Edge superconductivity in multilayer WTe2 Josephson junction

Abstract WTe2, as a type-II Weyl semimetal, has 2D Fermi arcs on the (001) surface in the bulk and 1D helical edge states in its monolayer. These features have recently attracted wide attention in condensed matter physics. However, in the intermediate regime between the bulk and monolayer, the edge states have not been resolved owing to its closed band gap which makes the bulk states dominant. Here, we report the signatures of the edge superconductivity by superconducting quantum interference measurements in multilayer WTe2 Josephson junctions and we directly map the localized supercurrent. In thick WTe2 (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\sim 60{\rm{\ nm}})$\end{document}, the supercurrent is uniformly distributed by bulk states with symmetric Josephson effect (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$| {I_c^ + ( B )} | {=} | {I_c^ - ( B )} |\ $\end{document}). In thin WTe2 (10 nm), however, the supercurrent becomes confined to the edge and its width reaches up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$1.4{\rm{\ \mu m\ }}$\end{document}and exhibits non-symmetric behavior \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$| {I_c^ + ( B )} | \ne | {I_c^ - ( B )} |$\end{document}. The ability to tune the edge domination by changing thickness and the edge superconductivity establishes WTe2 as a promising topological system with exotic quantum phases and a rich physics.

The thickness of WTe2 flakes can be reliably determined by atomic force microscopy (AFM). As an example, Figure S2a-b shows AFM image of device #5 with the clearly-defined step-edges. Furthermore, we measured the thickness along the two edges to ensure the homogeneity of samples as displayed in Supplementary Fig. 2c. Cross-sectional plots in Supplementary Fig. 2d demonstrate that the thickness fluctuation is less than 1 nm. Since the current density can be affected by the roughness at each position along the y-direction, the AFM results confirm the uniform roughness in the channel.  The thickness is measured by AFM. The length and width are obtained through SEM and Optical microscopy. The normal resistance , critical current , transition temperature can be extracted from the R-T and I-V curves. The superconducting gap and coherence length are estimated by = . and = ℏ , respectively.

III. Josephson junction properties in other samples
Temperature dependences of four typical junctions' resistance are shown in Supplementary Fig. 3. We can identify two transition temperatures in all 4 devices: 1~8 K is for the Nb superconducting transition temperature; and 2 varies from 0.5 K to 1 K for the Josephson effect.

IV. Josephson junction in different regimes
We choose two typical devices (#1 and #8) to discuss their characterizations here. The Fermi velocity , mobility and Fermi vector are evaluated from the Hall effect and Shubnikov-de Haas (SdH) quantum oscillation measurements performed before [1]. The extracted mobility is = 1.16 m 2 ⋅ −1 ⋅ −1 and = 0.26 m 2 ⋅ −1 ⋅ −1 for thick and thin WTe2, respectively. The Fermi vector is = 0.052Å −1 and = 0.057Å −1 for thick and thin WTe2, respectively. The Fermi velocity is 2.0 × 10 5 m/s and 2.5 × 10 5 m/s for the thick and thin WTe2, respectively.
Then, the superconducting coherence length is estimated to 340 nm and 480 nm for device # 8 and #1, respectively, which is close to the junction length . However, the virtual effective length is longer considering the London penetration depth [2] ~100 nm for Nb such that = + 2 , which can be confirmed by the Fraunhofer pattern in the SQI measurements. Then, < and it is in long junction regime which gives the relation of ∝ 1/ . Meanwhile, we also extract a mean free path = 1.3 μm and 0.15 μm for thick and thin WTe2, respectively. In thick WTe2, the mean free path is larger than the junction length > , suggesting that the junctions are in the ballistic regime. In contrast, the thin WTe2 is in the diffusive regime. To further confirm the ballistic or the diffusive behavior of the junctions, the critical current as a function of temperature, ( ), was measured for devices #1 and #8. For thick WTe2 device #8 which is a long ballistic junction, the critical current follows with [3]  ∝ exp(− / ) where ≈ ℏ /2 is expected to be independent of carrier density or mobility (as long as the junction remains ballistic).
is plotted on a semi-logarithmic scale and clearly shows exponential dependence at high-temperature as shown in Supplementary Fig. 4a. The little deviation at low temperature is common which has been observed in graphene long ballistic junction [3]. The energy scale can be extracted by the slope of linear fitting. Consequently the effective length = 0.62 μm is smaller than mean free path = 1.3 μm which accords with the ballistic condition. For thin WTe2 device #1 which belongs to a long diffusive limit, we used another model to fit as shown in Supplementary Fig. 4b. The model is divided into two temperature regimes. The high-temperature satisfies ≫ ℎ (or Δ), where ℎ = ℏ / 2 represents the Thouless energy. ℎ is estimated to 8 meV in device #1 and consequently > 0.1 K. Combined with the Usadel equations, the critical current is [4] ∝ ∑ Δ 2 exp(− / ) where is the normal metal resistance, is the temperature-dependent superconducting gap and = √ℏ /2 . As shown in the red curve, the fitted long diffusive model is in excellent agreement with the experimental data at high temperatures. In low-temperature regime, the numerical solution can be approximated by [4] ∝ (1 − − /3.2 ) where and are coefficients. As shown by the blue curve in Supplementary Fig. 4b, the equation fits the data well. Therefore, our thin WTe2 satisfies the long diffusive junction.

