Intrinsic supercurrent non-reciprocity coupled to the crystal structure of a van der Waals Josephson barrier

Non-reciprocal electronic transport in a spatially homogeneous system arises from the simultaneous breaking of inversion and time-reversal symmetries. Superconducting and Josephson diodes, a key ingredient for future non-dissipative quantum devices, have recently been realized. Only a few examples of a vertical superconducting diode effect have been reported and its mechanism, especially whether intrinsic or extrinsic, remains elusive. Here we demonstrate a substantial supercurrent non-reciprocity in a van der Waals vertical Josephson junction formed with a Td-WTe2 barrier and NbSe2 electrodes that clearly reflects the intrinsic crystal structure of Td-WTe2. The Josephson diode efficiency increases with the Td-WTe2 thickness up to critical thickness, and all junctions, irrespective of the barrier thickness, reveal magneto-chiral characteristics with respect to a mirror plane of Td-WTe2. Our results, together with the twist-angle-tuned magneto-chirality of a Td-WTe2 double-barrier junction, show that two-dimensional materials promise vertical Josephson diodes with high efficiency and tunability.

The non-reciprocal behavior in dissipative current flows, known as the diode effect, has played a central role in modern electronic devices and circuits 1,2 .In conventional schemes, nonreciprocity along the current direction arises from spatial inhomogeneity 1,2 .Recently, it has been shown that when inversion and time-reversal symmetries are both broken, and in combination with a spin-orbit interaction (SOI), even spatially homogeneous systems can provide for diode functionality [3][4][5] .By implementing this magneto-chirality with superconductors (SCs) and matching the superconducting gap with the SOI energy 6,7 , one can achieve a directional, non-dissipative, supercurrent flow, which is a prerequisite for the realization of future superconducting quantum devices.To date, several methods have been developed to intrinsically and/or extrinsically break the inversion symmetry.For example, via ionic-liquid gating of two-dimensional (2D) MoS2 flakes, via the formation of non-centrosymmetric V/Nb/Ta superlattices or using a few layers of 2H-NbSe2 have enabled the observation of non-reciprocal critical currents near their corresponding superconducting transition temperatures Tc [8][9][10] .Notably, non-reciprocity in Josephson supercurrents have also been realized in lateral Josephson junctions (JJs) with structures: Al/InAs 2D electron gas (2DEG)/Al 7 , Nb/proximity-magnetized Pt/Nb 11 and Nb/NiTe2 type-II Dirac semimetal/Nb 12 .To break the inversion symmetry, the first two utilize a Rashba(-type) superlattice and barrier, respectively, whereas the last exploits topological surface states in a NiTe2 barrier.
Very recently, a field-free Josephson diode effect has been observed in vertical NbSe2/Nb3Br8/NbSe2 junctions 13 , but the non-reciprocal supercurrents appear in the direction along which the inversion symmetry is broken and the diode polarity is, moreover, independent of an applied magnetic field, making investigations of other devices highly desirable to unravel the role of van der Waals barriers.
Here, we utilize Td-WTe2 single-crystal exfoliated flakes as an inherently inversion symmetry breaking barrier in van der Waals (vdW) JJs and through extensive measurements of the Td-WTe2 flake thickness, magnetic field strength/angle and temperature dependences, we demonstrate that the supercurrent non-reciprocity along the vertical direction is intimately connected and is highly dependent on the crystal structure of the Td-WTe2 barrier, thereby allowing a clear distinction between intrinsic and extrinsic mechanisms 7,11,12 .Our study establishes that the co-existence of the magneto-linearity, the magneto-chirality and the thickness and temperaturescaleable diode efficiency constitute the intrinsic supercurrent non-reciprocity, coupled to the crystal structure of a vdW barrier.
