalent motion (ratchet effect) of plastic vortex flow to the current directions.
The pinning potentials are likely of trigonal symmetry, reflecting the intrinsic crystal structure.
Although this pinning potential is believed to be weak in gate-induced superconductivity in MoS2 24 , it should be finite and affects the vortex motions sensitively. 7 For further understanding of the relation between vortex motion and nonreciprocal resistance at the low current region, we plot ω xx R and 2ω xx R at I = 5 A against B in Fig. 4a and b, respectively, with schematic images of vortex motion in each magnetic field region in the inset. In this particular device, the zero-resistance (ZR) state is observed at finite magnetic field up to 0.25 T (orange triangle), above which ω xx R increases in a nonlinear manner with B. This region is assigned as the quantum metallic phase 6

Discussion
We note that the electron temperature in the high current regions might be increased more or less in Fig. 4c. However, the plastic vortex flow region seems not to shrink abruptly with increasing current in our measurement range, implying the stable vortex dynamic state without quenching to normal state. Therefore, the heating effect on the nonreciprocal signal is not so significant if it exists and the dynamical phase diagram is not qualitatively altered. The suppression of nonreciprocal signals in the low current and low magnetic field region supports the scenario of quantum tunneling of vortices for the quantum metallic state 6 , contrary to the recent study 13 , which attributes a quantum metallic state to the heating effect by the extrinsic noise. Our results thus suggest that the nonreciprocal transport is a powerful tool to investigate the vortex dynamics in noncentrosymmetric 2D superconductors. Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Methods
where R0 is a resistance under zero magnetic field and  the coefficient of nonreciprocal resistance in normal state [1][2][3] . It is noted that nonreciprocity emerges as a term in R proportional to both B and I. However, when nonreciprocal transport stems from vortex flow rectified by asymmetric pinning potentials 4 , R is expressed by where R (1) is the resistance of first order, ' the coefficient of nonreciprocal resistance in this case and ˆB B B = the sign of magnetic field. It is noted that R (1) shows linear-like behavior but is even as a function of B. In this formula, the nonreciprocal term in R is also proportional to B as well as I. This definition is reasonable because nonreciprocity in vortex flow region is governed by the number of vortices, resulting in its linear dependence of R (1) on B. When the AC bias current (source-drain current) with a frequency of  ( = 0 cos ) is applied, the out-put voltage is expressed as follows: We further approximated R  and R  in eq. (6) by the peak value ( 2ω peak R ) and the R  value at the peak position ( ω peak R ), respectively. This definition is different from one used in previous study 5 (approximated as 2ω peak ω peak peak 0 2R RBI  = , where Bpeak is the magnetic field at the peak position). In the previous study, we thought of an effect of paraconductivity on nonreciprocal transport and thus expanded the concept of nonreciprocal transport in normal state 1-3 , where nonreciprocal resistance is proportional to both B and I. However, in the present system, the contribution of not particles but vortices is dominant. Therefore, we use the definition of ' value shown above. We compare it with  used in previous study in Fig. S1. The sudden 20 increase in  just below Tc described by the paraconductivity effect doesn't exist in ' because of decrease in Bpeak as T approaches to Tc. In the present case, i.e., low temperature region, where R  and R 2 show linear dependence on B because the characteristics of the system is governed by number of vortices, the definition of ' might be more plausible.

II. Symmetrization and antisymmetrization
To analyze nonreciprocal transport, we symmetrized or antisymmetrized the raw data obtained in the present experiment. We measured the magnetoresistance for both positive and negative magnetic field ( RB are even and odd as a function of B, respectively. We adopt these symmetrization or antisymmetrization for R  and R 2 in the main text. We provide typical raw data and antisymmetrized data in Fig. S2. The linear background component observed in Fig. 2e in the main text can be attributed to the Hall effect originating from the nonlinear component of source-drain current (I 2 ), namely a component which is proportional to the square of in-put AC current in the current flowing through the sample. This I 2 possibly comes from geometrical asymmetry of source and drain electrodes in the sample. When out-of-plane magnetic field is applied, I 2 generates the transverse second harmonic resistance in the similar manner as the normal Hall effect in the first harmonic component, which cannot be distinguished from the intrinsic nonreciprocal resistance in the normal state. However, the anomalous increase of 2 component with the peak in the superconducting states cannot be explained by the geometrical asymmetry, and thus we can safely conclude that this behavior intrinsically arises from the crystal symmetry of MoS2.