V. Analysis of the current density profile
In a Josephson junction immersed in a magnetic field B ( ⊥ ), the magnitude of the maximum critical current ( ) depends strongly on the supercurrent density between the Nb electrodes. Here, we convert our measured interference patterns to their originating supercurrent density profiles by the method developed by Dynes and Fulton [5].

/ 14
( ) is the magnitude of this summation: ( ) = | ( )|. We use the even ( ( )) and odd part ( ( )) extracted from the ( ). Then the ( ) can be expressed as The observed critical current ( ) = √ 2 ( ) + 2 ( ) is therefore dominated by ( ) except at its minimum points. Approximately, ( ) is obtained by multiplying ( ) by a flipping function that switches sign between adjacent lobes of the envelope function ( Supplementary Fig. 5a-b). When ( ) is minimal, the odd part ( ) dominates the critical current. ( ) can then be approximated by interpolating between the minima of ( ), and flipping sign between lobes ( Supplementary Fig.  5c). A Fourier transform of the resulting complex ( ), over the sampling range of , yields the current density profile (Supplementary Fig. 5d):

VI. Fits of the critical current-magnetic field relation
Since the Josephson current density is non-uniform (edge-stepped nonuniform) by edge modes, the model of the nonuniform supercurrent density provides a much better fit to the data than that of a uniform supercurrent density. Here we consider the normalized supercurrent density is and 1 in edge and bulk, respectively. This produces an edge-stepped nonuniform supercurrent density ( ) along the direction, as schematically shown in Supplementary Fig. 6a and is given by and are the edge thickness and junction width. In general, can be expressed as [6] ( ) = |∫ ( ) d 0 | where = 2 /Φ 0 . Combining the above two equations, the magnetic field dependence of is given as [6] ( ) Here, 0 is the width of each lobe. At the high magnetic field, the value 0 increases sufficiently beyond 1 and the second cosinusoidal term pre-dominates, which represents the typical cos ( 0 )-type B modulation of similar to a SQUID pattern.
In device #2, 0 = 0.21 mT and = 9 μm. Since the supercurrent density at the two sides may be different, we use the model to fit the negative and positive magnetic field data as shown in Supplementary Fig. 6b-c. The -field modeulation of critical 8 / 14 current agrees well with this model both in its magnitude and field periodicity. The length of the edge is fitted as ~0.70 and 1.65 μm at two sides and the average value is 1.18 μm and close to the Fourier imaging method in Fig. 2g. The edge/bulk supercurrent density ratio is 3.0 and 3.6 at two sides. The good fit of Supplementary  Fig. 6b-c