As illustrated in Fig. 1a, we fabricate vdW vertical JJs, in which a Td-WTe2 flake (2−60 nm thick) is sandwiched between two superconducting 2H-NbSe2 flakes, in an inert atmosphere glovebox using dry transfer techniques (see Methods for details).Here, we employed NbSe2 flakes thicker than 10 nm (16-17 × the monolayer thickness) to preclude the unintended contribution of Ising Cooper pairing 14 to the non-reciprocal transport properties of WTe2 JJs.Note that Td-WTe2 itself exhibits highly interesting physical properties including a type-II Weyl semi-metallic behavior 15 , higher-order topological hinge states 16 and quantum spin-Hall edge states 17 .In the present study, we focus on how the crystal inversion symmetry of Td-WTe2 is reflected in a vertical Josephson diode effect.Due to the lack of a screw rotational symmetry in Td-WTe2 from the Te atoms, the orthorhombic phase Td-WTe2 is non-centrosymmetric 16,18,19 .The mirror-symmetry plane of Td-WTe2 along the b-axis (green plane in Fig. 1b) creates a polar axis, that is, an internal crystal electric field  !" 18 and together with the heavy W atoms provide a strong SOI.This, in conjunction with an external in-plane (IP) magnetic field  #  ∥ , that breaks the time reversal symmetry, allows for an anomalous phase  # to be created in the current-phase relationship (CPR) of the vertical JJ [20][21][22] .As previously discussed theoretically (19-21), the presence of a finite  # is essential to generate a rectified Josephson supercurrent, namely, unequal positive and negative Below, we will separate a crystal-asymmetry-driven intrinsic diode effect from other extrinsic possibilities in the following manner.When the dc bias current I is applied along the vertical direction (// c-axis) and the IP magnetic field is applied in the a-b plane of the WTe2 (blue plane in Fig. 1b), the Josephson diode efficiency  = Here  *+ is the relative angle between the polar axis (// b-axis) of WTe2 and the applied [20][21][22]  ) for several values of the thickness of WTe2 (2, 7, and 60 nm) (Fig. 2, a-f), three notable features are revealed.First, for all the JJs,  increases linearly with increasing  #  ∥ up to a certain critical field, above which  starts to decay (Fig. 2, d-f).Second, when this field coincides with the first-order minimum field of the Fraunhofer interference pattern,  decays in a complex way with an accompanying sign change (Fig. 2, b and c, e and f).Third, the slope of the low-field ( #  ∥ ) data (Fig. 2, d-f) depends critically on the orientation of the polar axis of the WTe2 with respect to the applied  #  ∥ .While the first and second features are qualitatively similar to previous studies on Al/2DEG/Al 7 and Nb/NiTe2/Nb 12 lateral JJs, the third, signifying the magneto-chirality of  [∝ sin( *+ )] , is a new finding from the NbSe2/WTe2/NbSe2 vertical JJs.Note that no signature of fast oscillations, often attributed to topological edge states of WTe2 23 , are detected in measurements of the magnetic-field interference patterns for our vertical JJs (Fig. 2a-c).This is in agreement with theoretical considerations that topological edge states, that reside on the a-b planes of WTe2 24 , may contribute to lateral transport 25,26 but not to vertical transport.This indicates the crystallographic origin of the WTe2 barrier for the magneto-chirality that we report here.
To confirm this distinctive magneto-chirality, the  #  ∥ angular dependence of  is measured for each JJ.Note that this measurement is conducted in the low-field regime where the magneto-linearity of  (∝  #  ∥ ) holds (see Fig. 2, d-f).As summarized in Fig. 3, a-c, the measured () data are all well fitted by a sine function, suggesting a magneto-chiral origin of the Josephson diode effect in the vdW WTe2 JJs.Furthermore, there exists a visible shift between the sine fits to the () data for different WTe2 JJs.This indicates that the crystal structure of the WTe2 barrier indeed governs the Josephson supercurrent non-reciprocity in our system.The magneto-chirality described above predicts  to be maximized (minimized) when the applied  #  ∥ is aligned along the a-axis (b-axis).To identify the a-and b-axis of each WTe2 barrier in the fabricated junctions, angle-resolved polarized Raman spectroscopy was carried out: this technique can readily characterize the crystal orientation of low-dimensional materials 16,18 .Figure 3, d-f, show the polarization angle dependent relative intensities of two distinct Raman peaks at ~160 cm -1 and ~210 cm -1 .The extracted a-and b-axis of the WTe2 from the Raman data (Fig. 3, d-f, see Supplementary Information S3 for the detailed analysis) agree with those from the () data (Fig. 3, a-c).These results strongly support an intrinsic-crystal-structure-reflected vertical Josephson diode effect.For completeness, we also fabricated and measured control vdW JJs, where the noncentrosymmetric WTe2 barrier is replaced with a centrosymmetric 27 1T¢-MoTe2 barrier.In addition, the effect of direct tunneling between the top and bottom superconducting NbSe2 electrodes in JJs without any vdw-WTe2 (Supplementary Information S6) were fabricated.None of the magnetolinearity and the magneto-chirality effects are found in these experiments (see Supplementary Information S4), thereby pinpointing the critical role of the Josephson barrier's crystal structure for the supercurrent non-reciprocity in our WTe2 JJs.
