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. 2025 Jan 13;19(3):3878–3885. doi: 10.1021/acsnano.4c15893

Superconductive Coupling Effects in Selectively Grown Topological Insulator-Based Three-Terminal Junctions

Gerrit Behner †,‡,*, Abdur Rehman Jalil †,‡, Alina Rupp †,‡, Hans Lüth †,‡, Detlev Grützmacher †,‡, Thomas Schäpers †,‡,*
PMCID: PMC11781021  PMID: 39806296

Abstract

graphic file with name nn4c15893_0006.jpg

The combination of an ordinary s-type superconductor with three-dimensional topological insulators creates a promising platform for fault-tolerant topological quantum computing circuits based on Majorana braiding. The backbone of the braiding mechanism are three-terminal Josephson junctions. It is crucial to understand the transport in these devices for further use in quantum computing applications. We present low-temperature measurements of topological insulator-based three-terminal Josephson junctions fabricated by a combination of selective-area growth of Bi0.8Sb1.2Te3 and shadow mask evaporation of Nb. This approach allows for the in situ fabrication of Josephson junctions with an exceptional interface quality, important for the study of the proximity-effect. We map out the transport properties of the device as a function of bias currents and prove the coupling of the junctions by the observation of a multiterminal geometry-induced diode effect. We find good agreement of our findings with a resistively and capacitively shunted junction network model.

Keywords: topological insulators, multiterminal Josephson junction, superconductivity, shadow mask, Josephson diode effect, proximity effect

Three-dimensional topological insulators are a class of materials which recently raised a lot of interest due to its promising applicability in the field of topological quantum computing.1−3 The material class exhibits strong spin–orbit coupling. This, in turn, leads to band inversion in the bulk electronic band structure. As a consequence, gapless surface states appear, which are protected by a time-reversal symmetry. Proximizing a topological insulator nanoribbon with an s-type superconductor and aligning a magnetic field along the nanoribbon give rise to Majorana zero modes.4 Braiding of these Majorana zero modes is the essential computation operation in topological quantum computing.5−9 For this process, multiterminal structures are necessary in which the superconducting phase of the different electrodes needs to be adjusted. The three-terminal Josephson junction therefore represents an important building block for these networks.10−16 It is crucial to understand the transport in these devices for further use in topological quantum computing applications. Generally, hybrid devices with multiple connections lead to rich physics in terms of transport properties, with a huge parameter space to be probed.17

In recent years, the field of multiterminal Josephson junctions and their unique properties have attracted a lot of interest in the scientific community, e.g., the emergence of n-1-dimensional topological properties from an n-terminal Josephson junction made from conventional superconductor or the study of the synthetic Andreev band structure in the two-dimensional phase space.18−22 While multiterminal Josephson junctions have extensively been studied in epitaxially grown semiconductor-superconductor hybrid structures,22−26 not much has been reported on these devices in the field of topological materials. A flux-controlled three-terminal junction based on Bi2Te3 revealed the opening and closing of a minigap.27,28 Furthermore, a three-terminal junction based on the topological insulator Bi4Te3 was investigated, which did not show the expected signatures of the multiterminal Josephson effect but rather those of a resistively shunted network of two Josephson junctions.29

Recently, the superconducting diode effect has attracted a lot of attention.30 A characteristic of the diode effect is that the magnitude of the critical supercurrent is dependent on the direction in which the current is driven. The diode effect occurs when both inversion and time-reversal symmetry are broken. For Josephson junctions with a semiconducting31−34 or topological insulator35 weak link, this can be accomplished by the presence of spin–orbit coupling in conjunction with an external magnetic field for the time-reversal symmetry breaking. Alternatively, the inversion symmetry can be broken by the device layout itself. This can be achieved, for example, by a superconducting quantum interference device, where each of the two junctions of the interferometer has a different current phase relation.36 More recently, the asymmetry in a multiterminal Josephson junction has led to a diode effect, either by keeping one of the superconducting electrodes floating25 or by phase biasing using superconducting loops connecting pairs of electrodes in the junction.26

We present low-temperature measurements of three-terminal Josephson junctions fabricated by a combination of selective-area growth of the topological insulator Bi0.8Sb1.2Te3 and shadow mask evaporation of Nb as the superconductor (see Figure 1).37,38 This approach allows for the in situ fabrication of Josephson junctions with very high interface transparency, important for the study of the superconducting proximity-effect. The transport properties of the junction are mapped out as a function of the bias current and magnetic field. The bias current maps show several interesting transport phenomena, e.g., an extended superconducting area and multiple Andreev reflections (MAR), indicating the successful fabrication of a fully coupled three-terminal junction. The measured results for the junction appear to be in good agreement with a resistively and capacitively shunted junction (RCSJ) model. Intrinsic asymmetries of the device and their effect on transport in the junctions are also very well reproduced. The coupling of the junctions is emphasized by the observation of a multiterminal geometry-induced diode effect as a result of an externally applied magnetic field.

Figure 1.

