Gate-tunable superconducting diode effect in a three-terminal Josephson device

The phenomenon of non-reciprocal critical current in a Josephson device, termed the Josephson diode effect, has garnered much recent interest. Realization of the diode effect requires inversion symmetry breaking, typically obtained by spin-orbit interactions. Here we report observation of the Josephson diode effect in a three-terminal Josephson device based upon an InAs quantum well two-dimensional electron gas proximitized by an epitaxial aluminum superconducting layer. We demonstrate that the diode efficiency in our devices can be tuned by a small out-of-plane magnetic field or by electrostatic gating. We show that the Josephson diode effect in these devices is a consequence of the artificial realization of a current-phase relation that contains higher harmonics. We also show nonlinear DC intermodulation and simultaneous two-signal rectification, enabled by the multi-terminal nature of the devices. Furthermore, we show that the diode effect is an inherent property of multi-terminal Josephson devices, establishing an immediately scalable approach by which potential applications of the Josephson diode effect can be realized, agnostic to the underlying material platform. These Josephson devices may also serve as gate-tunable building blocks in designing topologically protected qubits.

The value of critical current fluctuates over time as seen in Supplementary Fig.1a, this is believed to be due to fluctuation in the external magnetic field as the fluctuations are correlated for the positive and the negative direction. This results in the observed punch through errors. However, only 3 minor punch through errors , as seen in Supplementary   Fig. 1 b, are observed over a time scale of 2hr, at an applied square wave of frequency 0.1Hz.

II. DATA FROM BIASING TERMINAL 2 AND 0
Here we show data when source terminal number is 2 and drain terminal is 0. In this case I 1 = 0 and I 2 = I and V 2 = V . The Fraunhofer-like interference lobes are tilted in the opposite direction, as seen in Supplementary Fig. 2a, w.r.t. current axis compared to the case of terminal 1 being the source terminal as shown in Figure 3 a. For B just above just zero the Q < 0 as seen in supplymentry Fig. 2b, compared to the case of terminal 1 being source where Q > 0 for B just above 0. This is consistent with the expected behaviour from the network model.

III. RECTIFICATION IN DEVICE 2
Here we present data from a second device which is lithographically identical to Device 1.
We measure this device with source terminal 2 and drain terminal 0, the observed Fraunhofer pattern (Supplementary Fig. 3a) is consistent with the device 1 at zero gate voltage. We can further access the two-terminal limit on this device by selectively gating the legs of the Yshaped junction. By applying -7 V on V g,1 and V g,3 we pinch off the junction region between terminal pair 0-1 and terminal pair 1-2. While keeping the junction region between terminal pair 2-0 open by setting V g,2 at 0 V. This effectively drives the device into a two-terminal regime as only one junction of the network is open. The observed Fraunhofer pattern is consistent with what has previously been observed for two-terminal Josephson junctions, as the lobes are not tilted ( Supplementary Fig. 3b). The period in B also increases as the effective area of the junction has decreased. This is further clarified by evaluating δI c for both the gate configurations as shown in Fig. 3c. Normalized δI c (normalized by the critical current at zero applied field) is nearly zero in the two terminal regime as shown in red, contrasted with the zero gate voltage case shown in blue.
We demonstrate of a square current wave (shown in yellow in Supplementary Fig. 3d) with an amplitude of A = 0.2 µA, such that I − c < A < |I + c |. The device remains superconducting (V ∼ 0 V) for the positive cycle of the square wave and has a finite voltage (V ∼ −8 µV) drop for the negative cycle. A single punch through error is observed over 5000 cycles. The critical current shows similar fluctuations in a similar way as seen for Device 1 ( Supplementary Fig. 3e).

IV. MODEL FOR EVALUATING CRITICAL CURRENT
Transport in our Three-terminal device can be viewed more simply in terms of connected here t is the width of the junction, B is the applied magnetic field, and ϕ 0 is the flux quantum. Similarly for the other two Josephson junctions integration around C 2 and C 3 gives: Integration around the contour C e (shown in Fig. 4a) of area A gives the following equation: If source terminal is S 1 and drain terminal S 0 we have from current conservation:  Table I. Table of parameter values used to produce diode efficiency curves in Figure   4.
From Eq. 2,3,6: To proceed in our calculations we assume J c (x 2 ) = J c (x 3 ), then we must have ϕ(x 2 = 0) = ϕ(x 3 = 0), this give Eq. 5 as: The critical current can be evaluated by maximizing in ϕ(x 3 = 0) in Eq.8, the code used  Supplementary Fig. 4 c,d,e,f. The code necessary for these calculations is provided with the manuscript.

V. TEMPERATURE DEPENDENCE OF DIODE EFFICIENCY
We also study temperature dependence of the diode effect by performing temperature sweep on Device 2 as shown in Supplementary Fig. 6. The device is measured in the positive polarity. The diode efficiency remains constant until the supercurrent itself starts decreasing with temperature. This shows that the thermal fluctuations should not affect the performance of diode based upon multi-terminal Josephson junctions.

VI. DATA FROM DEVICE 4
Here we show data from Device 4 measured in a Bluefors dilution refrigerator at a base temperature of 8 mK. The junction dimensions are lithographically identical to Device 1 and 2, but no metal gates were deposited. The device is measured by biasing terminal 1 and 0, hence I 1 = I and I 2 = 0 with V 1 = V . Obtained data is similar to what is obtained for device 1 (Supplementary Fig. 7 and Figure 3). This shows high degree of reproducibility and robustness of the diode effect in our three-terminal Josephson devices.