Martin Endres,Artem Kononov,Hasitha Suriya Arachchige,Jiaqiang Yan,David Mandrus,Kenji Watanabe,Takashi Taniguchi,Christian Schönenberger
Current–Phase Relation of a WTe2 Josephson Junction
WTe2,topological superconductivity,higher-order topological insulators,edge states,current−phase relation,asymmetric SQUID
When a topological insulator is incorporated into a Josephson junction, the system is predicted to reveal the fractional Josephson effect with a 4π-periodic current–phase relation. Here, we report the measurement of a 4π-periodic switching current through an asymmetric SQUID, formed by the higher-order topological insulator WTe2. Contrary to the established opinion, we show that a high asymmetry in critical current and negligible loop inductance are not sufficient by themselves to reliably measure the current–phase relation. Instead, we find that our measurement is heavily influenced by additional inductances originating from the self-formed PdTex inside the junction. We therefore develop a method to numerically recover the current–phase relation of the system and find the 1.5 μm long junction to be best described in the short ballistic limit. Our results highlight the complexity of subtle inductance effects that can give rise to misleading topological signatures in transport measurements.
Current–Phase Relation of a WTe2 Josephson Junction

When a topological insulator is incorporated into a Josephson junction, the system is predicted to reveal the fractional Josephson effect with a 4π-periodic current–phase relation. Here, we report the measurement of a 4π-periodic switching current through an asymmetric SQUID, formed by the higher-order topological insulator WTe2. Contrary to the established opinion, we show that a high asymmetry in critical current and negligible loop inductance are not sufficient by themselves to reliably measure the current–phase relation. Instead, we find that our measurement is heavily influenced by additional inductances originating from the self-formed PdTex inside the junction. We therefore develop a method to numerically recover the current–phase relation of the system and find the 1.5 μm long junction to be best described in the short ballistic limit. Our results highlight the complexity of subtle inductance effects that can give rise to misleading topological signatures in transport measurements.

Topological insulators (TIs) belong to a unique class of materials that are insulating in their bulk while hosting gapless boundary states that are protected by time-reversal symmetry. The class of three-dimensional TIs has recently been extended, realizing that a d-dimensional TI of order n can develop (d – n)-dimensional hinge or corner states. A promising candidate of this novel material class that is predicted to host topological hinge states is the semimetallic transition-metal dichalcogenide WTe2. While bulk states dominate transport in the normal state, hinge states become the governing transport channel over long distances in the superconducting state, as they can carry a higher critical current due to their reduced dimensionality. Therefore, Josephson junctions (JJs) formed by a TI as the weak link between two superconducting electrodes provide an ideal platform to probe the topological nature in a transport experiment. Topological hybrid systems are of great interest, as they are predicted to host unconventional superconductivity, the fundamental building block of a potential topologically protected quantum bit. The fingerprint of a JJ is the current–phase relation (CPR), the dependence of the supercurrent on the phase difference φ between the superconducting electrodes. The measurement of the CPR directly reflects the underlying transport mechanism with which the Cooper pairs are shuttled across the weak link. For a topological weak link, perfect Andreev reflection is expected, since spin-momentum locking in the hinge states prohibits normal electron reflection at the interface to the superconductor. In the long junction limit, the supercurrent is carried by 4π-periodic Andreev bound states of opposite parity, σ+ and σ–, that give rise to a characteristic sawtooth-shaped Ic. Parity conservation prohibits the recombination to the lower energy branch and results in a multivalued Ic with a distinct diamond shape, as plotted in Figure 1a. For comparison, the expected CPR of a trivial ballistic JJ in the long junction limit is plotted as a black dashed line in the same figure. Accordingly, the literature often infers ballistic topological states from the observation of a sawtooth-shaped flux dependence of the critical current. Recently, high-quality JJs formed in WTe2 on palladium (Pd) bottom contacts have been reported and provided evidence of a