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T. Wakamura, M. Hashisaka, S. Hoshino, M. Bard, S. Okazaki, T. Sasagawa, [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), [K. Watanabe](https://orcid.org/0000-0003-3701-8119), K. Muraki, N. Kumada

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[Gate-tunable giant superconducting nonreciprocal transport in few-layer <math>  <mrow>    <msub>      <mi>T</mi>      <mi>d</mi>    </msub>    <mtext>−</mtext>    <msub>      <mi>MoTe</mi>      <mn>2</mn>    </msub>  </mrow></math>](https://mdr.nims.go.jp/datasets/652f441d-0705-4501-ac48-7739370eca72)

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Gate-tunable giant superconducting nonreciprocal transport in few-layer $T_{d}\textrm {-}{\rm MoTe}_2$PHYSICAL REVIEW RESEARCH 6, 013132 (2024)Gate-tunable giant superconducting nonreciprocal transport in few-layer Td-MoTe2T. Wakamura ,1,* M. Hashisaka,1,2 S. Hoshino,3 M. Bard,1 S. Okazaki,4 T. Sasagawa ,4 T. Taniguchi ,5K. Watanabe ,6 K. Muraki ,1 and N. Kumada 11NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan2Institute for Solid State Physics,University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa 277-8581, Japan3Department of Physics, Saitama University, Shimo-Okubo, Saitama 338-8570, Japan4Laboratory for Materials and Structures, Tokyo Institute of Technology, Nagatsuta 226-8503, Japan5International Center for Materials Nanoarchitectronics, National Institute for Materials and Science, 1-1 Namiki, Tsukuba 305-0044, Japan6Research Center for Functional Materials, National Institute for Materials and Science, 1-1 Namiki, Tsukuba 305-0044, Japan(Received 16 January 2023; revised 30 August 2023; accepted 12 December 2023; published 1 February 2024)We demonstrate gate-tunable giant field-dependent nonreciprocal transport (magnetochiral anisotropy) in anoncentrosymmetric superconductor Td-MoTe2 in the thin limit. Giant magnetochiral anisotropy (MCA) witha rectification coefficient (or a figure of merit) γ = 3.1 × 106 T−1 A−1 is observed at 230 mK, below thesuperconducting transition temperature (Tc). This is one of the largest values reported so far and may be attributedto the reduced symmetry of the crystal structure. The temperature dependence of γ indicates that ratchetlikemotion of magnetic vortices is the origin of the MCA, as supported by our theoretical model. For bilayer (2 L)Td-MoTe2, we can successfully modulate γ by gating. Our experimental results provide a new route to realizingelectrically controllable superconducting rectification devices in a single material.DOI: 10.1103/PhysRevResearch.6.013132I. INTRODUCTIONRecent intensive studies on nonreciprocal transport haverevealed the potential of using noncentrosymmetric materialsor inversion-symmetry-breaking multilayer structures to de-velop novel rectification devices based on superconducting orJosephson diode effect [1–3]. In systems with broken inver-sion and time-reversal symmetries, Onsager’s reciprocal theo-rem allows the electrical resistance to be different for oppositecurrent directions. This is called magnetochiral anisotropy(MCA), which leads to the rectification effect [4–6].Broken inversion symmetry is more beneficial in super-conductors. Rectification via ratchetlike motion of magneticvortices was reported more than a decade ago for supercon-ductors with asymmetric artificial magnetic nanostructuresor with asymmetric antidots as an asymmetric pinning po-tential [7–12]. These previous works revealed that as theasymmetry of the pinning potential for magnetic vorticesbecomes stronger, rectification becomes more efficient. Re-cent studies have pointed out that the ratchetlike motionof magnetic vortices is also possible in unpatterned non-centrosymmetric superconductors and provides large MCA[13–16]. In such systems, the asymmetry of the crystal struc-ture intrinsically induces asymmetric pinning potential. Incontrast to superconducting films with artificial structures,*Corresponding author: taro.wakamura@ntt.comPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.noncentrosymmetric superconductors do not require com-plex fabrication processes, offering more facile accessibilityto nonreciprocal transports. However, previous reports onMCA in noncentrosymmetric superconductors has mostlyconcentrated on those with trigonal symmetry [14,15,17–19],and other crystal symmetries have been poorly investigated.Toward more efficient rectification via MCA, it is essential toexplore noncentrosymmetric superconductors with differentsymmetries, especially with lower crystal symmetry than trig-onal symmetry. Furthermore, regarding future technologicalapplications, facile electrical control of nonreciprocal signalsis required, but no previous reports have addressed gate mod-ulation of superconducting MCA.In this study, we demonstrate gate-tunable giant MCA ina noncentrosymmetric superconductor Td-MoTe2 in the thinlimit. Td-MoTe2 lacks inversion symmetry and, for thin layers,has only one mirror plane normal to the b axis as shownin Fig. 1(a). This reduced symmetry of the crystal struc-ture may make the pinning potential for magnetic vorticeshighly asymmetric, which can contribute to generating largeMCA. From the MCA measurements under a perpendicu-lar magnetic field in few-layer Td-MoTe2 samples below Tc,we obtain the rectification coefficient, i.e., a figure of meritγ = 3.1 × 106 T−1A−1 at 230 mK, one of the largest valuesamong those reported so far. The monotonic increase in γwith decreasing temperature indicates that the giant MCAis due to the ratchetlike motion of magnetic vortices in themixed state of the type-II superconductor [13]. Interestingly,despite that Td-MoTe2 is a semimetal, in the 2 L sample we cansuccessfully modulate the MCA via an external gate voltageand demonstrate the modulation of γ . This ability to producea large variation in the nonreciprocal resistance by changingthe gate voltage may provide key insights into the mechanisms2643-1564/2024/6(1)/013132(13) 013132-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-8353-4013https://orcid.org/0000-0003-0149-6696https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-0289-5496https://orcid.org/0000-0001-7826-6894https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.013132&domain=pdf&date_stamp=2024-02-01https://doi.org/10.1103/PhysRevResearch.6.013132https://creativecommons.org/licenses/by/4.0/T. WAKAMURA et al. PHYSICAL REVIEW RESEARCH 6, 013132 (2024)0 1 2 3 40100200300T [K]Rω [Ω]4 L2 L(a)(c)(b)Tc = 750 mK-0.4 -0.2 0 0.2 0.4-101B [T]R2ω [Ω]T = 230 mK I || a I || b(d)Mo TeTc = 2.2 KFIG. 1. (a) Top view of the crystal structure, in which brokeninversion symmetry is evident. For thin layers, only one mirrorplane is present. (b) Optical microscope image of a 4 L device. Athin Td-MoTe2 flake is deposited on metallic contacts prepared inadvance. (c) Temperature dependence of the resistance of 4 and 2 Lsamples. (d) Comparison of R2ω when current is parallel to the aaxis (red) and b axis (blue) taken of the 4 L sample. For the setupwith I ‖ a, we drive the current between 1© and 5© and measure thevoltage between 3© and 4©. For the I ‖ b configuration, the current isdriven between 3© and 7©, and the voltage is measured between 2©and 8© (see also Figs. 11).behind the giant MCA by associating it with modulation of thesuperconducting properties.II. METHODSThe few-layer Td-MoTe2 flakes are mechanically exfoli-ated from high-quality Td-MoTe2 crystals with a residual-resistivity ratio (RRR) ∼1000 grown via the flux growthmethod. The mechanical exfoliation is carried out insidean