4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions

The Josephson effect describes the generic appearance of a supercurrent in a weak link between two superconductors. Its exact physical nature deeply influences the properties of the supercurrent. In recent years, considerable efforts have focused on the coupling of superconductors to the surface states of a three-dimensional topological insulator. In such a material, an unconventional induced p-wave superconductivity should occur, with a doublet of topologically protected gapless Andreev bound states, whose energies vary 4π-periodically with the superconducting phase difference across the junction. In this article, we report the observation of an anomalous response to rf irradiation in a Josephson junction made of a HgTe weak link. The response is understood as due to a 4π-periodic contribution to the supercurrent, and its amplitude is compatible with the expected contribution of a gapless Andreev doublet. Our work opens the way to more elaborate experiments to investigate the induced superconductivity in a three-dimensional insulator.

rf irradiation at f = 13.2 GHz. The subharmonic structure (clearly visible in the differential resistance dV /dI in Supplementary Fig.3) is related to the appearance of subharmonic steps with index n = 1/2, 3/2.... Here in particular, one observes the steps n = 1/2 and n = 3/2.
Step amplitude            Step amplitude [I c ] Step amplitude [I All steps are clearly visible and appear one by one as I rf increases. As frequency f is decreased 1f J for C) the pinch-off of the supercurrent is moved to higher powers.
One also observes a decreasing period in the oscillating pattern.
Step amplitude [I c ] Step amplitude [I c ] Supplementary Figure    Norm Supplementary Figure   The excess current in our data is indeed strikingly present in all our samples. We To analyze more precisely this behavior, we plot in the right panel of Supplementary In contrast the excess current is present out to high voltages beyond 2∆ and perhaps beyond ∆. Careful experimental work is therefore needed to disentangle these two processes from the experimental data. Furthermore, the theory has been developed for nanowires (single-band and short junction limit) with a simple representation of induced superconductivity, by a unique gap ∆. The fact that our system is a 2D surface state rather than a nanowire is expected to have little consequence.
However, we expect to have a very different density of states due to the presence of two Supplementary Eq.(1) has three fit parameters: the tunneling strength Γ (or equivalently the induced gap ∆ i ), the barrier parameter Z and the number of modes N . As seen on Supplementary Fig.10A, the agreement below 1 K is very good for the L = 150 nm junction, but the fit diverges rapidly at high temperature. We obtain the following fitting In Supplementary Fig.3, we present datasets measured for a different sample than the one presented in the main text with length L = 400 nm. In particular, we show the transition from a doubled Shapiro step (at f = 3.34 GHz) to a regime in which the first step is fully recovered. At high frequencies (for f ≥ 6.2 GHz), we observe the appearance of new half-integer steps, at voltages given by the step indexes n = 1/2, 3/2, 5/2, .... To visulaize more clearly the subharmonic steps, we introduce a different way of visualizing our data. The differential conductance dI/dV is plotted as a colorscale, as a function of the dc voltage V and rf current drive I rf . Thus, Shapiro steps appear as maxima (dI/dV diverges) for constant voltages, similarly to what is seen in the bin counts presented in the main text. In such plots, the information on the step amplitude is lost, but subharmonic steps of small amplitudes become more visible.
At f = 3.34 GHz, the first step (n = 1) is partially suppressed, but is recovered as frequency increases. The change in the oscillatory pattern is also clearly visible: at low frequency, fast oscillations follow a long region in which step n = 0 (supercurrent) is present. As f is increased, oscillations start earlier and with a larger period. These in the main text (Fig.2). Aging of the sample has been observed in several samples and could explain this discrepancy. This could constitute evidence of a missing or suppressed third step, clearly observed only in one sample. However, other irregularities in the higher order steps are sometimes visible (see Fig.3 in the main article), so that this evidence has to be considered carefully.

