Evidence for an anomalous current phase relation in topological insulator Josephson junctions

Josephson junctions with topological insulator weak links can host low energy Andreev bound states giving rise to a current phase relation that deviates from sinusoidal behaviour. Of particular interest are zero energy Majorana bound states that form at a phase difference of $\pi$. Here we report on interferometry studies of Josephson junctions and superconducting quantum interference devices (SQUIDs) incorporating topological insulator weak links. We find that the nodes in single junction diffraction patterns and SQUID oscillations are lifted and independent of chemical potential. At high temperatures, the SQUID oscillations revert to conventional behaviour, ruling out asymmetry. The node lifting of the SQUID oscillations is consistent with low energy Andreev bound states exhibiting a nonsinusoidal current phase relation, coexisting with states possessing a conventional sinusoidal current phase relation. However, the finite nodal currents in the single junction diffraction pattern suggest an anomalous contribution to the supercurrent possibly carried by Majorana bound states, although we also consider the possibility of inhomogeneity.

We analyze both single junctions and dc SQUIDs based on lateral Josephson junctions on the surface of bismuth selenide. We focus on one particular SQUID (illustrated in Fig. 1a, with sample I − V s shown in Fig. 1b), with similar results observed in other devices. The SQUID is formed from three leads on the surface of the TI, separated by 100 nm. Recently, there has been much experimental progress in realizing and studying the Josephson effect in TIs [8][9][10][11][12][13][14]. Although most such devices appear to have a significant bulk contribution to the normal state conductance, there is evidence that the majority of the supercurrent is carried by surface states. This is demonstrated in the SQUID at zero field, where we observe a sharp drop in the critical current with top gating (Fig. 1c). This signals a topological phase transition that occurs when the conventional 2DEG originating from band-bending at the top surface is nearly depleted, which exposes the helical surface states that carry the majority of the supercurrent to greater disorder [14]. As shown in Fig. 1d, across the transition we find a qualitative change in the temperature dependence, as the junction acquires a more diffusive character [15]. We have observed very similar behavior in nearly all of our TI junctions, independent of TI film thickness (from 7 to 86 nm), suggesting that the supercurrent is dominated by surface effects. However, we emphasize that our interpretations of interferometric measurements in this paper can be made independent of exact knowledge of the role played by trivial states in the bulk or the surface. This assertion is justified because the topological states can coexist with such trivial states [16]. Theoretical studies of doped topological superconductors [17][18][19][20] also suggest that the bulk can be gapped by superconductivity, permitting the observation of surface physics.
Magnetic flux inserted into the junction induces phase-winding along the width of the junction, leading to interference effects that modulate the critical current. In a Josephson junction with a uniform current density and a sinusoidal CPR, this results in a Fraunhofer diffraction pattern, characterized by vanishing of the critical current from destructive interference whenever an integer number of flux quanta are enclosed by the junction. These nodes remain zero for non-sinusoidal CPRs that are 2π-periodic. In contrast, it has been proposed [7] that Josephson vortices could stabilize pairs of Majorana bound states in TI junctions, leading to a residual critical current at integer flux quanta. While anomalous diffraction patterns from TI junctions have been reported [11], interpretation of such node-lifting must be done carefully due to the possibility of trivial effects, such as inhomogeneity, disorder, and screening effects of large supercurrents.
Alternatively, one may analyze quantum interference between two junctions interrupting a superconducting loop (SQUID). Here, flux threaded within the loop imposes a phase difference between the two junctions, generating interference that is less sensitive to junction details. We combine both approaches (single junction diffraction pattern and SQUID oscillations) to probe the CPR of TI JJs. These approaches can be more sensitive to anomalous signals than a direct measurement of CPR [21] by focusing on regimes where conventional components are canceled out by destructive interference.
In Fig. 2a, we show the magnetic field dependence of the critical current for two different top gate biases. We observe rapid SQUID oscillations with a period of ≈ 0.21 mT, consistent with the lithographic area of SQUID loop and an estimate of flux focusing by the superconducting film. The SQUID oscillations are enclosed in an envelope reflecting the diffraction pattern of the individual junctions. Nodes in this envelope correspond to integer flux quanta enclosed by an individual junction. We observe that the critical current does not completely vanish at these field values; instead, the current remains stuck at a finite value which is essentially independent of gate bias. Similar node-lifting is observed in a single-junction device and other SQUIDs (see Supplementary Materials). We also observe lifting of the nodes in the SQUID oscillations, shown in greater detail in Fig. 2b. While the maximum (antinodal) supercurrent varies dramatically across the topological phase transition, the nodal current remains fixed at a value of ≈ 150 nA.
What is the nature of the node-lifting? We focus first on the SQUID nodes because there are a number of well-known phenomena that can lift the nodes of SQUIDS, particularly finite inductance of the SQUID loop, junction asymmetry, skewness of the CPR, and parallel conductance mechanisms, i.e. shorts in one of the junctions. A superconducting short is ruled out because the node current decays with field much like the anti-nodes, indicating that it is a Josephson effect spread through the junction length. Because the SQUID inductance parameter β L = LIc/Φ 0 ≈ 10 −3 is much less than 1, circulating currents are unlikely to be the cause of the observed node-lifting [22].
