Introduction

Superconductor-semiconductor heterostructures have been studied both experimentally and theoretically over the past few decades, motivated by their potential to realize topologically protected quantum states1,2,3,4,5,6,7,8,9,10,11,12,13 or gate-tunable quantum bits14. Such states may have applications in fault-tolerant quantum information processing15,16,17,18. Multi-terminal Josephson junctions (MTJJs) may provide a novel platform for realizing higher dimensional artificial band structures formed by the Andreev bound states (ABS) present in the junction. In a Josephson device with N superconducting terminals, the ABS spectrum depends on the N − 1 independent phase differences between terminals, ϕ1, ϕ2, . . . , ϕN−1, which act as quasimomenta, as well as on the scattering matrix \(\hat{S}\) of the interstitial junction region. Furthermore, the ABS spectra of MTJJs are predicted to host topologically protected Weyl nodes and higher-order Chern numbers19,20,21,22,23. The energy gap between different ABS bands depends on the number of conductance modes between terminals, with theoretical efforts focusing on the case of unity or near-unity number of interterminal modes19,20,24. Approaching this condition necessitates the independent control of interterminal conductance modes in an MTJJ.

MTJJs may also find application as circuit elements for coupling multiple qubits14,25,26,27. Additionally, they have shown rich transport features such as the coexistence of superconducting and dissipative currents28, multi-terminal fractional Shapiro steps29,30, generalizations of multiple Andreev reflections (MAR)31,32, multi-loop superconducting interferometry33,34 and exotic Cooper quartet transport35,36,37,38.

Previous experiments on MTJJs28,32,39 have discussed the current-space differential resistance maps in three- and four-terminal devices and its dependence on parameters such as magnetic field and a single global gate voltage, but in the regime of many conductance modes. Theoretical proposals for topological ABS spectra outline the need for a small central scattering region through which the superconducting terminals are coupled in the regime of a few quantum modes, however, a global gate is not ideal for implementing this experimentally. Rather, a split-gate quantum-point-contact-like design where the junction legs can be independently depleted is necessary for the transport to be localized in a central common region (Fig. 1a).

Fig. 1: Device geometry.
figure 1

a Schematic depiction of transport in Device 1 and Device 3. The junction area under the gates, shown in yellow, can be fully depleted of carriers, leaving a central scattering region supporting a few conductance modes connected to the superconducting contacts. b False-color scanning electron microscope (SEM) image of Device 1, a three-terminal Josephson junction with individually tunable QPC gates, showing measurement schematic. The etched junction area is visible as the dark lines under the gates (gold-colored). c SEM image of Device 2, which has a central top gate that can be used both to form QPCs and to gate the central scattering area of the three-terminal junction. d 3D schematic of Devices 1 and 3 showing layered heterostructure.

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In this work, we utilize a split-gate quantum-point contact (QPC) geometry, which allows selective gating of each leg of a Y-shaped three-terminal junction. With this approach, we demonstrate control over conductance modes between pairs of terminals, along with access to the single-mode regime in the junction, coexisting with superconductivity. This establishes a potential platform for the exploration of the tunable ABS spectra of MTJJ devices. We present detailed results from two device designs with different junction dimensions and different split-gate geometries.

Results

Device architecture

The devices are fabricated on InAs quantum well heterostructures featuring a two-dimensional electron gas (2DEG) proximitized by an epitaxial aluminum layer. High interface transparency between Al and InAs (leading to induced gap comparable to the bulk gap of Al) and coherent ballistic transport in this heterostructure have been demonstrated6,40,41 making it an ideal platform to realize MTJJs. The heterostructure was grown on a semi-insulating InP(001) substrate using molecular-beam epitaxy. From the bottom, the heterostructure consists of a graded buffer of InxAl1−xAs with x ranging from 0.52 to 0.81, 25 nm In0.75Ga0.25As super-lattice, 10.72 nm In0.75Ga0.25As bottom barrier, 4.54 nm InAs quantum well, 10.72 nm In0.75Ga0.25As top barrier. Finally, there is a 10 nm layer of epitaxial aluminum deposited on the surface of the sample. The carrier concentration and mobility of the InAs 2DEG were measured using a Hall bar geometry and found to be n = 1.22 × 1012 cm2 and μ = 9920 cm2 V−1 s−1 in the absence of gating (see Supplementary Fig. 1), resulting in a mean free path of ℓ ~ 180 nm.