Ar-filled glovebox containing concentrations of O2 andH2O below 0.5 ppm. Independently from the flakes, we pre-pare metallic electrodes by using typical electron-beam (EB)lithography and EB evaporation on a SiO2/Si substrate (forthe 4 L sample) or hexagonal boron-nitride (h-BN) exfoliatedon a SiO2/Si substrate (for the 2 L sample). We use Au (for the4 L sample) or Pt (for the 2 L sample) with Ti as a buffer layerfor the electrodes, and the total thickness of the electrodesis set to less than 10 nm to avoid exerting extra strain onthe flake. The exfoliated Td-MoTe2 is transferred with h-BNpicked up by using polydimethylpolysiloxane (PDMS) cov-ered with polycarbonate (PC) film [20]. The h-BN flake onTd-MoTe2 also acts as a protective layer because Td-MoTe2easily deteriorates when it is exposed to air. The transferprocess is also performed inside the glovebox. After takingthe sample out of the glovebox, it is immediately mountedon the sample holder of the 3He insert and encapsulated bythe internal vacuum can then cooled down. Raman scatteringand atomic-force microscope (AFM) measurements are car-ried out at room temperature and in the atmosphere after thetransport measurements.III. GIANT SUPERCONDUCTING NONRECIPROCALTRANSPORT DRIVEN BY RATCHETLIKEVORTEX MOTIONIn materials under broken inversion and time-reversalsymmetries, Onsager’s reciprocal theorem allows the linearlongitudinal resistance to be different for opposite currentdirections [4]. Rikken et al. heuristically found a generalformula for the nonreciprocal transport, also called the MCA,expressed as [21]R = R0(1 + γ BI ), (1)where γ is the rectification coefficient, which quantifies theefficiency of generating the nonreciprocal resistance. R0, Band I are the linear resistance, magnetic field, and excitationcurrent, respectively. Substituting Eq. (1) into Ohm’s law V =RI leads toV = R0I + γ BR0I2. (2)The first term is the typical linear voltage response to thecurrent and the second term is related to the nonreciprocaltransport. Thus, the nonreciprocal response is obtained as asecond harmonic signal for the ac excitation current Iω ∝sin(ωt ).First, we show the temperature dependence of the resis-tance for the four layer [4 L, see Fig. 1(b)] and bilayer (2 L)samples in Fig. 1(c). While Tc is low (∼100 mK) for bulkTd-MoTe2 [22], that for the 4 L and 2 L samples is 750 mKand 2.2 K, respectively. This large enhancement in Tc for thinlayers is consistent with previous studies [23,24]. Note thathere Tc is defined as the temperature where the resistancebecomes half of that in the normal state.Now let us focus on measuring the nonreciprocal transportin the superconducting state. Figure 1(d) shows the second-harmonic longitudinal resistance R2ω for Iω ‖ b and for Iω ‖ aat 230 mK. a and b are the crystal axes as defined in Fig. 1(a),and the b axis is orthogonal to the mirror plane. A clear peakand dip are observed in R2ω for Iω ‖ b. The field-asymmetricR2ω signals are in agreement with the MCA in Eq. (1) and areconsistent with previous experimental results [14,17,18,25].The suppression of R2ω at higher fields is due to the suppres-sion of superconductivity by a magnetic field. Note that thenonlinearity of the resistance due to the transition betweenthe normal and superconducting state is symmetric in B, soit is excluded as the origin of R2ω. In contrast to the case forIω ‖ b, R2ω for Iω ‖ a is dramatically suppressed. This is alsoconsistent with the geometry of MCA, where the symmetryplane, the directions of the magnetic field, and generatedsecond-harmonic voltage are all perpendicular to each other[4]. Note that the finite signal for Iω ‖ a is due to misalignmentof the electrodes to the crystal axis [see Fig. 1(b) and alsoAppendices B, F, and G]. Below we focus on the geometrywhere Iω ‖ b.The top part of Fig. 2(a) displays R2ω as a func-tion of perpendicular magnetic field measured at different013132-2GATE-TUNABLE GIANT SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 6, 013132 (2024)-101R2ω [Ω] 230 mK 300 mK 400 mK 500 mK 600 mK 700 mK 800 mK 1 KIω = 100 nA18 Hz4 ML(a)(c)(b)JBvE = B x vU(y)U-0.4 -0.2 0 0.2 0.40100200300Rω [Ω]B [T]y/l1 2f1/2-0.5 0 0.5-0.100.1Idc [μA]Vdc [mV]T = 230 mKB = 0 T4 L-0.4 -0.2 0 0.2 0.4104105106107B [T]γ [A-1T-1] 230 mK 300 mK 400 mK 500 mK 600 mK 700 mK 800 mK 1 KFIG. 2. (a) Experimental data from the 4 L sample. Top: Nonreciprocal resistance (R2ω = V2ω/Iω, Iω = 100 nA) measured at differenttemperatures. Middle: Rω signals measured simultaneously with R2ω. Bottom: γ as a function of B at different temperatures. The B valueswhere R2ω shows a peak (Bpeak) are marked with open triangles. Inset in the middle figure: Iω − Vω curve at B = 0 T and T = 230 mK. Zeroresistance is observed around Iω = 0, but the shape of the curve is somewhat rounded, partly because of the 2D nature of superconductivity.(b) Temperature dependence of γ taken from the 4 L sample. The orange curve shows the fit based on equation (3). Inset: Experimental dataof γ as a function of temperature obtained from the 2 L sample with the fit. Yellow shaded regions in the main figure and the inset representthe quantum metal (QM) phase (see also Appendix E). (c) Top: Schematic illustration of the motion of magnetic vortices with the velocity vdriven by an external current J, which generates an electric field E = B × v. Bottom: Image of a sawtooth potential assumed as the ratchetpinning potential in Eq. (3).temperatures. The amplitude of the signals monotonicallydecreases with increasing temperature. Above Tc, R2ω is com-pletely suppressed, indicating that the effect is related tosuperconductivity. The middle part of Fig. 2(a) shows the Rωsignals measured simultaneously with the R2ω signals. Theinset is Iω − Vω curve measured at B = 0 T and T = 230 mK,showing the exactly zero resistance state at low Iω. At Iω =100 nA, Rω starts to deviate from zero immediately after amagnetic field is turned on.Now that we have obtained R2ω and Rω, we can estimatethe value of the rectification coefficient γ = 2R2ω/(RωBIω )[4,13,17]. As shown in the bottom of Fig. 2(a), γ depends onB and increases rapidly as B → 0, particularly at the lowesttemperature. This is due to the decrease in Rω and B in thedenominator. In the following, we use γ at B (≡ Bpeak), atwhich R2ω is at a peak [triangles in the bottom of Fig. 2(a)]as a representative value. This definition of representative γis often used in previous studies and useful for quantitativediscussion [14,15,17,18,26]. Figure 2(b) shows that γ con-tinues to increase with decreasing temperature and reachesγ = 3.1 × 106 T−1 A−1 at 230 mK, the lowest measurementtemperature. This value is two to three orders of magnitudelarger than that of other two-dimensional superconductors,such as MoS2 and NbSe2 as we will discuss later. In the insetof Fig. 2(b), we also plot the temperature dependence of γ forthe 2 L sample, which shows the similar trend with slightlysmaller amplitudes. The smaller γ in the 2 L sample is dueto larger Bpeak caused by more robust superconductivity (seeAppendix D).So far, several mechanisms have been proposed to explainthe MCA in the superconducting state [13,16]. The tempera-ture dependence of the signals and the direction of the appliedmagnetic field are clues for identifying the mechanism. Forexample, paraconductivity is a mechanism proposed as anorigin of MCA under an in-plane magnetic field [13,25]. Sinceit is relevant to thermal fluctuations of the superconductingorder parameter, the nonreciprocal signal is slightly enhancedabove Tc and suppressed much below Tc. However, ratchet-like