If the two junctions are asymmetric, then perfect destructive interference will not occur at the nodes either. We test for asymmetry by measuring the nodal supercurrent at elevated temperature and when the device is in the topological regime (V T G = −18 V), illustrated in Fig. 3. While the antinodal current is only weakly temperature dependent up to 800 mK and then only slowly declines, we find that the nodal current quickly collapses, becoming essentially zero beyond 850 mK. This reduction is faster than expected for thermal noise, considering that the noise parameter Γ = 2ek B T /I ch = 0.22 for T = 800 mK and I c = 150 nA. Such thermal noise should reduce the apparent critical current for an underdamped Josephson junction by a factor of ∼ 2 [23], but we observe at least a factor of 8 reduction.
Indeed, when the antinodal critical current is suppressed to 150 nA at higher magnetic fields due to phase-winding along the junction length (e.g at B = 2.44 mT), the detected critical current falls by 50% between 20 mK and 800 mK, as expected.
Thus, the vanishing of nodal current at high temperature is inconsistent with asymmetric junctions. Instead, this suggests that the nodes are lifted at low temperature because of a non-sinusoidal CPR, which then becomes conventional at higher temperatures. The nonsinusoidal behavior is expected in junctions in which some of the supercurrent is carried by low energy ABSs with high transparency. In Fig. 4a, we superimpose a theoretical currentflux relation based on the formalism in Ref. [2] with the observed SQUID oscillations at 20 mK, using a CPR shown in Fig. 4b. Here, the individual ABSs can be labeled by their transverse momentum q. States with large |q| contribute an essentially sinusoidal component to the CPR while states with small |q| give a highly forward-skewed component. The two q = 0 states are identified as the MBSs. The nodes are lifted due to the highly forwardskewed CPR component from low q and thus low energy ABSs, illustrated by the red curve in Fig. 4b. We emphasize that this is a toy model that ignores scattering, finite junction length, and temperature [3].
At 800 mK, shown in Fig. 4c, the SQUID oscillations are well described by two identical junctions with sinusoidal CPR, suggesting that the anomalous components are suppressed by temperature. At higher temperature either dephasing or quasiparticle poisoning causes the CPR to become conventional, consistent with direct measurements of CPR in SNS devices [24]. Our direct measurements of the CPR in TI junctions also show evidence of slightly forward skewness that disappears with temperature (see Supplementary Materials).
The independence of the anomalous states with gating suggests that the top gate primarily suppresses conventional states.
One might claim that the node-current comes from some separate component with a sinusoidal CPR, such as current through the bulk or the bottom layer. This component could conceivably be asymmetric between the two junctions, not affected by the top gate, and much more susceptible to thermal fluctuations (due to lower mobility or phase coherence).
However, such a component should also contribute a sizable amount to the antinode current (much larger than the 150 nA observed at the node). But at V T G = −18 V, the antinode current is largely temperature independent from base temperature up to 800 mK.
Having established the existence of non-sinusoidal CPR from low energy Andreev bound states, we return to the diffraction patterns to consider the possibility of Majorana bound states. Skewed CPR components from q = 0 modes can lift the nodes of the SQUID oscillations, but would still undergo completely destructive interference when an integer number of flux quantum are within the junction. An inhomogeneous current distribution could lift such nodes due to incomplete cancellation of supercurrent, but the SQUID oscillations at elevated temperature suggest that the device is very nearly uniform. Even if we model the T = 0 node-lifting in the SQUID oscillations with a conventional supercurrent that spatially varies, we find that the resulting nodes in the single-junction diffraction pattern (relative to the height of the first side-lobe) are a factor of two too small to explain the nodes in the observed diffraction pattern. Odder still, we find that at 800 mK (Fig. 4d) the first and third nodes remain significantly lifted but the second is suppressed, leading to an even-odd effect that has been previously observed at lower temperatures in single junctions fabricated on TI films. The lifted diffraction pattern nodes are consistent with Majorana fermions bound to vortices [7], which hybridize at the junction edge. The phase-dependent energy splitting of the MBSs generates a supercurrent at the nodes. The magnitude of the observed nodal supercurrent agrees with the predicted value of I ≈ ∆ 0 /Φ 0 ≈ 100 nA, where ∆ is the niobium superconducting gap. The independence of the nodal current with respect to top gate bias is also consistent with the robustness of MBSs. Careful measurements of I c near the diffraction pattern nodes (inset of Fig. 2a) also reveal a sharp rise in the height of the SQUID node as one approaches a magnetic flux quantum per junction, consistent with the highly non-sinusoidal CPR proposed by Ref. [7]. It is conceivable that such anomalous supercurrent localized in vortices could better protected against thermal effects than current delocalized along the junction width. The observed even-odd effect is not consistent with Ref. [7] and is suggestive of a 4π-periodic CPR. Although the sin (φ/2) component derived at low fields in Ref. [2] is expected to revert to 2π-periodicity due to quasiparticle poisoning [25], the hybridization of MBSs between adjacent vortices could suppress the anomalous supercurrent predicted by Ref. [7] for an even number of vortices. Thermal fluctuations will likely enhance such hybridization, leading to the suppression of the second node at high temperature in Fig. 4d.
In conclusion, we have probed the CPR of TI Josephson junctions. We find an anomalous supercurrent at the nodes in the SQUID oscillations that is independent of gate bias but disappears at elevated temperature, demonstrating evidence for a skewed CPR from low energy Andreev bound states. The lifting of the nodes in the single junction diffraction pattern is consistent with Majorana states bound to vortices in the junctions. Our results explore the components necessary for a topological quantum computer. nm of Nb at room temperature. Brief Ar ion milling is employed before metallization in situ to ensure good contact between the Bi 2 Se 3 and the leads. A top gate may be created by covering the sample with 30 nm of alumina via ALD and deposition of Ti/Au over the exposed Bi 2 Se 3 . The devices were thermally anchored to the mixing chamber of a cryogenfree dilution refrigerator equipped with a vector magnet and filtered wiring. We perform current-biased transport measurements with standard lockin techniques, typically with a 4