The proportionality of  # to the barrier length is another characteristic feature of the intrinsic Josephson diode effect [20][21][22] , so one can anticipate that the thicker the WTe2 barrier, the greater the diode efficiency.To test this, we next investigate how the field-strength-normalized , measured at qMC = 90°, scales with the WTe2 thickness (Fig. 4a).Note that since  ∝  #  ∥ in the low-field regime,  * (= ) allows for a more quantitative comparison.
As the WTe2 thickness is increased from 2 to 23 nm, we find that  * increases linearly.This linear scaling behavior is, in fact, predicted theoretically for a ballistic JJ 20 .When the thickness of the WTe2 is increased to 60 nm, which is larger than the coherence length (~30 nm, Supplementary Information S5) and the c-axis mean free path (~40 nm) 28 ,  * deviates from the linear thickness dependence.Given the distinctively different barrier-thickness dependence of  # on whether the junction is in the ballistic or diffusive regime, such a deviation is likely related to a ballistic-todiffusive (long-junction) transition [20][21][22] .
The temperature T dependent evolution of  * , gives additional information about the intrinsic Josephson diode properties.Figure 4b shows the T/Tc-dependent  * for several WTe2 JJs.
It is especially noteworthy that the  * (/ ! ) data can be well described by a =1 − , as predicted in several earlier theoretical studies [20][21][22] .For ,  * tends to saturate, which is qualitatively similar to recent related experiments 7,10 .A very recent theory predicts a rather complicated T-dependence of the superconducting diode effect in diffusive Rashba-type SCs 29 as the low-T limit diode efficiency is affected sensitively by structural disorder and impurity scattering.On the other hand, for Al/2DEG/Al ballistic JJs 7 , the field-strength-dependent supercurrent non-reciprocity is explained by considering contributions from first and second order harmonics in the JJ CPR.If we apply this analysis to our  * (/ ! ) data, both harmonics seem to become constant at , as is consistent with the calculation in Ref. 7.
To further show the importance of our vertical JJ platform, that goes beyond recent studies 7- 12 , we have fabricated vertical JJs with a twisted WTe2 double-barrier (Methods).Note that the twist angle  34563 between two distinct WTe2 layers uniquely forms an artificial polar axis of the entire Josephson barrier in a controllable manner (Extended Data Fig. 1).As summarized in Extended Data Fig. 2, the magneto-chirality of supercurrent non-reciprocity in such JJs is not only determined by the polar axis of the top WTe2 but also by that of the bottom WTe2, demonstrating a tunable magneto-chirality via twist-angle engineering and opens an avenue for the development of twistable 30 active Josephson diodes.
We have carefully fabricated vdW vertical JJs with an inherently inversion symmetry breaking Td-WTe2 barrier.It enables us to demonstrate the intrinsic Josephson non-reciprocity 20- 22 , namely, the crystal structure of the Josephson barrier governs overall properties of the Josephson diode, which is unprecedently achieved.We believe that our result help understand better the barrier's role in the realization of rectified Josephson supercurrents in vdW heterostructures 13 .Our approach can be extended to other low-dimensional, low-symmetric and twisted vdW systems [31][32][33] for accelerating the development of two-dimensional superconducting devices and circuits.

Methods
Device fabrication.The NbSe2/WTe2/NbSe2 van der Waals (vdW) heterostructures (Fig. S1, a-d) were formed using standard dry transfer techniques.All the needed processes, including mechanical exfoliation, pick-up and dry transfer of the vdW flakes, were performed inside a nitrogen gas filled glovebox to prevent the flakes from oxidizing in ambient air.First, NbSe2 (≥ 10 nm thick) and WTe2 (2−60 nm thick) flakes were exfoliated on to a 300-nm-thick SiO2/p++ Si substrate.Each of the flakes needed to form the heterostructure was chosen by optical microscopy examination.The chosen flakes were picked up sequentially using a polycarbonate (PC) film coated dome-shaped polydimethylsiloxane (PDMS) stamp.The flakes were aligned one on top of each other and them the entire stack was released and then placed on top of a set of pre-patterned gold electrodes on a second 300-nm-thick SiO2/p++ Si substrate.The release process was performed at 200 C°, after which the entire stack in the Si substrate was immersed in a chloroform solution to remove any PC residue.Using this fabrication process flow, we also prepared three distinct types of JJs: 1) a control vdW JJ, in which the non-centrosymmetric WTe2 barrier 16,18,19 is replaced by a centrosymmetric 27 1T¢-MoTe2 barrier (Supplementary Information S4), 2) a reference vdW JJ with no WTe2 barrier (Supplementary Information S6), and 3) a twisted WTe2 double-barrier JJ (Extended Data Fig. 1-2), where one WTe2 is twisted in-plane relative to the other WTe2.For the formation of the twisted WTe2 double-barriers, we first identified the crystal orientation of two chosen WTe2 flakes through their elongated shapes 16 (which tends to be along the a-axis), and then carefully twisted and stacked one on top of the other to realize  34563 » 90°.