motion of magnetic vortices enhances the MCA belowTc under a perpendicular magnetic field [13]. In the mixedstate of type-II superconductors, magnetic fluxes penetrate thesuperconductor, and they are usually trapped by pinning po-tentials induced by disorder such as underlying discrete latticestructure, defects, and impurities. External current can drivethe magnetic fluxes through the Lorenz force as schematicallyshown in the top image of Fig. 2(c), if it is large enough013132-3T. WAKAMURA et al. PHYSICAL REVIEW RESEARCH 6, 013132 (2024)TABLE I. Summary of γ , γW , γW t , and γ Bpeak for different 2D superconductors.Material Symmetry Bpeak [T] R2ω[�] W [µm] γ [A−1T−1] γW [A−1T−1m] γW t [A−1T−1m2] γ Bpeak [A−1] Ref.MoTe2 2 L Cs 0.35 1.8 3 6.7 ×105 2.0 2.8 ×10−9 2.34 ×105 This studyMoTe2 4 L Cs 0.050 1.3 6 3.1 ×106 19 1.1 ×10−7 1.54 ×105 This studyMoS2a C3v 0.50 0.65 3 4.6 × 103 1.4 × 10−2 – 2.32 × 103 [16]NbSe2 5 L D3hb 3.0 0.062 5 2.8 × 102 1.4 × 10−3 4.7 × 10−11 8.49 × 102 [18]SrTiO3a – 0.050 0.29 80 3.2 ×106 2.6 × 102 – 1.60 × 105 [14]aBecause superconductivity is induced by the ionic liquid gating close to the surface, the exact value of t is difficult to define.bThe point group of bulk NbSe2 is D6h. However, D3h is the point group for thin NbSe2 with the odd number of layers.to overcome the pinning potential [27,28]. In superconduc-tors with broken inversion symmetry, the asymmetry of thecrystal structure locally affects the shape of the pinning po-tentials, making them asymmetric [21,29]. In this case, themagnetic vortices can exhibit ratchetlike motion, where theleftward and rightward motion of the vortex is not equiva-lent [7–11,13–15,18]. This asymmetry provides a source fornonreciprocal transport. The ratchetlike motion of magneticvortices provides increasing γ with decreasing temperaturebecause thermal fluctuations of the magnetic vortices insidethe pinning potential, which disturb the ratchetlike motion,are suppressed with decreasing temperature, and also thecoherence length, which determines the diameter of the vor-tex, becomes smaller, making the vortex more sensitive to thepinning potentials.We next discuss the temperature dependence of γ in moredetail by comparing the experimental data with a theoreticalmodel based on the ratchetlike motion of magnetic vortices.Solid lines in Fig. 2(b) represent the theoretical curve follow-ing the expression (see Appendix H for details) [13,19]:γ = φ∗0β�W Bg2(βU )g1(βU ), (3)where W is the width of the sample, φ∗0 = h/2|e| is the fluxquantum and β = 1/kBT is the inverse temperature. � andU are the mean periodicity and the height of the pinningpotential for a vortex, respectively. Here, to account for vortexdynamics qualitatively, we phenomenologically introduce thepinning potential and employ the Langevin dynamics. Wetake the simple potential shape shown in the bottom figure ofFig. 2(c), where the dimensionless parameter f controls theasymmetry of the potential. g1 and g2 are dimensionlessfunctions determined from the linear- and second-order re-sponses. The ratio is given by g2(βU )g1(βU ) ∼ f (βU )3180 for a movingvortex regime with a small ratchet potential. The curves fol-low the experimental data qualitatively in the intermediatetemperature region as shown in Fig. 2(b), which supportsthe ratchetlike motion of magnetic vortices as the dominantmechanism for the giant MCA in this system. In the theoret-ical curves, there are two fitting parameters α and f , and theformer defines the exponent in the temperature dependenceof the ratchet potential U ∼ U0[(Tc − T )/Tc]α with U0 asU (T = 0). Fitting the experimental results in the intermediatetemperature region provides α and f . By using these valuesand U estimated from the critical current density jc at a smallmagnetic field, we obtain U0 = 0.10 (0.17) meV for the 4 L(2 L) sample. Note that these U0 values are consistent withU0 = 0.10 (0.56) meV, alternatively obtained from the tem-perature dependence of the resistance under different mag-netic fields (see Appendix E).In contrast to the intermediate temperature region, the fitssubstantially overestimate γ at lower temperatures. This canbe explained by appearance of the quantum metal phase,in which quantum tunneling of vortices through the pinningpotential suppresses the ratchetlike motion and thus MCA[14,30]. Indeed, the temperature range where the experimentaldata deviate from the theoretical curve corresponds to theregion for the quantum metal phase [yellow shaded region onFig. 2(b)] in the vortex phase diagram (see Appendix E). Notethat the current density in the present measurements is smallenough to preserve the quantum metal phase [14]. At highertemperatures, however, rectification mechanism is replaced bysuperconducting fluctuation and normal contributions [13,25].We then compare the amplitude of γ with those in differentmaterials. The summary of γ in different noncentrosymmetricsuperconductors is shown in Table I. Td-MoTe2 exhibits largerγ by several orders of magnitude than those for trigonalsuperconductors such as MoS2 and NbSe2. The only valuecomparable to ours reported in the previous studies is thatfrom a SrTiO3 Rashba superconductor under an in-plane mag-netic field [25]. Therefore, the value obtained in our study isone of the largest reported so far [14,15,17–19].In Table I, we also show other quantities taking into ac-count the difference in the sample geometry and also Bpeak.Because γ depends on the sample width and thickness, wecompare γW and γW t , where W and t are the width andthe thickness of the sample, respectively. It is evident that thevalues for Td-MoTe2 surpasses substantially those for othernoncentrosymmetric superconductors, except for SrTiO3 dueto the much larger sample width. The values of γ Bpeakare also compared, which consider the difference in Bpeak.Td-MoTe2 shows much larger values than those for MoS2 andNbSe2, and is comparable or slightly larger than the value forSrTiO3. These comparisons corroborate the enhanced MCAin Td-MoTe2 among van der Waals superconductors. Note thatwhile our lowest measurement temperature (≡ Tmeas) is lowerthan that in previous studies for MoS2 or NbSe2 [14,17,18],Tc of Td-MoTe2 is lower, and the energy scale (kBTmeas) rela-tive to the superconducting gap is comparable. Thus we canrule out lower measurement temperature as an origin for thesubstantial nonreciprocal signals.Difference in the crystal symmetry is likely to be the originof the large variation in γ . In comparison with other two-dimensional trigonal superconductors, Td-MoTe2 has reducedsymmetry with only one mirror plane for thin layers. This013132-4GATE-TUNABLE GIANT SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 6, 013132 (2024)FIG. 3. (a) Gate voltage (Vg) dependence of Tc for the 2 L sample taken at 230 mK. The inset shows the R-T curves for the different Vg.