D. Hysteresis, bias instability and Shapiro steps
In Supplementary Fig.6, we present two datasets measured on the same junction as the one presented in section C at two different temperatures. To emphasize the role of the control parameters (dc and rf currents I and I rf ), we now plot the differential resistance dV /dI as a colormap, as a function of these two parameters. On the left panel, measurements were obtained at f = 3 GHz at the base temperature of the dilution refrigerator (T = 12 mK). On the right panel, measurements were obtained in the same conditions (f = 3 GHz) except for the temperature, here set high enough to suppress the hysteresis (T 800mK).
Shapiro steps are identified as black regions where dV /dI 0, while blue lines between black regions emphasize transitions between the different plateaus. A simultaneous read-ing of the voltage V gives access to the step index n (a few of them are indicated directly on Supplementary Fig.6). The dc current is swept in the direction indicated by the white arrow (from negative towards positive bias). At T = 12 mK, the very clear asymmetry at low rf excitations is a signature of the hysteresis observed in the I-V curve presented in the main text. The bistable dynamics that leads to hysteresis prevents the development of the phase-locked dynamics responsible for the Shapiro steps. Consequently, the latter are missing in the hysteretic region. In contrast, all steps are clearly visible at higher temperature (T 800 mK), except for the n = 1 step which is fully suppressed at It is possible to make a clear distinction between missing steps due to hysteresis and the missing n = 1 step attributed to a 4π-periodic supercurrent. First, as can be seen on Supplementary Fig.6, hysteretic switchings present vertical tangents (similar to ref. 10) while the doubled Shapiro step always exhibits a finite slope. Second, hysteresis is characterized by an asymmetry depending on the sweep directions, which is not the case of the missing step in our measurements. Third, the anomalous splitting of the n = 1/n = 2 steps beyond the pinch-off of the supercurrent (corresponding to the dark fringe in the bin counts, discussed in the main text) that accompanies a missing n = 1 step remains visible at all temperature regardless of the presence of hysteresis for low rf powers. For these reasons, it appears clear that one can safely neglect hysteresis as the origin of the missing n = 1 step. To avoid any problem, most measurements were performed at a temperature high enough to suppress the hysteresis (typically 450 to 800 mK).

E. Shapiro steps on a shunted device
Furthermore, it is in fact possible to rule out bias instabilities as a possible mechanism for the missing n = 1 step. By adding a shunt resistance in parallel with the junction, one can indeed suppress hysteresis 11 and work in a configuration that approaches the voltage bias regime. To do so, we add a 10 Ω resistor in series with the junction, and shunt these two elements with a 1 Ω resistor. Thus, the current flowing through the junction is accessed by measuring the voltage across the 10 Ω resistor, together with the voltage across the junction.
Results obtained at the base temperature of the fridge (12 mK) and with an rf excitation at 4.2 GHz are shown in Supplementary Fig.7. First, the black solid line in Supplementary Fig.7A represents the I − V curve in the absence of rf irradiation. We do not detect in this configuration any instability. At low voltages, one sees that dV /dI < 0.
In this region, a pure current bias generates bias instability and hysteresis, which is here suppressed by the shunt resistance. When the rf irradiation is switched on (colored plain lines), Shapiro steps become visible in the I − V curves. As previously, the step n = 1 is clearly suppressed at low rf excitation (blue lines). Increasing the rf drive (from blue to yellow lines), one sees the n = 1 step is recovered at high drive amplitude as previously.
Supplementary Fig.7 B and C present the voltage histograms and differential resistance dI/dV as a function of the normalized DC voltage V and rf current I rf . Shapiro steps are visible as previously as maxima following horizontal lines. As for the measurements shown in the main text, the first step n = 1 is fully suppressed up to the oscillating regions. Though the contrast is not as good, the "dark fringe" at finite voltage described in the main text is also visible.