Single crystals of Bi
nA AC excitation at f = 73 Hz. The doped silicon substrate can act as an electrostatic back gate, but we found that the critical current was only very weakly tuned by back gate bias.
The device featured in the main part of this paper was 9 nm thick and possessed a normal state resistance of 37 ohms, which was only weakly dependent on top or back gate bias.

COMPETING FINANCIAL INTERESTS
The authors report no competing financial interests.   For rounded I-Vs, use "excess current" method to determine critical current.

II. SECOND DC SQUID WITH GATE INDEPENDENT NODES
We also studied another dc SQUID for which the nodes were gate independent, as summarized in Fig. S2. This device was constructed on a 19 nm thick, 300 nm wide flake of Further details behind our method are given in Ref. [1]. For a sinusoidal CPR, the circuit becomes hysteric if β L = 2πLI c /Φ 0 > 1. A non-sinusoidal CPR makes the non-hysteric condition harder to satisfy [1].
Most of our direct measurements of the CPR in TI JJs have been plagued by hysteresis.
Combined with noise-rounding, this can result in an apparently backward skewed CPR, as seen at the lowest temperature in Fig. S3b. We emphasize that this backward skewness is due to a measurement artifact and does not reflect the intrinsic CPR of the junction.
However, we do find that at higher temperature the hysteresis becomes less problematic because I c is reduced. For example, at 250 mK we find evidence for a small amount of forward-skewness. At still higher temperature, the CPR reverts to a non-skewed, sinusoidal form.
The magnitude of forward-skewness is small in Fig. S3c, particularly in comparison with the skewness that we use to model the lifted SQUID nodes in the main body of this paper.
However, this preliminary CPR measurement was performed at nominal density (i.e. larger density of conventional states) and at somewhat elevated temperature. Noise-rounding can also obscure features at φ = π, thus hiding the contribution of individual, highly forwardskewed components in the CPR. Thus, the direct measurement of CPR is not inconsistent with the theoretical CPR we used to model the SQUID nodes.