(b) R2ω as a function of B at different Vg at 230 mK. (c) γ as a function of Vg. Inset: Gate voltage dependence of R2ω.reduced symmetry affects the asymmetry of the pinning po-tential. Since the symmetry of the pinning potential is crucialfor the vortex dynamics, as reported previously [11,12,31,32],the lower symmetry in the pinning potentials may generatelarger nonreciprocal signals. Further theoretical study is re-quired to scrutinize the effect of the symmetry reduction onthe amplitude of nonreciprocal signals.IV. GATE MODULATION OF GIANT SUPERCONDUCTINGNONRECIPROCAL TRANSPORTFinally, we demonstrate the gate modulation of the MCAfor the 2 L Td-MoTe2. While gate control of the MCA in thenormal state has been studied in a BST topological nanowire[6] and at the LaAlO3/SrTiO3 interface [33], it has not beenreported yet in superconductors. The primary reason is thatthe concentration of charge carriers in a superconductor istypically high, making it challenging to employ a conventionalsolid gate to regulate superconducting characteristics due tothe electric field screening on the nanometer scale within thematerial. We can overcome this problem by thinning downTd-MoTe2 to a thickness comparable to the screening length[34,35]. Note that the screening length is estimated to be ∼0.4 nm, the same order of the length as the one layer thicknessof Td-MoTe2. Figure 3(a) displays the gate dependence ofTc obtained from the 2 L sample. Here the gate voltage (Vg)is applied through a h-BN (34 nm in thickness) as a gateinsulator. Tc is successfully modulated by Vg, and at Vg = 8 Vit is larger by around 20 % compared with at Vg = −8 V.In addition to the variation of Tc, the MCA signals are alsomodulated by Vg [Fig. 3(b)]. We find that not only the height ofthe peak but Bpeak is also modulated by Vg, suggesting that thegating largely affects the vortex dynamics. Figure 3(c) plots γas a function of Vg, showing the large variation of γ .The gate voltage can modulate some parameters relevantto superconductivity, such as Tc, Bc2, the magnetic penetrationlength λ and the coherence length ξ . λ is a characteristic scalefor vortex-vortex interaction, whose crucial role was previ-ously pointed out in the ratchetlike motion [31,36], whereasnonreciprocal signals in the superconducting state are largelyaffected by the variation of Tc [13,16,19]. As the gating mod-ulates these parameters in a complex manner, at present wecannot identify the dominant contribution to the large varia-tion of γ . We hope that our results stimulate further theoreticalas well as experimental investigations to reveal the role of thecrystal symmetry for MCA and vortex dynamics in noncen-trosymmetric superconductors.V. CONCLUSIONIn conclusion, we showed giant superconducting nonrecip-rocal transport (MCA) in thin samples of the noncentrosym-metric superconductor Td-MoTe2, which has only one mirrorplane. We obtained 3.1 × 106 T−1 A−1 at 230 mK, oneof the largest values of γ recorded so far. The temperaturedependence of γ supports the ratchetlike motion of magneticvortices as the origin of the nonreciprocal transport. The re-duced symmetry of the crystal structure of Td-MoTe2 maycontribute to the large nonreciprocal signals. We also demon-strated gate modulation of the MCA in the superconductingstate. In the 2 L Td-MoTe2, we obtain a substantial modulationof γ using a typical solid gate. Simultaneous demonstration)b()a(10 μm10 μmFIG. 4. Optical microscope images of thin Td-MoTe2 flakes ob-tained via mechanical exfoliation. (a) 4 L, (b) 2 L.013132-5T. WAKAMURA et al. PHYSICAL REVIEW RESEARCH 6, 013132 (2024)0 100 200 300 400-276-275[nm][nm]2 ML0 100 200 300 400161162163164165[nm][nm]4 ML(a) (b)~1.4 nm = 2 L~2.7 nm = 4 L2 L4 LFIG. 5. Height profiles obtained from AFM scan of the 4 L(a) and 2 L (b) samples.of the gigantic MCA and its gate modulation in the supercon-ducting state reveals that Td-MoTe2 is a potential candidate forrealizing electrically tunable efficient superconducting rectifi-cation devices.ACKNOWLEDGMENTSWe gratefully acknowledge M. Imai, S. Sasaki, H. Muro-fushi, and S. Wang for their support in the experiments.This project is financially supported in part by the JPSJKAKENHI (Grants No. JP21H01022, No. JP21H04652, No.JP21K18181, No. JP21H05236, No. JP20H00354, and No.JP19H05790).APPENDIX A: THICKNESS IDENTIFICATION VIAATOMIC FORCE MICROSCOPE (AFM)Here we show the AFM data on the thickness of the thinTd-MoTe2 flakes. Figures 4 show some examples of exfoliatedflakes a few layers in thickness on a SiO2(285 nm)/Si sub-strate. As shown in Figs. 5(a) and 5(b), the two samplesemployed in this study are 4 and 2 L. Note that AFM measure-ments are performed after the transport measurements, thusTd-MoTe2 flakes are encapsulated by hBN.APPENDIX B: DETERMINATION OF THE CRYSTAL AXESBY RAMAN SCATTERINGSince Td-MoTe2 is a highly anisotropic material, it isimportant to determine the crystal axes of the sample andassociate them with its transport properties. The polarization-angle dependence of the Raman intensity is a powerful toolfor identifying the crystal axes [37,38]. We perform Ramanscattering measurements using HeNe laser light at 633 nmand measure the polarization-angle dependence of the Ramanintensity at 163 cm−1 Raman shift. Previous studies on angle-resolved Raman scattering measurements reported that theRaman intensity at 163 cm−1 exhibits a maximum when thepolarization is parallel to the Mo-zigzag chain (b axis), and aminimum when it is perpendicular to it (a axis). Figure 6(a)displays the angle dependence of the Raman intensity at 163cm−1 for the 4 L sample. Due to the reduced thickness ofthe sample in comparison with bulk, the otherwise typicaltwo-lobe structure is deformed, but the in-plane anisotropyand the orientation of the axes are clearly discernible. The cor-responding orientation of the 4 L sample is shown in Fig. 6(b).We can see that the electrodes are slightly misaligned from thedirection of the principal crystal axes, which is the reason fora finite R2ω for I ‖ a [≡ R2ω(I ‖ a)] shown in the main text.The misalignment angle is estimated to be around 15◦, leadingto sin(−15◦) ∼ −0.26. This value is in close agreement withthe ratio R2ω(I ‖ a)/R2ω(I ‖ b) = −0.23, taking into accountthat the peak (dip) is observed in the positive magnetic fieldfor R2ω(I ‖ a) [R2ω(I ‖ b)]. Figures 6(c) and 6(d) display theangular dependence of the Raman intensity and the samplepicture for the 2 L sample. Misalignment between the elec-trodes and the crystal axes is larger, probably leading to aslight suppression of the nonreciprocal signals compared withthose for the 4 L sample.APPENDIX C: CURRENT AND FREQUENCYDEPENDENCE OF THE NONRECIPROCAL RESISTANCEThe main text shows R2ω as a function of B with Iω at18 Hz. One may assume that the second harmonic signalsare due to some spurious effect such as a capacitive couplingof the sample to the surrounding conductive environment. Torule out this possibility, we carry out R2ω measurements bydriving Iω at different frequencies. Figure 7(a) displays theresults, demonstrating that R2ω signals are independent ofthe frequency of Iω. This is corroboration that R2ω signalswe measure derive from the intrinsic transport properties ofTd-MoTe2, namely, magnetochiral anisotropy (MCA).As mentioned in the main text, the efficiency of gen-erating the nonreciprocal resistance is evaluated as γ =2R2ω/(RωBIω ). Because of the definition of γ , one mightnaively think that γ is divergent in the limit of Iω → 0.