nonsinusoidal CPR. Here, we combine such JJs based on Pd-induced superconductivity in WTe2 with external superconducting leads, as illustrated in Figure 1b. A superconducting quantum interference device (SQUID) formed out of two such JJs is expected to reflect the CPR, provided that the critical current amplitudes of the two JJs are highly different and the loop inductance is negligible. Based on this method, we observe a switching current distribution with 4π-periodicity that resembles a topological JJ. Contrary to the topological interpretation, we provide an alternative explanation based on inductance effects, which could be relevant for a number of previous experiments. Importantly, the inductance contribution originates from the JJs themselves and exceeds the loop inductance. We further developed a numerical model based on the maximization of the supercurrent in the SQUID loop that allows us to exclude screening effects from the additional inductances and recover the real CPR. The calculations suggest that the critical current of our 1.5 μm long junction is best reproduced in the short ballistic limit, despite its long physical length. We begin with the fabrication of an asymmetric SQUID out of WTe2. In our experiment, both JJs are formed in the same needle-shaped WTe2 flake of width w = 1.5 μm. The asymmetry in critical current Ic of the two involved JJs is achieved by a different spacing between the Pd bottom contacts lw = 1.5 μm and lr = 0.5 μm for the weak and reference junction, respectively, as sketched in Figure 2a. The superconducting loop is formed by etching through the top hBN into WTe2 and sputtering niobium (Nb) on top. The Nb leads are between 2.2 and 3 μm wide and 100 nm thick. A detailed description of the fabrication process can be found in the Supporting Information. Figure 2a displays an optical image of the finished devices. In the following we will focus on the lower SQUID, enclosed by the dashed line. The critical current of the SQUID is given by the sum of the individual currents Iw and Ir through the two branches of the loop, defined by the critical current Ici and the normalized CPR fi of the ith Josephson junction. The total flux Φtot threading the loop connects the phase differences across the two JJs φw – φr = 2πΦtot/Φ0 = ϕtot, with ϕtot denoting the external phase. In the absence of inductances in the loop, ϕtot is simply defined by the external phase ϕx. The asymmetry Icr ≫ Icw pins φr = φrmax at a fixed phase, for which Icr is maximized. The normalized CPR of the weak junction fw can then be deduced from the measurement ofIn the experiment, Φx = BAo is controlled by an applied perpendicular magnetic field B threading the effective loop area Ao = 186 μm2. The Meissner screening of the enclosing superconducting loop was taken into account by including half of its width into the loop area. The final device is probed in a quasi four-terminal configuration by sourcing an ac current and monitoring the voltage drop over the SQUID. We use the counter technique, as described in the Supporting Information, to measure the switching statistics of the device. Figure 2b presents the measured switching statistics of the SQUID in an extended magnetic field range at base temperature T = 30 mK of the cryostat. Visible is a multivalued Ic that oscillates periodically around the offset current Icr = 160 μA of the reference junction. Given the great difference in critical current amplitudes Icr/Icw ≈ 40, we expect the SQUID to be highly asymmetric and the measured signal to reflect the CPR of the weak JJ. A second set of oscillations appears for field values from B = −100 μT upward and is explained in the Supporting Information by the behavior of the reference junction. The extracted oscillation period of a fixed parity branch, as defined in Figure 1a, is equal to δB = 23.2 μT and therefore twice the value δB = Φ0/Ao = 11.1 μT expected for the enclosed loop area Ao. The data represent striking resemblance to the two parity states of a topological JJ in the long junction limit with a 4π-periodicity in flux. However, the amplitude of the signal deems this explanation unlikely. A single channel in the topological ballistic long junction can carry a current Ic,4π = EThe/ℏ = vFe/l, with ETh being the Thouless energy, vF the Fermi velocity, and l the length of the weak link. Ic in our junction would therefore have to be carried by at least 116 perfectly ballistic channels in parallel, assuming vF = 3.09 × 105 m s–1. WTe2 is expected to host a pair of hinge states on opposite edges of the crystal. Conducting states have been found to reside at step edges in the crystal where the number of vdW layers changes, but we do not observe such crystal steps in optical microscopy for the used