The Y-shaped three-terminal devices presented in this work have different junction widths and different split-gate geometries. Device 1 has a nominal contact spacing between superconducting electrodes of 50 nm, with three split gates as shown in Fig. 1b. These split gates can deplete the 2DEG underneath, forming a few-mode central region coupling each superconducting terminal. Device 2 has a nominal contact spacing of 200 nm, three split gates forming QPC-like constrictions, and also a central top gate for independent gate control of the central scattering region (Fig. 1c). Device 3 is similar in shape to Device 1, but with an electrode spacing of ~120 nm. We begin by discussing the transport properties of Device 1 and Device 2 and demonstrate the selective gate tunability of Device 2. We then show the accessibility of the single-mode regime coexistent with superconductivity in these devices.

Transport properties

We perform DC current-bias measurements in a dilution refrigerator on all three devices using the configuration shown in Fig. 1b, c. The superconducting data for Device 1 and Device 3 were taken at fridge temperature T ~ 40 mK, and Device 2 at temperature T ~ 90 mK. We independently control the current inputs into the epitaxial aluminum terminals 1 (I1) and 2 (I2) while terminal 0 is grounded. We simultaneously measure the voltages of terminals 1 (V1) and 2 (V2) relative to terminal 0. In a typical measurement, we step I2 from negative to positive, and sweep I1 from negative to positive at each value of I2. We then calculate differential resistances dV1/dI1 and dV2/dI2 by discrete differentiation. The differential resistance maps show a central superconducting region where both V1 and V2 (Fig. 2a–c) are zero. Beyond this central region, superconducting arms are also observed approximately along I2 = − 2I1 (Fig. 2a) where only V1 is zero, and I1 = − 2I2 (Fig. 2b) where only V2 is zero. A third superconducting arm is observed approximately along I1 = I2. This feature is due to super-current being present between terminals 1 and 2 (Fig. 2a–c), while the other two arms have a nonzero resistance. The slopes of these superconducting arms in the I1, I2-plane can be understood by a resistor network model (see Supplementary Information).

Fig. 2: Three-terminal differential resistance maps.
figure 2

a Measurement of the differential resistance dV1/dI1 on Device 1 with a small perpendicular magnetic field. MAR along the lines V1 = 2Δ/n are highlighted by cyan lines and along V1 − V2 = 2Δ/n highlighted by dashed cyan lines. b Measurement of dV2/dI2 on Device 1 with small perpendicular field. MAR along V2 = 2Δ/n are shown by cyan lines. The lower resistance feature along V1 = − V2 is shown by a dashed ellipse, which can be attributed to Cooper quartet transport. Here I1 is stepped and I2 is swept. c Measurement of dV1/dI1 on Device 2 at magnetic field B = 0, showing MAR resonances.

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The differential resistance maps exhibit rich MAR patterns. We can observe MAR as features of lower resistance along the lines V1 = 2Δ/n and V1 − V2 = 2Δ/n, where n is an integer and Δ ~ 145μV is the induced superconducting gap (estimated by fitting MAR at V1 = 2Δ). Figure 2a shows these MAR lines highlighted in cyan for n = 2, 4, 6. In Fig. 2b we highlight MAR along V2 = 2Δ/n for n = 2, 4 in the differential resistance dV2/dI2. These three sets of MAR signatures can be understood as independent Andreev reflections between all three pairs of terminals. We also observe a signature of Cooper quartet transport35,36,37,38,42, indicated by a lower resistance feature along the line V1 = − V2. The differential resistance maps can also be plotted as a function of V1, V2 where the quartet signature is clearly visible along the V1 = − V2 diagonal (see Supplementary Fig. 2). This places Device 1 in the phase-coherent quasiballistic regime and opens up interesting possibilities for investigating cross-terminal quantum correlations.