(a) (b)θFIG. 6. (a) Angular dependence of the Raman intensity at the 163 cm−1 Raman shift for the 4 L sample. (b) Optical image of the 4 Lsample. The orientation of the sample corresponds to the orientation of the angle in panel (a). The scale bar is 5 µm. (c) Angular dependencefor the 2 L sample. (d) Sample picture of the 2 L sample, with 5 µm scale bar.013132-6GATE-TUNABLE GIANT SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 6, 013132 (2024)-0.4 -0.2 0 0.2 0.4-0.100.1B [T]V2ω [μV]T = 230 mKIω = 100 nA  18 Hz  77 Hz 177 Hz0 100 200 300 400 50000.51Iω [nA]R2ω [Ω])b()a(FIG. 7. (a) R2ω at different frequencies. No frequency dependencies are observed. (b) Driving current (Iω) dependence of R2ω. Iω = 100 nAprovides the largest R2ω. The data in panels (a) and (b) are both from the 4 L sample.However, this would be incorrect considering that the originof the nonreciprocal transport is the ratchetlike motion of themagnetic vortices, because the vortices are not driven whenthe Lorenz force exerted by Iω does not overcome the pinningforce. However, superconductivity is suppressed if Iω is toolarge. Therefore, it is expected that there is an intermediatevalue of Iω which provides the largest signal of R2ω. Indeed,as shown in Fig. 7(b), R2ω is suppressed as Iω → 0, and thereis an optimal value to obtain the largest R2ω, which is 100 nAfor the 4 L sample. R2ω diminishes when Iω is larger than thisvalue. Therefore, Iω = 100 nA was used for the nonreciprocaltransport measurements shown in the main text. Note that theoptimum value of Iω for the 2 L sample is 200 nA reflectinghigher Tc.APPENDIX D: MCA FOR THE 2 L MoTe2 SAMPLEAND COMPARISON OF γ AS A FUNCTION OF BWhereas in the main text we principally shows the datafrom the 4 L sample, similar nonreciprocal transport resultsare also obtained for the 2 L sample. Figure 8(a) shows R2ωcurves taken at different temperatures from the 2 L sample.Increasing R2ω with decreasing temperature is visible explic-itly, while the value of Bpeak is different from that of the 4 Lsample because of the higher Tc.In the main text we discuss the value of γ using the valuesof R2ω and Rω at Bpeak. With this estimate, γ for the 2 L sampleis slightly smaller than the 4 L sample. By contrast, we canalso plot γ as a function of B as we show in the bottom panelof Fig. 2(a). Figure 8(b) displays the evolution of γ with Bfrom the 2 and 4 L samples. It is evident that at any B, γ fromthe 2 L sample is larger than that from the 4 L sample. Thisindicates that the smaller γ for the 2 L sample is mainly dueto the larger Bpeak.APPENDIX E: VORTEX PHASE DIAGRAMAND ESTIMATION OF U0Since giant MCA observed in Td-MoTe2 is likely attributedto the ratchetlike motion of magnetic vortices, we show thevortex phase diagram in Fig. 9(a) to identify the vortex stateat each temperature and magnetic field where we observe alarge nonreciprocal signal. At lower temperatures and lowermagnetic fields, vortices are in the “quantum metal” phase, inwhich a finite resistance remains even much below Tc undera magnetic field [39]. Vortices are in the thermal creep (orthermally assisted flux flow) regime at higher temperatures-0.4 -0.2 0 0.2 0.4103104105106107108109B [T]γ  [A-1T-1] 2 L 4 L(b)FIG. 8. (a) MCA signals obtained from the 2 L sample at different temperatures. The characteristics are similar to those for the 4 L sample.(b) Comparison of the relation between γ and B at 230 mK between the 2 L (at Vg = 0) and 4 L sample.013132-7T. WAKAMURA et al. PHYSICAL REVIEW RESEARCH 6, 013132 (2024)FIG. 9. (a) Vortex phase diagram obtained from the 4 L sample. Thermal creep (thermally activated flux flow) phase transits into thequantum metal phase at lower temperatures. Blue squares are determined from the upper critical field Bc2, and red triangles are obtained fromthe temperature dependence of the resistance under perpendicular magnetic field. Orange cross marks express the points at which R2ω takesa peak. All the cross marks apart from two of them at lower temperatures are inside the thermal creep phase, supporting the vortex dynamicsplays a central role for giant nonreciprocal signals. (b) Arrhenius plot of the temperature dependence of the resistance. The boundary pointbetween the quantum metal and thermal creep phase is determined as a point (the yellow point) at which the Arrhenius plot deviates from theexponential decay (black solid line). This yellow point corresponds to the yellow point in panel (a). (c) The slope shown by the solid line inpanel (b) provides the activation energy U (B) at different magnetic fields. From the slope of U (B)/kB as a function of B (orange solid line),we can obtain U0.and higher magnetic fields up to Tc and the upper criticalmagnetic field (Bc2), where vortices are mobile and canplastically flow under an excitation current. Ratchetlikemotion of magnetic vortices is effective in the thermalcreep regime. The experimental data points which separatesthe quantum metal phase and thermal creep phase areobtained following [39]. Arrhenius plots of the temperaturedependence of the resistance are prepared under a magneticfield, and the temperature at which the plot deviates from theexponential decay is defined as the boundary temperature [seeFig. 9(b)]. Repeating this procedure for different magneticfields provides sets of data (T , B) for the boundary betweenthe thermal creep and the quantum metal phase, as plottedin Fig. 9(a). The cross marks in Fig. 9(a) compose sets oftemperature and magnetic field condition at which the peaksin R2ω are observed. Most of the points are contained inthe thermal creep region, except for the two points at lowertemperature, giving another evidence that vortex dynamicsaffected by the ratchetlike potential plays a principle rolefor giant nonreciprocal signals. In the phase diagram it isvisible that the quantum metal phase extends to relativelylarger magnetic fields at lower temperatures. The remainingtwo points overlap this region, indicating that quantumeffects may affect the nonreciprocal signals. This explains thesuppression of γ obtained from experiments in comparisonto the theoretical fit based on the ratchetlike motionof magnetic vortices as we discuss in the maintext.Temperature (T ) dependence of the resistance under amagnetic field also enables to estimate the potential heightU0. In the thermal creep regime, the resistance R dependson the activation energy U (B) as R = R0 exp[−U (B)/kBT ]with the normal state resistance R0 [39]. Thus the slope ina logarithmic plot of R with 1/T provides U (B) [see alsoFig. 9(b)]. Measuring T -dependence of R under different Bprovides the relation between U (B) and B, from which we canobtain U0 = 0.10(0.56) meV by using the relation U (B) =U0 ln(B0/B) for the 4 L (2 L) samples [Fig. 9(c)] [39].APPENDIX F: AXES DEPENDENCE OF THENONRECIPROCAL SIGNALIn the main text we show the experimental results of MCAfrom the 2 and 4 L samples. While these samples providea sufficient amount of data sets, the crystal axes are notperfectly aligned parallel to the current direction, making itdifficult to identify the axes dependence of the nonreciprocalsignals. Here, we provide additional data from other sampleswhose crystal axes are almost parallel to the current direction.Figures 10(a)–10(c) are from another 4 L sample (4 L#2)and the current direction is parallel to the b axis (I ‖ b). Thecrystal axes are determined from the Raman intensity profileshown in Fig. 10(b). As seen in Fig. 10(c), clear peak anddip structures are observed. Figures 10(d)–10(f) display thesecond harmonic signal from the 6 L sample where I ‖ a.We can easily find in Fig. 10(f) that while slowly oscillatingbackground is visible, no peak and dip structures typical forMCA are observed. This also corroborates that peak and dipstructures that we observe in the other samples arise fromMCA.APPENDIX G: OTHER POSSIBLE EFFECTSTO GENERATE NONRECIPROCAL SIGNALSWhile we have shown a number of additional experi-mental data which support MCA as the origin of the giant013132-8GATE-TUNABLE GIANT SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 6, 013132 (2024)FIG. 10. (a) Optical microscope image of the 4 L#2 device. The directions of the crystal axes and excitation current are denoted. (b) Angulardependence of the Raman scattering intensity. The sample is set as shown in panel (a), thus 0 degree corresponds to the b axis of the sample.