flake. We conclude that it is unlikely for the current to be carried purely by ballistic hinge states. Instead, multivalued CPR measurements have previously been reported in devices containing a superconducting weak link with large kinetic inductance that is responsible for strong screening effects. In strong contrast to previous experiments, however, the device studied here contains a normal conducting weak link and has negligible loop inductance, as will be shown later. The schematic in Figure 2c introduces a set of additional inductances, Lr and Lw, that are placed in series to the reference and weak junction, respectively, while potential mutual inductances are assumed to be negligible. In general, the total phasecan differ strongly from the phase created by the external flux ϕx = 2πΦx/Φ0, due to the contribution induced by the currents Ir and Iw passing through the inductances in the SQUID arms, Lr and Lw. We note that while a screening current can distort the flux dependence of the critical current, it does not change its periodicity. Deducing the CPR from the inductive SQUID measurement requires the knowledge of the phase dependences of Ir(ϕtot) and Iw(ϕtot) themselves. In order to bypass this recursive constraint, we make an assumption about the CPRs of the JJs that is based on the experimental data. We are then going to use this information in the next step to calculate , for which Ic(ϕtot) is maximized according to eq 1. Last, we are going to include screening effects in the model and obtain the relation Ic(ϕx), which is placed in context to the experimental data. Our choice of CPR for the weak junction is based on two experimental observations. First, the rising slope of Ic(ϕx) is nonlinear, suggesting the same should hold for the CPR. Second, Ic(ϕx) has self-crossings, implying that the phase of the reference junction does not remain fixed, contrary to the established expectation for a highly asymmetric SQUID. The behavior is possible if the CPR of the weak junction contains abrupt changes, such as is the case in the ballistic limit of the CPR. We illustrate this behavior later in the text and provide an additional discussion in the Supporting Information. Iw(φw) is modeled to be in the 2π-periodic, ballistic short junction limitscaled by the amplitude of Icw. The exact CPR of the reference junction plays little role in the further discussion, yet, since it is formed in the same material but with reduced length, we also model it as a short ballistic junction, scaled by Icr. Independently, we have used length-dependent measurements in the Supporting Information and the analysis of the PdTex diffusion profile in ref (18) to verify that both junctions behave indeed as JJs and are not shorted by PdTex. Having made this initial assumption, we continue to calculate the resulting Ic by maximizing the current through the SQUID. The top panel in Figure 3b illustrates a visual method to maximize the critical current as a function of ϕtot. The individual currents through the SQUID arms Ir and Iw follow eq 3. The currents evolve in opposite flux direction, due to the connection φr – φw = ϕtot. We start at the configuration ϕr = ϕrmax and ϕw = ϕwmax, when the currents through the reference and weak junctions are at their maximum, Icr and Icw, respectively. Moving from this point in the negative direction of ϕtot, Ic follows Iw, plotted as a red curve. In the opposite direction toward positive values of ϕtot, Iw faces a sudden drop. Instead of following Iw, a higher Ic is obtained by following Ir, drawn in dark blue, until the point when Ir intersects with Iw. This creates a small flux range δφtot ∝ πIcw/Icr ≪ π, in which Ic is maximized by following Ir rather than Iw, resulting in a changing φr, while ϕwmax = π remains fixed. The situation is illustrated in the lower panel of Figure 3b. Once the two current branches cross with evolving flux, φr returns to its maximum value ϕrmax = π and Ic(ϕtot) follows the flux dependence of Iw. The well-known behavior of the asymmetric SQUID is restored. While δϕtot remains small in the above scenario, it can extend significantly in the experiment, due to the inductance effects, as described by eq 2. Given that φr does not necessarily remain fixed in an asymmetric SQUID, we introduce next a numerical procedure to transfer the above model from the dependence of total flux ϕtot to the external flux ϕx, the quantity applied in the experiment. The slope of Ic(ϕx) is directly related to the inductance of the current carrying arm in the SQUID, via , with LJi being the Josephson inductance and i = r, w, depending on the considered data mapping the CPR of the reference or the weak junction. First, we numerically maximize the expressionwith respect to φr(ϕtot) for