These features are also observed for Device 2 as shown in Fig. 2c, despite the junction width being nearly four times larger. This is possible due to the highly transparent interface between the epitaxial aluminum and InAs quantum well of the heterostructure, and displays the robustness of our fabrication process for MTJJs and the high degree of reproducibility. The central superconducting region is not current-symmetric in the differential resistance maps for Device 1. This indicates the presence of a small residual magnetic field resulting in asymmetric critical current32,43, as verified in Device 2 by correcting for this residual field in our external superconducting magnet. We can observe the disappearance of this asymmetry when the perpendicular magnetic field, B, vanishes, as shown in Fig. 2c.

Selective control of conductance in three-terminal Josephson junctions

A distinctive feature of these devices is their independent split top gates, enabling individual control of each leg of the Y-shaped junction. In order to demonstrate local control of the Josephson junctions formed between each pair of terminals, we can examine the results of gate-depleting carriers in each of the legs selectively. Negative voltage gating of a leg results in narrowing of the width of the superconducting arm associated with it in the differential resistance map. Additionally, the slopes of the lines change in the I1,I2-plane about which superconducting features are centered. These slope changes are due to an increase in the normal state resistance (Rn) of the leg, which affects the division of dissipative currents between the three terminals. When the normal state resistances in the resistor network are Rn,1, Rn,2, Rn,3, the feature due to super-current between terminals 1 and 0 (V1 = 0) falls along the line \({I}_{2}=-\left({R}_{{{{{{{{\rm{n}}}}}}}},3}/{R}_{{{{{{{{\rm{n}}}}}}}},2}+1\right){I}_{1}\). For super-current between terminals 2 and 0 (V2 = 0), this relation is \({I}_{2}=-{\left({R}_{{{{{{{{\rm{n}}}}}}}},3}/{R}_{{{{{{{{\rm{n}}}}}}}},1}+1\right)}^{-1}{I}_{1}\) and between terminals 1 and 2 (V1 − V2 = 0) it lies along I2 = (Rn,1/Rn,2)I1. Thus, we can demonstrate truly selective gating in our devices by examining the narrowing of superconducting features and their modified slopes.

As a starting point, we measure the differential resistance dV1/dI1 with the same applied voltage on all four independent gates in Device 2 (three gates on the legs and one central gate) with Vg = − 5 V (Fig. 3a). This voltage is applied to amplify the effect of selective gating, since the superconducting features become more sensitive to gating at sufficiently negative gate voltages. Although the applied voltage is the same, we can see a minor asymmetry of features compared to the plot at the zero gate (Fig. 2c). We then decrease the voltage further only on the gate between terminals 1 and 2 (Vg,3) (Fig. 3b). This results in a distinct change in the differential resistance map as can be seen in Fig. 3c. The superconducting arm due to supercurrent between terminals 1 and 2 (V1 − V2 = 0) dramatically decreases in width. The slope of the superconducting arm where V1 = 0 has tilted toward the line I1 = 0, and the V2 = 0 superconducting arms has tilted toward the line I2 = 0. The slope of the narrowing arm (V1 − V2 = 0) has remained unchanged. This is consistent with the limiting cases of the equations in the previous paragraph for Rn,3 ≫ Rn,1, Rn,2. We have studied the effect of selective gating on the other two legs as well, and the slope changes were found to be consistent with this resistor network model (see Supplementary Information).

Fig. 3: Selective gating.
figure 3

a Measurement of dV1/dI1 on Device 2, B = 0, T ~ 90 mK with Vg,c, Vg,1, Vg,2, Vg,3 = − 5 V. b Schematic of gate configuration for the selective gating of the junction leg between terminals 1 and 2. c Measurement of dV1/dI1 at Vg,c, Vg,1, Vg,2 = − 5 V and Vg,2 = − 6 V. d RCSJ simulation of dV1/dI1 with parameters tuned to reproduce the features of experimental data.