VII.
Additional devices We also reproduce the Fraunhofer and SQUID pattern transition in the other three WTe2 devices (#5, #3 and #1) as shown in Supplementary Fig. 7.
The 2 (device #3) does not oscillate with the magnetic field as shown in Supplementary Fig. S8. On the contrary, with a small magnetic field 0.25 T, the superconducting behavior disappears. The center lobe (black line) can be simulated by Fraunhofer pattern as We can estimate the width 2 for 2 is about 1.9 μm, which corresponds well with the actual junction width 1.7 μm as shown in Fig. 3a.
Even the second lobe is hard to be distinguished in 2 . We think it is related to the incomplete superconductivity (Josephson effect) in WTe2. The resistance decrease in 2 is only smaller than 10% indicating that the proximity Josephson effect just starts and a very small magnetic field can quench the superconductivity in WTe2. We also performed the control experiment on a similar device structure as sample # 3 by two-probe configuration is shown in Supplementary Fig. 9a. Sample #9 is made by different channels as shown in Supplementary Fig. 9a. 1 and 2 represent the Josephson channel across edge or not, respectively. The width of the 1 and 2 are 10 μm and 7 μm, respectively. Both channels' lengths are about 200 nm which is smaller than device #3 in Fig. 3. The shorter and wider channel of 2 helps to increase the possibility of bulk superconductivity. We observed the Josephson effect in both channels as shown in Supplementary Fig. 9b. The critical current is 1 = 6.0 μA and 2 = 0.67 μA for 1 and 2 , respectively and indicates the huge reduction of 2 because of the elimination of edge superconductivity in 2 . We obtain a period of ~0.25 mT for 1 , which yields the effective length of = Φ 0 / ( )~0.8 μm. The central lobe length is about ~0.3 mT which is just a little larger and two times smaller than the oscillation period. Therefore, we can regard the pattern of 1 as a mixture of Fraunhofer and SQUID-like while traditional Fraunhofer pattern in 2 in Supplementary Fig. 9c-d. The supercurrent distribution by the inverse Fourier transform illustrates edge and uniform superconductivity for 1 and 2 , respectively as shown in Fig. R9e. The corresponding supercurrent distribution is highly edge-like with small bulk density and bulk-dominated for 1 and 2 , respectively. The 1 widths of the supercurrent-carrying edge channels are estimated to be 1.2 − 1.3 μm. On the contrary, the 2 shows no edge superconductivity because of the lack of edge channels. Furthermore, the asymmetric critical currents at different current directions as shown in Supplementary Fig. 9f which indicates a non-symmetric behavior + ( ) ≠ − ( ), where + anddenote the sweep direction of the bias current and + andare the magnetic field directions.
The two-probe configuration method on thick WTe2 (#10) is shown in Supplementary Fig. 10a. We briefly summarize the results next. Different from the edge superconductivity in #9, the thick WTe2 exhibits the traditional Fraunhofer pattern ( Supplementary Fig. 10b), uniform supercurrent density (Supplementary Fig. 10c) and  Fig. 10d). The effective length is estimated to be ~1.0 μm which is also larger than the channel length ~0.25 μm.

VIII. Discussion on non-uniform supercurrent density
There is no doubt that thinner films are more susceptible to fluctuations and affected more by the SiO2 substrate or the capping layer.
However, the fluctuations should be random along the in-plane direction. It is quite difficult to believe that the fluctuations are not uniform and gathered only at the edges. Moreover, four thin devices behave large edge supercurrent density while three thick devices show the Fraunhofer pattern which indicates good repeatability. Actually, we simulate the effect by random fluctuations as shown in Supplementary Fig. 11. The supercurrent density fluctuations are assumed to be ±10% and ±30% in Supplementary Fig. 11a. The − relation can be estimated by The result is simulated as shown in Supplementary Fig. 11b, which is a not ideal Fraunhofer pattern but almost similar to conventional unchanged supercurrent density.
The trap sites in the SiO2 substrate or the capping layer. This effect also possibly  exists in our systems. However, such substrate or the capping layer induced nonuniform supercurrent density should affect obviously the out-of-plane direction rather than in-plane direction. Both the substrate and the capping layer can induce fluctuations or trapped carriers at the top and bottom surface of WTe2 rather than non-uniform distribution in-plane. Therefore, this effect should be much smaller in our experiments. Furthermore, a lot of Josephson junction works based on other thin materials have been reported before and exhibit the normal Fraunhofer pattern [7][8][9].
In summary, based on the three reasons listed above, we conclude that the fluctuations and effect by SiO2 cannot result in the non-uniform supercurrent density.

Supplementary Figure 11. Expected effect on SQI measuring by contaminations. (a)
The supposed supercurrent density by fixed value and fluctuating value for no and random contamination effect, respectively. (b) The estimated magnetic field dependent supercurrent oscillation from (a) and indicates whether small fluctuating supercurrent density will almost not change the SQI results.
In graphene, "fibre-optic" modes exist at the edge due to the band bending [10]. This trivial effect will also result in the SQUID interference pattern [10]. However, this edge-mode-dominated current flow can be observed only near the Dirac point, not at a higher density of states. The conductance of thin WTe2 is larger than the quantum conductance ( ⋅ 2 2 ℎ , is the thickness) indicating a non-negligible bulk contribution and a high Fermi level. Thus, the edge superconductivity observed in WTe2 does not seem to be caused by "fibre-optic" edge modes.