(c) R2ω signal as a function of perpendicular magnetic field B for I ‖ b. Large peak and dip structures are clearly visible. Inset shows the linerresistance Rω simultaneously measured with R2ω. (d) Optical microscope image of another sample with 6 L in thickness. Here the directionof a current I is parallel to the a axis, determined from the angular dependence of the Raman scattering intensity shown in panel (e). (f) R2ωfor I ‖ a. There is a slowly oscillating background, but no peak and dip are observed. Rω signal is displayed in the inset. Note that becauseof the problems in the electrodes, we cannot perform similar measurements driving I parallel to the b axis in this sample. In panels (a) and(d) Td-MoTe2 is highlighted by a yellow dotted line, and the scale bar corresponds to 5 µm.nonreciprocal signals, some may still wonder other effects canbe considered to explain those nonreciprocal resistances. Letus rule out some other possibilities as an origin of the secondharmonic signals.1. Thermal effectsIf there existed a thermal gradient ∇T , the Nernst effectwould generate an electric field E ∝ B × ∇T under a mag-netic field B. The thermal gradient may be due to the Jouleheating effect, therefore ∇T ∝ j2, where j is current density,and ∇T should be parallel to j. Indeed, such a superconduct-ing Nernst effect which generates a second harmonic voltage(V2ω) was observed in another van der Waals superconductorNbSe2, using a thermal gradient driven by a heater mountedclose to the sample [40]. V2ω exhibits similar magnetic fielddependence as those observed in our samples, derived fromthe (vortex) Nernst effect [41]. We can rule out the Nernsteffect as a possible origin of MCA due to the followingreasons: (i) Temperature gradient assumed above should notexist considering that our Td-MoTe2 is highly crystalline sothat excitation currents pass almost homogeneously throughthe sample. (ii) V2ω from the Nernst effect is reduced tozero at lower temperatures, opposite to our observations.Furthermore, the Nernst signal persists even above Tc, incontrast to the suppressed nonreciprocal signals above Tc.(iii) V2ω induced by the Joule heating should be orthogonalto j, inconsistent with our observations of large longitudinalnonreciprocal signals.2. Geometrical effectsVortex rectification effect induced by the asymmetric sam-ple geometry (e.g., asymmetric edge shape of the sample)was previously proposed [42] and experimentally confirmed[43,44]. The main idea is that since magnetic vortices alwaysenter from the edge (or surface in the case of three dimen-sional superconductors) of the sample and cannot nucleateinside the superconductor, the edge asymmetry between theopposite sides of the sample generates the asymmetric sur-face potential barrier for vortices. Because the edge selectedfor the vortex entry depends on the polarity of the current,the inequivalency in the potential barrier between the edgesleads to the different current condition for the vortex entrythus rectification. One may wonder that the nonreciprocalsignals observed in this study are attributed to the asymme-try between the edges of the sample. It is true that slightasymmetry between the edges is inevitable in our samples em-ploying mechanically exfoliated Td-MoTe2 flakes. If the edgeasymmetry is a dominant contribution in our samples, largernonreciprocal signals are expected for samples with moreasymmetric edges. We do not see the correlation between theedge asymmetry and the amplitude of the nonreciprocal signalfor different samples with different edge shapes. As an exam-ple, the edge asymmetry is more peculiar in the sample shownin Fig. 10(d) than in Fig. 10(a), but the nonreciprocal signalis much suppressed. Moreover, in our samples electrodes arealways aligned symmetrically, ruling out the possibility of theasymmetry induced by electrodes. Therefore, we can drawa conclusion that vortex rectification due to the geometrical013132-9T. WAKAMURA et al. PHYSICAL REVIEW RESEARCH 6, 013132 (2024)asymmetry in the sample is not an origin for the giant nonre-ciprocal signals.3. Surface barrier effectIn relation to the geometrical effect, one may argue that theasymmetry in the surface barrier for magnetic vortices is theorigin of MCA. As for the dynamics of magnetic vortices insuperconductors, we must consider two effects: Bulk pinningand surface barrier. The important point is which effect isdominant in a certain condition. We highlight that the surfacebarrier effect plays a major role in the vortex dynamics onlywhen the temperature is close to Tc, at which the bulk pinningbecomes extremely weak. This important point has been al-ready demonstrated by many previous studies [45–48]. Thisbehavior is inconsistent with our observation, indicating thatbulk pinning effect is dominant in our samples.APPENDIX H: THEORETICAL DESCRIPTION USINGRATCHET MODELHere we describe the nonreciprocal transport signal orig-inating from the vortex motion [13,19]. For simplicity, weregard the vortex as a point particle. The relation betweenvelocity and force is written asvx = q1xxFx + q2xxyFxFy, (H1)vy = q1yyFy + q2yxxF 2x + q2yyyF 2y , (H2)where the mirror symmetry along the y axis is assumed (fora trigonal symmetry, there are the relations q1xx = q1yy andq2xxy = −2q2yxx = 2q2yyy). If we take the configuration withFx = 0, only the y direction is involved. Hence, we considerthe one-dimensional equation of motion for the estimation ofq2yyy:ηẏ = Fy − ∂U (y)∂y+ ξ (t ), (H3)where η = 1.45π h̄2σn2e2ξ 2 is a damping coefficient with the nor-mal conductivity σn and in-plane coherence length ξ [49].The random force ξ (t ) represents a thermal noise satisfying〈ξ (t )ξ (t ′)〉 = 2ηkBT δ(t − t ′) where the bracket indicates therandom average. The asymmetric potential U (y) for vortexis responsible for the nonreciprocal transport signal. We takethe periodic potential of the height U and the periodicity �given in Fig. 12(a). The response coefficients are given byq1yy = 1ηg1(βU ) and q2yyy = β�ηg2(βU ). The functional formsof g1,2 are explicitly given byg1(x) = x22(cosh x − 1), (H4)g2(x) = f x (4 + x2 − 4 cosh x + x sinh x)2(cosh x − 1)2. (H5)The parameter f (� 12 ) controls the asymmetry of thepotential.The force acting on the vortex is given by Fy = jφ∗0 wherej is a current density along x direction and φ∗0 = h2|e| is the fluxquantum for superconductors. The number density of vorticesis given by n = Bφ∗0. The voltage along x direction is then given(a) (b) (c)VVVFIG. 11. (a) The current and voltage probes configuration forI ‖ a in the 4 L sample. (b) The current and voltage probes configu-ration for I ‖ b in the 4 L sample. (c) The current and voltage probesconfiguration in the 2 L sample.by the Josephson relation Vx = φ∗0 Lnvy = R1I + R2I whereI = jW is a current with the sample width W and length L.The linear and nonlinear transport coefficients areR1 = φ∗0 LBηWg1(βU ), R2 = (φ∗0 )2ILB�ηkBTW 2g2(βU ). (H6)Since both the signals are proportional to the vortex numberdensity, their ratio is written in a simple formγ ′ = R2R1I= φ∗0�W kBT· g2(βU )g1(βU ), (H7)which is determined from the profile of the potential of vor-tices. Note also the relations Rω = R1 and R2ω = R2/2.Now we discuss characteristic parameters used in thismodel. For the characteristic length �, we consider the valueof the magnetic field Bpin at which all the pinning centersare occupied at low temperature. The length scale is thengiven by �(T = 0) ∼ √φ∗0/Bpin. The potential height U isestimated by the critical current density jc at small magneticfield by the relation U = jcφ∗0�( 12 + f ). We also consider thetemperature dependence of these parameters