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Additionally, we performed simulations of the system using a three-terminal resistively and capacitively shunted junction (RCSJ) network model by solving coupled differential equations obtained by a multi-terminal generalization of the RCSJ model30. Details of the simulation and model parameters can be found in the Supplementary information. This network model consists of three nodes with RCSJs between each of them, and thus contains nine independent parameters, namely the critical currents Ic,i, normal state resistances Rn,i and capacitances Ci. The effect of gating was modeled by increasing the normal state resistance as well as decreasing the critical current of the RCSJ between two nodes relative to the others. Tuning the resistance and critical current parameters allows us to precisely reproduce the features seen in current-biased differential resistance data for preferential gating along each of the three legs (Fig. 3d). The striped MAR features are not reproduced, as this is a quantum phenomenon not captured by the semiclassical RCSJ model. This conclusively shows independent control of conductance modes in each leg of the MTJJ.

Few-mode three-terminal josephson junction

To demonstrate tunability of conductance modes in our devices, we perform DC voltage-biased measurements on Device 1. We apply a DC source-drain voltage bias Vsd between a pair of terminals, with the third terminal electrically floating, and measure the resultant DC current Imeas. The voltage drop across the device Vmeas is also monitored simultaneously to exclude the effect of series resistance. We can then compute the differential resistance dVmeas/dImeas and differential conductance G = dImeas/dVmeas by discrete differentiation.

Figure 4a shows a map of differential resistance between terminals 1 and 2 in Device 1 as a function of Vsd and gate voltage applied to all three split gates, Vg. The critical current countours are observed as areas of zero resistance, and MAR is observed as areas of reduced resistance for Vsd ≲ 2 mV. These features show periodic oscillations as a function of gate voltage. These oscillations indicate Fabry-Pérot interference, which has been observed before in a two-terminal graphene Josephson junction44. This results from the interference of supercurrent trajectories that travel ballistically from one contact to the other, conclusively showing ballistic transport between the two terminals. Supercurrent is present between the two measured terminals at the conductance values ~1.0 G0, where G0 = 2e2/h is the conductance quantum. We also observe conductance plateaus as a function of Vg (Fig. 4b). The step height of these plateaus differs from the conductance quantum G0, and the quantization weakens for higher values of conductance. This is likely due to the effect of finite source-drain bias on the conductance. At finite bias, the value of the conductance steps is determined by the number of quasi-1D subbands falling within the bias window set by Vsd41,45,46. Such finite-bias conductance measurements are necessary due to the presence of superconductivity and MAR resonances below Vmeas < 2Δ46,47,48.

Fig. 4: Single-mode MTJJ.
figure 4

a Differential conductance as a function of source-drain bias Vsd and gate voltage Vg for terminal pair 1 and 2 for Device 1 at B = 0 and T ~ 40 mK. b Differential conductance as a function of gate voltage for different Vsd for Device 1 at B = 0 and T ~ 40 mK. The curves correspond to Vsd values between 2.0 mV and 5.0 mV (shown by dashed black lines in a in increments of 0.125 mV, and are each offset along the Vg axis (arrow indicating direction of increasing Vsd) by 3 mV for clarity.

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To measure conductance at zero-bias, an out-of-plane magnetic field can be applied to eliminate superconducting effects. We measure Device 3 (lithographically identical to Device 1) in this regime. Conductance measurements are performed for terminal pair 2 and 0 using standard lock-in techniques. The waterfall plot of conductance in Fig. 5a shows the bunching of lines at zero-bias at values of ~0.5, 1, 1.5, and 2.0 G0 due to spin splitting of the subbands. At finite bias, the waterfall plot shows bunching of curves at conductance values between integer multiples of e2/h as previously observed in two-terminal superconducting QPCs with magnetic field46. This can cause the step heights to differ from the conductance quantum G0, as observed for Device 1 at B = 0 T. In a separate measurement, conductance scans are performed by varying the magnetic field and keeping Vsd = 0 V. As shown in the red curve in Fig. 5b, conductance steps are observed near the conductance values where lines bunch at B = 1 T. Fig. 5b further shows that the plateaus become more well-defined as B is increased. Additionally, resonances in the conductance data are smoothed by application of magnetic field, attributed to the suppression of coherent backscattering due to the Aharanov-Bohm phase contribution. Conductance measurements at B = 0 T are also performed for all three pairs of the terminals and are consistent with those on Device 1 (Supplementary Fig. 5). The voltage range Vg,2 is different between Fig. 5a, b due to gate hysteresis.