F. Magnetic field dependence
Further investigations of the anomalous Shapiro response have been carried out in the presence of perpendicular-to-plane magnetic fields. First, when the I-V curve is measured without rf irradiation, a Fraunhofer-like diffraction pattern of the critical current is observed (plotted as a red line in Supplementary Fig.8, upper panel). In the junction presented here, the pattern is slightly distorted (probably due to flux-trapping in the magnet). The periodicity in the magnetic field has been evaluated from undistorted patterns in various other samples. It corresponds to a conventional periodicity as previously reported on similar samples 9 , and as expected for ballistic systems with such aspect ratios 12,13 . to have no spin-degeneracy and therefore the level crossing at phase differences π, 3π, etc.
should persist even in the presence of a magnetic field on the TI. The graphene flake exhibit densities between −2.5 × 10 12 cm −2 and 2.5 × 10 12 cm −2 , which can be tuned by the means of a back-gate. In the left panel of Supplementary   Fig.11, we present the extracted normal state resistance R n of the device as a function of the back-gate voltage V g . The mobility approaches 3000 -8000 cm 2 V −1 s −1 . Close to the Dirac point, this corresponds to a mean free path of around 50 nm. Consequently, these devices are not in the ballistic limit but are relatively close to it.
In the right panel of Supplementary Fig.11, we present the differential resistance dV /dI of one junction as a colormap, as a function of the back-gate voltage V g and dc drive current In this section we present two typical sets of data taken on the graphene-based junctions (see Supplementary Fig.12). The differential conductance dI/dV is plotted as a colorscale, In the lower panels, the step amplitudes for n = 0, 1, 2, 3, 4 are plotted as a function of the rf current amplitude I rf . Again, one observes the effect of the excitation frequency f .
As f decreases, the width of the first lobe gets much larger than the widths of the other lobes. Moreover, these graphs also show that the step amplitudes (hence their visibility) decrease (with respect to the critical current). This phenomenon, clearly observable in our measurements (lower panels of Fig.3, main text), limits our measurements to f ≥ 2 GHz.
B. Extended RSJ model with a 4π-periodic contribution: Simulations have also been carried out when adding a 4π-periodic contribution following Dominguez et al. 15 . We present in this section our results : the appearance of the doubled step is qualitatively well described as in the previous reference, but a quantitative agreement has not been obtained yet. The results are presented below in the following manner. First, we show that the addition of a 4π-periodic contribution I 4π sin φ/2 to a sinusoidal 2π-periodic current-phase relation (CPR) I 2π sin φ is responsible for the disappearance of all odd steps at low frequency f < f 4π , in a comparable way to what we experimentally observe on the first step n = 1. Then the marginal effect of the 2π-periodic CPR is illustrated by comparing I-V curves in the presence of rf irradiation for a few different CPRs. No generic 2π-periodic CPR is found to show missing odd steps, while the addition of a small 4π-periodic term enforces the disappearance of odd steps regardless of the 2π-periodic term.
a. Effect of frequency on a 2π +4π-periodic supercurrent: We simulate the effect of a small 4π-periodic contribution I 4π sin φ/2. In Supplementary Fig.14 it happens for f = f 4π (= 0.15f J in the case of Supplementary Fig.14). One also observes that the oscillatory pattern (at high power) is also progressively modified from a 2π-to a 4π-dominated pattern. In particular, odd steps show a very pronounced first minimum.
The dark fringe we experimentally observe in the oscillatory pattern is understood as the result of the progressive towards a pattern with a halved number of oscillations, thus yielding progressively suppressed lobes.
b. Effect of the CPR Though the presence of a small 4π-periodic contribution I 4π sin φ/2 is found necessary to observe vanishing odd steps, the exact description of the 2π-periodic supercurrent does not influence much the Shapiro response. To illustrate this finding, we focus on three different CPR. The first one is a standard I 2π sin φ contribution Andreev levels have not been explored yet. We thus use a single mode approximation 15 , and crudely assume that one level (with lowest gap δ) has a predominant role in Landau-Zener processes.
Following Dominguez et al. 15 , we include in our simulations stochastic Landau-Zener transitions at the anticrossings, with a probability P , and partially reproduce their results.
The exact motion of the phase difference is hard to picture, but the general trend can be understood on a heuristic basis. The phase will undergo shifts of 4π (per period of the drive) in the case of a Landau-Zener transition and only 2π in the absence of transitions.
On average this results in a non-universal Shapiro step that is neither hf /2e nor hf /e.
These two values are recovered for P = 0 and P = 1 respectively. The results of our simulations are presented in Supplementary Fig.17A, in which a close-up of the n = 2 Shapiro step is presented, for different values of P . As Dominguez et al., we observe a splitting of the step for P close to 1, with voltage plateaus that deviate from the quantized value hf /e until the split steps are eventually less discernible for P 0.7. Our experimental results do not show any splitting or deviation to the quantized value, with an accuracy of a few percents. This indicates that the probability must be P > 0.97 (P = 1 being equivalent to having a fully gapless mode).
From the previous model, one can estimate the importance of the Landau-Zener transitions. First, we solve the RSJ equation, and obtain the phase φ(t) and its derivativė φ(t) ∝ V (t) as a function of time t. The parameters are chosen such that the junction lies on the first Shapiro step (V = V (t) = hf 2e ). In that case, a high probability of Landau-Zener transitions would lead the junction to exhibit a doubled step.
A typical plot of φ(t) and V (t) is shown in Supplementary Fig.17B, with φ(t) as a red line and V (t) ∝φ(t) as a blue line for the following parameters: I = 0.5 I c , I rf = 0.8 I c , f = 0.2 f J . One first observes that the phase φ follows an anharmonic motion synchronized with the excitation drive at frequency f : during one period of duration 1/f , the phase φ increases by 2π, yielding an averaged voltage V = hf /2e as expected for the first Shapiro step. Equivalently, one can calculate the average of V (t) and obtain V = V (t) = hf 2e . Then, we access the time t for which φ reaches the anticrossing (for φ(t) = 3π for example) and read the derivative of the phaseφ| 3π at this point or equivalently the voltage V | 3π .