a given ϕtot. Ir and Iw correspond to the currents through the respective SQUID arms, i.e. and . The two amplitudes Icr = 160 μA and Icw = 3.75 μA are determined from the fixed current background and the oscillation amplitude of the intertwined branches. The lower panel in Figure 3b plots the obtained φr and φw in blue and red, respectively. Based on the choice of the CPR function in eq 3, φr is mostly fixed at the maximum value ϕrmax = π, while φw evolves linearly in ϕtot, according to ϕw = φtot + ϕrmax. However, a small range δϕtot exists, where φr changes in flux while φw remains fixed, in agreement with the graphical method introduce above in the upper panel of Figure 3b. Using φr(ϕtot), it is now possible to extract the inductance effects and recalculate ϕx = ϕtot – 2π(LrIr – LwIw)/Φ0. Depending on the magnitude of the incorporated inductances and critical currents, self-inductance effects in the loop cause the connection ϕtot(ϕx) to become multivalued, as is visible from Figure 3c. Finally, Ic(ϕx) is plotted in red in Figure 3a. Despite the long physical length Lw = 1.5 μm of the junction, we find the bending of the gradually rising slope dIc/dϕx to be well reproduced by fw being in the 2π-periodic, ballistic short junction limit. In both cases, the magnetic field dependences of the current amplitudes are assumed to be negligible for the given field range. Importantly, despite the great difference in critical current amplitudes of the embedded junctions, the model confirms that φr does not remain fixed in flux. The experimental CPR of the SQUID is composed of the weak and the reference junction CPR. Even though the CPR of the weak junction is not necessarily uniquely in the short junction limit, it has to have a sharp transition in flux and therefore be close to ballistic in order to ease the shift between the fixed φr and φw. Further, we extract the inductances Lr = 60 pH and Lw = 220 pH, by matching the rising and falling slopes of the fit to the data. An important result that distinguishes ours from previous experiments is that Lr and Lw by themselves exceed the sum of geometrical inductance Lgeo ≈ 27.0 pH and kinetic inductance Lkin ≈ 5.5 pH for the Nb SQUID loop. Possibly, additional JJs can form at the interface between the sputtered superconducting leads and the self-formed superconducting PdTex, yet given that their critical current has to be larger than Icr, little inductance contribution is to be expected. Instead, we attribute the origin of additional inductance and its asymmetry between the SQUID arms to the superconducting PdTex that has self-formed at the interface between WTe2 and Pd. Further support of this interpretation is provided in the Supporting Information, including the comparison of the data to different initial CPR assumptions. Finally, the multivalued Ic can also be explained in the framework of excited vorticity states in an inductive SQUID. Using the parameters obtained from our fit, we calculate the magnetic screening factor 1, reflecting that an additional flux quantum can be created by the maximum circulating current through the weak JJ. The result is a multivalued ϕtot as a function of ϕx, as was shown in Figure 3c for the given device parameters. The above behavior can differ strongly even on a single sample chip. While the second SQUID loop formed on the same WTe2 flake (compare Figure 2a) reveals the same behavior of the reference junction with multiple branches, we do not observe higher-vorticity states in the SQUID oscillations. The absence of the feature is most likely connected to the overall smaller Icw, despite the shorter junction length with lw = 1.2 μm. In conclusion, the established assumption of a fixed reference junction phase in flux does not hold for highly transparent junctions, even in the case of highly asymmetric critical current amplitudes. Furthermore, we have shown the complexity of subtle inductance effects that reach beyond the standard consideration of the loop inductance and might create misleading topological features. It is therefore crucial for a correct CPR measurement to consider potential inductance contributions from the interfaces between the embedded junctions with the SQUID loop. The origin of such additional inductances can go beyond the diffusion of PdTex presented here and may include defects implanted at the interface through various fabrication steps. Despite these limitations, the fitting routine presented here allows reproduction of the experimental data closely. The best result was obtained by placing the weak junction in the short ballistic limit, as presented in the Supporting Information. Our results establish WTe2 as a promising platform for further experiments toward topological superconductivity.