Fig. 5: Zero-bias conductance measurements.
figure 5

a Waterfall plot of differential conductance as a function of Vsd for a range of gate voltages from Vg,2 = − 5.26 V to Vg,2 = − 5.5 V with a step size δVg = 1.3 mV, for Device 3. b Zero-bias differential conductance for Device 3 for different values of out-of-plane magnetic field. For these measurements we have set Vg,1 = − 6 V and Vg,3 = − 3 V. The curves are offset on the gate voltage axis by 0.02 V, 0.06 V and 0.1 V for B = 0.8 T, B = 0.5 T, and B = 0.1 T, respectively for clarity. All measurements were taken at T = 40 mK.

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This demonstrates that transport between the two measured terminals takes place via a few conductance modes. Only the first few modes are individually resolved by our measurements, which could be due to a non-ideal potential profile in the central region of the junction, where the confining potential may be weakened due to screening by the Al contacts. However, to resolve the ABS spectra of MTJJs only the first few conductance modes are necessary19,26. Moreover, theory predicts that the quantized transconductance signatures of Weyl nodes can only be resolved in the single-mode limit24.

Similar data are obtained for all three pairs of terminals for Device 2 as well (Supplementary Fig. 6), and the single-mode regime is accessible, coexistent with superconductivity. However, the conductance quantization is less robust than that seen in Devices 1 and 3 (Supplementary Fig. 6). This can be attributed to the mean free path in the InAs QW (ℓ ~ 180 nm) being comparable to the junction width of 200 nm, increasing the susceptibility to scattering relative to Devices 1 and 3. It should be noted that the maximum measured resistance saturates at ~100 kΩ for Device 1 and ~10 kΩ for Device 2. For the Device 3 data in Fig. 5, we subtract this conductance contribution (~40 kΩ) at each value of the magnetic field. These residual resistances can be attributed to trivial edge modes of InAs present in the etched mesa. These surface modes do not respond to a top gate and are difficult to eliminate in InAs49,50,51,52, but not expected to be detrimental to the investigation of the ABS.

Discussion

We demonstrate phase-coherent quasiballistic transport in three-terminal split-gated Josephson devices, with access to the single-quantum-mode regime independently in each leg. This is the first demonstration of the accessibility of all theoretical constraints necessary to observe topologically protected states formed in ABS of MTJJs. This presents a promising alternative platform to realize topological quantum states, complementary to the much-explored Majorana zero modes. Realization of topologically protected states in the ABS spectra of MTJJs also requires fine-tuning of the scattering matrix of the central region, which can in principle be achieved with geometries similar to that of Device 2. Devices with more than three terminals can be explored on the same material platform with a similar gate structure, making detection of these exotic states more likely using a range of proposed approaches.

Methods

Device fabrication

Standard electron-beam lithography (EBL) and wet etching techniques were used to fabricate a mesa and the Y-shaped junction area. Approximately 40 nm of Al2O3 dielectric was deposited using thermal atomic layer deposition. Using EBL, split gates are defined over the junction area and electrodes are deposited using electron-beam evaporation of Ti/Au (5 nm/25 nm). In separate lithography step thicker gold contacts (Ti/Au, 5 nm/200 nm) are made to the gate electrodes53.

Measurement details

Differential resistance maps on both devices and conductance quantization data on Device 1 were obtained by low-noise DC transport measurements in a 3He/4He dilution refrigerator. For the conductance quantization data on Device 2 (in Supplementary Material), standard low-frequency lock-in techniques were used with a small excitation voltage and a frequency of 19 Hz. For the AC conductance measurements, the raw data are corrected by subtracting the series filter and the ammeter resistances which combine to give 6.6 kΩ. Low-pass Gaussian filtering was used to smooth numerical derivatives.