After completing the transport measurements, we finally confirmed the respective crystal that was needed to break the time-reversal symmetry, was applied within the a-b plane of the WTe2 single crystalline barrier.Note that  is the angle of  #  ∥ relative to one edge of the Si wafer on which the exfoliated layer stack was placed: the wafer had been purposely cut into a rectangular shape to define a clear reference direction.The typical Josephson penetration depth 34 of our junctions (of at least a few µm) is much larger than the WTe2 thickness, so we can exclude orbital magnetic field effects (or Meissner demagnetizing supercurrents) as a possible extrinsic source for of the Josephson diode effect in our JJs.We measured the field strength dependent diode signals (Fig. 2, A-C) by applying  #  ∥ along a given  direction.The angle dependent diode signals (Fig. 3, A-C) were measured at a constant  #  ∥ using a vector field magnet.(green) mT.Note that  is the angle of  #  ∥ relative to an edge of the rectangular Si wafer on which the JJ was formed.

Atomic
orientations of the top and bottom WTe2 flakes by a means of angle-resolved polarized Raman spectroscopy.Electrical measurements.The current-voltage (I-V) curves of the fabricated NbSe2/WTe2/NbSe2 vdW vertical Josephson junctions (JJs) were measured in a BlueFors dilution refrigerator or a Quantum Design Physical Property Measurement System with base temperatures of ~20 mK and ~1.8 K, respectively.The four-point I-V measurements were carried out using a Keithley 6221 current source and a Keithley 2182A nanovoltmeter.An external in-plane magnetic field ( #  ∥ ), force microscopy.To characterize the thickness of each WTe2 barrier, we conducted atomic force microscopy (AFM) measurements on the fabricated vdW vertical JJs.The measured AFM image and height of each WTe2 barrier layer are shown together with the corresponding optical micrograph of the JJs in Fig. S1.

Fig. 1 .
Fig. 1.Vertical rectified supercurrents in a van der Waals WTe2 Josephson junction.a, Schematic of a NbSe2/Td-WTe2/NbSe2 van der Waals vertical Josephson junction (JJ).The combination of intrinsic inversion symmetry breaking within the Td-WTe2 barrier and timereversal symmetry breaking by an applied magnetic field give rise to the (vertical) supercurrent non-reciprocity.b, Schematic diagram of the crystal structure of WTe2.Orange (blue) symbols represent the W (Te) atoms.qMC is defined as the relative angle between the polar axis along baxis, the mirror plane of Td-WTe2 (green plane), and the direction of the in-plane (IP) magnetic field  #  ∥ applied within the a-b plane of Td-WTe2 (blue plane).c, Current-voltage I-V curves of a van der Waals JJ with a 23 nm thick Td-WTe2 barrier for  #  ∥ = 0 (black), +6 (orange) and -6

Fig. 2 .Fig. 3 .Fig. 4 . 1 .Extended Data Figure 2 .)Top WTe 2 ,
Fig. 2. Scaling of Josephson supercurrent non-reciprocity with magnetic field strength.a, b, c, Positive and negative Josephson critical current  !% (green) and | !& | (orange) versus in-plane (IP) magnetic field  #  ∥ for (a) 2 nm, (b) 7 nm and (c) 60 nm thick Td-WTe2 barriers in a NbSe2/Td-WTe2/NbSe2 Josephson junction.The measurement was conducted at T = 200 mK for the 2 nm and 7 nm thick junctions and T = 20 mK for the 60 nm thick junction, respectively.d, e, f, . The magneto-linearity (∝  #  ∥ ) and the magneto-chirality [∝ sin( *+ )] of  are key measures of the intrinsic Josephson diode effect.Tc ≈ 5 K.  is the angle of  #  ∥ relative to an edge of the Si wafer on which the exfoliated layers were placed: the wafer had been cut into a rectangular shape to define a clear reference direction.It is clear that a critical current asymmetry (∆! =  !%− | !& | ≠ 0) is only developed when  #  ∥ is non-zero and that the polarity of ∆ ! is inverted when the direction of  #  ∥ is reversed.These features correspond to a Josephson diode effect. ∥ for a fixed  *+ ≠ 0 o ( = 0 o ).From a comparison of 8 !( #  ∥ )8 and ( #  ∥