since the size ofvortices changes as the coherence length varies with increas-ing temperature. We assume the temperature dependence ofU (T ) ∼ U0( Tc−TTc)αand �(T ) ∼ �0( Tc−TTc)α− 12, which results injc ∝ √Tc − T (α = 1 is used in Ref. [30]). Since the micro-scopic origin of the vortex potential is not clear at present, herewe take α as a parameter phenomenologically for a better fitto the experimental data. The temperature dependence of Uis more strongly reflected in the signal compared to that of �,because βU is the argument of the nonlinear function g2.Now we estimate the magnitude of the nonlinear transportsignal. Since the nonreciprocal signals in magnetic field de-pendence is maximized in a moving vortex regime, we assumethe expression for a small pinning potential g1(βU ) ∼ 1 andg2(βU ) ∼ f (βU )3180 , with which the vortices are not fixed in thepinning potential. We use the parameters for the 4 L (2 L)sample such as W = 5 (2.5) µm, Tc = 0.75 (2.2) K, Bpin(T =0) ∼ 0.025 (0.2) T, and Ic(T = 0) = jcW ∼ 200 (500) nA.Bpin is estimated from the magnetic-field dependence of theresistance, and Ic from the current-voltage characteristicsunder a magnetic field measured for each sample. To becompatible with experiments, we choose the vortex potentialparameters as f = 0.15 and α = 0.5, which are used for boththe 4 and 2 L samples. With these parameters, the pinning013132-10GATE-TUNABLE GIANT SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 6, 013132 (2024)FIG. 12. (a) Spatial dependence of the ratchet potential, where f (0 � f � 1/2) controls the asymmetry. (b) Temperature dependenceof (b1) γ ′ = γ B and (b2) R2ω for the 4 L sample. Analogous plots for the 2 L sample are shown in panels (c1) and (c2).potential height at zero temperature is estimated as U0 ∼0.1 (0.17) meV. Note that the estimated values of U0 are closeto those experimentally obtained from the temperature depen-dence of the resistance under a perpendicular magnetic field(see Appendix E). Since γ ′ is proportional to (Tc − T )4α− 12 inour model, the temperature dependence becomes more convexdownward if we take larger α.The temperature dependence of γ ′ = γ B is shown inFig. 12(b1) for the 4 L sample and in Fig. 12(c1) for the2 L sample. The vortex ratchet model with two adjustableparameters α and f reproduces the magnitude of signals ob-served in experiments. We note that the present model can bejustified in the middle temperature range. For high tempera-ture range T � Tc, the vortex picture should be replaced bysuperconducting fluctuation mechanism and/or normal con-tribution. At low temperatures, however, a quantum effect onratchetlike motion [30] is needed for more accurate estimate,as supported by the vortex phase diagram discussed above.With the above setup, we also estimate the values of R2ωby using the normal resistance Rn = LW σn ∼ 330 (380) �, thecoherence length ξ (T ) ∼ ξ0√Tc/(Tc − T ) with ξ0 ∼ 45 (24)nm, and the current I = 100 (200) nA. The results are shownin Fig. 12(b2) for the 4 L sample and Fig. 12(c2) for the 2 Lsample. Since the experimental data plotted here are obtainedusing different magnetic fields (0.04–0.05 T for 4 L and 0.25–0.42 T for 2 L), we show several lines for different magneticfields. We confirm that the present model can roughly repro-duce the magnitude R2ω in a middle temperature range belowTc. R2ω is decreasing function of temperature, which is alsoconsistent with experiments.[1] F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota,T. Moriyama, Y. Yanase, and T. Ono, Observation of supercon-ducting diode effect, Nature (London) 584, 373 (2020).[2] H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U.Filippozzi, S. S. P. Parkin, Y.-J. Zeng, T. McQueen, and M. N.Ali, The field-free Josephson diode in a van der Waals het-erostructure, Nature (London) 604, 653 (2022).[3] L. Bauriedl, C. Bäuml, L. Fuchs, C. Baumgartner, N. Paulik,J. M. Bauer, K.-Q. Lin, J. M. Lupton, T. Taniguchi, K.Watanabe, C. Strunk, and N. Paradiso, Supercurrent diode ef-fect and magnetochiral anisotropy in few-layer NbSe2, Nat.Commun. 13, 4266 (2022).[4] Y. Tokura and N. Nagaosa, Nonreciprocal responses from non-centrosymmetric quantum materials, Nat. Commun. 9, 3740(2018).[5] T. Ideue, K. Hamamoto, S. Koshikawa, M. Ezawa, S. Shimizu,Y. Kaneko, Y. Tokura, N. Nagaosa, and Y. Iwasa, Bulk rec-tification effect in a polar semiconductor, Nat. Phys. 13, 578(2017).[6] H. F. Legg, M. Rossler, F. Munning, D. Fan, O. Breunig,A. Bliesener, G. Lippertz, A. Uday, A. A. Taskin, D. Loss,J. Klinovaja, and Y. Ando, Giant magnetochiral anisotropyfrom quantum-confined surface states of topological insulatornanowires, Nat. Nanotechnol. 17, 696 (2022).013132-11https://doi.org/10.1038/s41586-020-2590-4https://doi.org/10.1038/s41586-022-04504-8https://doi.org/10.1038/s41467-022-31954-5https://doi.org/10.1038/s41467-018-05759-4https://doi.org/10.1038/nphys4056https://doi.org/10.1038/s41565-022-01124-1T. WAKAMURA et al. PHYSICAL REVIEW RESEARCH 6, 013132 (2024)[7] J. E. Villegas, S. Savel’ev, F. Nori, E. M. Gonzalez, J. V.Anguita, R. García, and J. L. Vicent, A superconductingreversible rectifier that controls the motion of magnetic fluxquanta, Science 302, 1188 (2003).[8] C.-S. Lee, B. Jankó, I. Derényi, and A.-L. Barabási, Reducingvortex density in superconductors using the “ratchet effect,”Nature (London) 400, 337 (1999).[9] C. C. de Souza Silva, J. Van de Vondel, B. Y. Zhu, M. Morelle,and V. V. Moshchalkov, Vortex ratchet effects in films with aperiodic array of antidots, Phys. Rev. B 73, 014507 (2006).[10] J. E. Villegas, E. M. Gonzalez, M. I. Montero, I. K. Schuller,and J. L. Vicent, Directional vortex motion guided by artifi-cially induced mesoscopic potentials, Phys. Rev. B 68, 224504(2003).[11] B. Y. Zhu, L. VanLook, V. V. Moshchalkov, B. R. Zhao, andZ. X. Zhao, Vortex dynamics in regular arrays of asymmetricpinning centers, Phys. Rev. B 64, 012504 (2001).[12] J. E. Villegas, E. M. Gonzalez, M. P. Gonzalez, J. V. Anguita,and J. L. Vicent, Experimental ratchet effect in superconductingfilms with periodic arrays of asymmetric potentials, Phys. Rev.B 71, 024519 (2005).[13] S. Hoshino, R. Wakatsuki, K. Hamamoto, and N. Nagaosa,Nonreciprocal charge transport in two-dimensional noncen-trosymmetric superconductors, Phys. Rev. B 98, 054510(2018).[14] Y. M. Itahashi, Y. Saito, T. Ideue, T. Nojima, and Y. Iwasa,Quantum and classical ratchet motions of vortices in atwo-dimensional trigonal superconductor, Phys. Rev. Res. 2,023127 (2020).[15] T. Ideue, S. Koshikawa, H. Namiki, T. Sasagawa, and Y. Iwasa,Giant nonreciprocal magnetotransport in bulk trigonal super-conductor PbTaSe2, Phys. Rev. Res. 2, 042046(R) (2020).[16] R. Wakatsuki and N. Nagaosa, Nonreciprocal current in non-centrosymmetric Rashba superconductors, Phys. Rev. Lett. 121,026601 (2018).[17] R. Wakatsuki, Y. Saito, S. Hoshino, Y. M. Itahashi, T. Ideue,M. Ezawa, Y. Iwasa, and N. Nagaosa, Nonreciprocal chargetransport in noncentrosymmetric superconductors, Sci. Adv. 3,e1602390 (2017).[18] E. Zhang, X. Xu, Y.-C. Zou, L. Ai, X. Dong, C. Huang, P.Leng, S. Liu, Y. Zhang, Z. Jia, X. Peng, M. Zhao, Y. Yang,Z. Li, H. Guo, S. J. Haigh, N. Nagaosa, J. Shen, and F. Xiu,Nonreciprocal superconducting NbSe2 antenna, Nat. Commun.11, 5634 (2020).[19] Y. M. Itahashi, T. Ideue, S. Hoshino, C. Goto, H. Namiki,T. Sasagawa, and Y. Iwasa, Giant second harmonic transportunder time-reversal symmetry in a trigonal superconductor,Nat. Commun. 13, 1659 (2022).[20] P. J. Zomer, M. H. D. Guimaraes, J. C. Brant, N. Tombros, andB. J. van Wees, Fast pick up technique for high quality het-erostructures of bilayer graphene and hexagonal boron nitride,Appl. Phys. Lett. 105, 013101 (2014).[21] G. L. J. A. Rikken, J. Folling, and P. Wyder, Electrical magne-tochiral anisotropy, Phys. Rev. Lett. 87, 236602 (2001).[22] Y. Qi, P. G. Naumov, M. N. Ali, C. R. Rajamathi, W. Schnelle,O. Barkalov, M. Hanfland, S.-C. Wu, C. Shekhar, Y. Sun,V. Suss, M. Schmidt, U. Schwarz, E. Pippel, P. Werner, R.Hillebrand, T. Forster, E. Kampert, S. Parkin, R. J. Cavaet al., Superconductivity in Weyl semimetal candidate MoTe2,Nat. Commun. 7, 11038 (2016).[23] D. A. Rhodes, A. Jindal, N. F. Q. Yuan, Y. Jung, A. Antony,H. Wang, B. Kim, Y.-C. Chiu, T. Taniguchi, K. Watanabe,K. Barmak, L. Balicas, C. R. Dean, X. Qian, L. Fu, A. N.Pasupathy, and J. Hone, Enhanced superconductivity in mono-layer Td-MoTe2, Nano Lett. 21, 2505 (2021).[24] Y. Gan, C.-W. Cho, A. Li, J. Lyu, X. Du, J.-S. Wen, and L.-Y.Zhang, Giant enhancement of superconductivity in few layersMoTe2, Chin. Phys. B 28, 117401 (2019).[25] Y. M. Itahashi, T. Ideue, Y. Saito, S. Shimizu, T. Ouchi, T.Nojima, and Y. Iwasa, Nonreciprocal transport in gate-inducedpolar superconductor SrTiO3, Sci. Adv. 6, eaay9120 (2020).[26] M. Masuko, M. Kawamura, R. Yoshimi, M. Hirayama, Y.Ikeda, R. Watanabe, J. J. He, D. Maryenko, A. Tsukazaki,K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura,Nonreciprocal charge transport in topological superconductorcandidate Bi2Te3/PdTe2 heterostructure, npj Quantum Mater.7, 104 (2022).[27] M. Tinkham, Introduction to Superconductivity, 2nd ed. (DoverPublications, Mineola, NY, 2004).[28] W.-K. Kwok, U. Welp, A. Glatz, A. E. Koshelev, K. J.Kihlstrom, and G. W. Crabtree, Vortices in high-performancehigh-temperature superconductors, Rep. Prog. Phys. 79, 116501(2016).[29] A. Fente, W. R. Meier, T. Kong, V. G. Kogan, S. L. Bud’ko,P. C. Canfield, I. Guillamón, and H. Suderow, Influence ofmultiband sign-changing superconductivity on vortex cores andvortex pinning in stoichiometric high-TcCaKFe4As4, Phys. Rev.B 97, 134501 (2018).[30] K. Hamamoto, T. Park, H. Ishizuka, and N. Nagaosa, Scal-ing theory of a quantum ratchet, Phys. Rev. B 99, 064307(2019).[31] A. Palau, C. Monton, V. Rouco, X. Obradors, and T. Puig,Guided vortex motion in YBa2Cu3O7 thin films with collectiveratchet pinning potentials, Phys. Rev. B 85, 012502 (2012).[32] D. J. Morgan and J. B. Ketterson, Asymmetric flux pinning ina regular array of magnetic dipoles, Phys. Rev. Lett. 80, 3614(1998).[33] D. Choe, M.-J. Jin, S.-I. Kim, H.-J. Choi, J. Jo, I. Oh, J. Park, H.Jin, H. C. Koo, B.-C. Min, S. Hong, H.-W. Lee, S.-H. Baek, andJ.-W. Yoo, Gate-tunable giant nonreciprocal charge transport innoncentrosymmetric oxide interfaces, Nat. Commun. 10, 4510(2019).[34] Q. Ma, S.-Y. Xu, H. Shen, D. MacNeill, V. Fatemi, T.-R. Chang,A. M. Mier Valdivia, S. Wu, Z. Du, C.-H. Hsu, S. Fang, Q. D.Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, E. Kaxiras,H.-Z. Lu, H. Lin, L. Fu, N. Gedik et al., Observation of thenonlinear Hall effect under time-reversal-symmetric conditions,Nature (London) 565, 337 (2019).[35] Z. Fei, W. Zhao, T. A. Palomaki, B. Sun, M. K. Miller, Z. Zhao,J. Yan, X. Xu, and D. H. Cobden, Ferroelectric switching of atwo-dimensional metal, Nature (London) 560, 336 (2018).[36] C. J. Olson, C. Reichhardt, B. Jankó, and F. Nori, Collectiveinteraction-driven ratchet for transporting flux quanta, Phys.Rev. Lett. 87, 177002 (2001).[37] Q. Song, H. Wang, X. Pan, X. Xu, Y. Wang, Y. Li, F. Song, X.Wan, Y. Ye, and L. Dai, Anomalous in-plane anisotropic ramanresponse of monoclinic semimetal 1T -MoTe2, Sci. Rep. 7, 1758(2017).[38] L. Zhou, S. Huang, Y. Tatsumi, L. Wu, H. Guo, Y.-Q. Bie, K.Ueno, T. Yang, Y. Zhu, J. Kong, R. Saito, and M. Dresselhaus,013132-12https://doi.org/10.1126/science.1090390https://doi.org/10.1038/22485https://doi.org/10.1103/PhysRevB.73.014507https://doi.org/10.1103/PhysRevB.68.224504https://doi.org/10.1103/PhysRevB.64.012504https://doi.org/10.1103/PhysRevB.71.024519https://doi.org/10.1103/PhysRevB.98.054510https://doi.org/10.1103/PhysRevResearch.2.023127https://doi.org/10.1103/PhysRevResearch.2.042046https://doi.org/10.1103/PhysRevLett.121.026601https://doi.org/10.1126/sciadv.1602390https://doi.org/10.1038/s41467-020-19459-5https://doi.org/10.1038/s41467-022-29314-4https://doi.org/10.1063/1.4886096https://doi.org/10.1103/PhysRevLett.87.236602https://doi.org/10.1038/ncomms11038https://doi.org/10.1021/acs.nanolett.0c04935https://doi.org/10.1088/1674-1056/ab457fhttps://doi.org/10.1126/sciadv.aay9120https://doi.org/10.1038/s41535-022-00514-xhttps://doi.org/10.1088/0034-4885/79/11/116501https://doi.org/10.1103/PhysRevB.97.134501https://doi.org/10.1103/PhysRevB.99.064307https://doi.org/10.1103/PhysRevB.85.012502https://doi.org/10.1103/PhysRevLett.80.3614https://doi.org/10.1038/s41467-019-12466-1https://doi.org/10.1038/s41586-018-0807-6https://doi.org/10.1038/s41586-018-0336-3https://doi.org/10.1103/PhysRevLett.87.177002https://doi.org/10.1038/s41598-017-01874-2GATE-TUNABLE GIANT SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 6, 013132 (2024)Sensitive phonon-based probe for structure identification of1T ′ MoTe2, J. Am. Chem. Soc. 139, 8396 (2017).[39] Y. Saito, Y. Kasahara, J. Ye, Y. Iwasa, and T. Nojima, Metallicground state in an ion-gated two-dimensional superconductor,Science 350, 409 (2015).[40] X.-Q. Li, Z.-L. Li, J.-J. Zhao, and X.-S. Wu, Electrical and ther-moelectric study of two-dimensional crystal of NbSe2, Chin.Phys. B 29, 087402 (2020).[41] K. Behnia and H. Aubin, Nernst effect in metals and super-conductors: A review of concepts and experiments, Rep. Prog.Phys. 79, 046502 (2016).[42] D. Y. Vodolazov and F. M. Peeters, Superconducting rectifierbased on the asymmetric surface barrier effect, Phys. Rev. B72, 172508 (2005).[43] D. Cerbu, V. N. Gladilin, J. Cuppens, J. Fritzsche, J. Tempere,J. T. Devreese, V. V. Moshchalkov, A. V. Silhanek, and J. V. deVondel, Vortex ratchet induced by controlled edge roughness,New J. Phys. 15, 063022 (2013).[44] J. Ji, J. Yuan, G. He, B. Jin, B. Zhu, X. Kong, X. Jia, L. Kang,K. Jin, and P. Wu, Vortex ratchet effects in a superconductingasymmetric ring-shaped device, Appl. Phys. Lett. 109, 242601(2016).[45] M. Konczykowski, L. I. Burlachkov, Y. Yeshurun, and F.Holtzberg, Evidence for surface barriers and their effect on irre-versibility and lower-critical-field measurements in Y-Ba-Cu-Ocrystals, Phys. Rev. B 43, 13707 (1991).[46] C. D. Dewhurst, D. A. Cardwell, A. M. Campbell, R. A.Doyle, G. Balakrishnan, and D. M. Paul, Determinationof the onset of bulk pinning and the low-temperature-irreversibility line in Bi2Sr2CaCu2O8+δ , Phys. Rev. B 53, 14594(1996).[47] M.-S. Kim, C. U. Jung, M.-S. Park, S. Y. Lee, K. H. P. Kim,W. N. Kang, and S.-I. Lee, Prominent bulk pinning effect in theMgB2 superconductor, Phys. Rev. B 64, 012511 (2001).[48] E. Zeldov, A. I. Larkin, V. B. Geshkenbein, M. Konczykowski,D. Majer, B. Khaykovich, V. M. Vinokur, and H. Shtrikman,Geometrical barriers in high-temperature superconductors,Phys. Rev. Lett. 73, 1428 (1994).[49] N. B. Kopnin, Theory of Nonequilibrium Superconductivity(Oxford University Press, New York, NY, 2000).013132-13https://doi.org/10.1021/jacs.7b03445https://doi.org/10.1126/science.1259440https://doi.org/10.1088/1674-1056/ab9614https://doi.org/10.1088/0034-4885/79/4/046502https://doi.org/10.1103/PhysRevB.72.172508https://doi.org/10.1088/1367-2630/15/6/063022https://doi.org/10.1063/1.4971835https://doi.org/10.1103/PhysRevB.43.13707https://doi.org/10.1103/PhysRevB.53.14594https://doi.org/10.1103/PhysRevB.64.012511https://doi.org/10.1103/PhysRevLett.73.1428