Figure 1

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(a) Scanning electron micrograph of the device. The electronic setup used for applying current and measuring voltage in the first three-terminal measurement configuration is also shown. The terminal names (0, 1, 2) that are used for naming the single junctions are indicated in red. The blue dashed line indicates a line cut through the layer stack which is presented in (b). (b) Schematic illustration of the line cut (dashed blue line) in the layer stack making up the device.

Results and Discussion

Two-Terminal Characteristics

Here, we discuss the basic characteristics of the effective Josephson junctions that are defined by contacting two of the three terminals, e.g., the bottom and the right superconducting arm. These terminals are indicated in red as 1 and 0 in Figure 1a. Henceforward, the junction will be termed JJeff10. The width of the topological insulator ribbons is about 100 nm. The distance between the Nb electrodes is about 70 nm between terminals 1 and 2 and 150 nm between the other pairs of electrodes. Figure 2a shows the DC-characteristics of all three effective junctions, i.e., JJeff10, JJeff20, and JJeff21 at a zero magnetic field. The junctions exhibit a hysteresis between the switching current Ic and retrapping current Ir. The current–voltage characteristics of Josephson junctions can be described by a RCSJ model. However, because of the coplanar junction geometry and the resulting very low junction capacitance, we can rule out that the hysteresis is due to an overdamped junction characteristics. We rather attribute the hysteresis to Joule heating effects.39 Note that JJeff20 and JJeff21 exhibit similar switching currents slightly below 10 μA, whereas JJeff10 shows a critical current larger by almost a factor of 2. We attribute this difference to variations in the interface properties. The switching currents, as well as the other characteristic parameters for all junctions, are summarized in Table 1. In the following, we will discuss the properties of the junction JJeff10 as an example.

Figure 2.

Figure 2

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(a) Current–voltage characteristics of the single junctions JJeff10, JJeff20, and JJeff21 in the device shown in Figure 1a. (b) Differential resistance of JJeff10 as a function of bias voltage V10. The position of the voltage bias for possible MAR is indicated by dashed lines, where the first two are labeled 2Δ and Δ. (c) Temperature dependence of the critical current of JJeff10.

Table 1. Single Junction Parameters: Ic Critical Current, RN Normal State Resistance, andτ Transparency.

  JJeff10 JJeff20 JJeff21
Ic 16 μA 9 μA 8 μA
RN 113 Ω 138 Ω 131 Ω
Iexc 19.1 μA 10.7 μA 12.6 μA
Τ 0.94 0.87 0.89
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The excess current Iexc of the junctions is determined by a linear regression of the junctions’ ohmic behavior at bias voltages larger than 2Δ, with Δ the superconducting gap energy. Here, 2Δ ≈ 2.6 meV is determined from the critical temperature Tc ≈ 8.5 K of the Nb film. The slope of the linear regression also determines the normal state resistance RN of the junction, which is 113 Ω for junction JJeff10. The excess current Iexc ≈ 19.1 μA is used to estimate the junction transparency τ of JJeff10. It can be gained by a fit to the analytical calculation of the excess current following the work of Niebler et al. based on the Octavio–Tinkham–Blonder–Klapwijk model.40−42 The junction transparency is a figure of merit to evaluate the interface quality. In fact, our junctions exhibit a large transparency, i.e., up to τ = 0.94 for junction JJeff10.

Figure 2b shows the differential conductance as a function of bias voltage. Here, features in the differential conductance reveal subharmonic gap structures.43−48 The signatures can in most cases be attributed to MAR. They are expected to appear at voltages of V = (2Δ)/(en), where e is the elementary charge and n is an integer.40,49 The shape of the features is determined by the transport characteristics in the junction.50−52 The positions of the voltage biases for possible MAR are indicated by vertical dashed lines, where the first two are labeled 2Δ and Δ. The peak located at roughly 1.7 mV bias voltage V10 could be a result of the 2Δ MAR resonance due to the induced gap in the TI weak link or in fact a result of a resistance change in one of the other junctions. The exact reason is hard to determine as all junction are evidently coupled to each other, influencing the behavior and measured results. Figure 2c shows the temperature dependence of the switching current of JJeff10. The temperature dependence indicates that the behavior of the weak link is dominated by diffusive transport.37 We attribute this to the fact that the supercurrent is carried not only by ballistic surface states but also by diffusive bulk states.

Three-Terminal Measurements

Due to the intrinsic asymmetry of the T-shape of our three-terminal junction, two different three-terminal configurations are probed. The first is schematically shown in Figure 1a. Here, the coupling between the two driving junctions, i.e., from terminals 1 and 2 to terminal 0 at ground, respectively, is mediated by the junction connecting the left and right arm. The characteristics of the three-terminal junction are probed by measuring the differential resistance dV20/dI20 between terminals 2 and 0 for the respective configurations. The corresponding bias maps for dV10/dI10 are shown in the Supporting Information.

Figure 3a shows the differential resistance dV20/dI20 measured in the configuration shown in Figure 1a as a function of applied currents I10 and I20. The measurement outcome can be summarized by three prominent features.17 The first and most distinct feature is an extended region of superconductivity in the center of the map, indicated by the dark blue area. This corresponds to the zero-voltage state between terminal 2 and 0 of the three-terminal junction in the (I10, I20) plane. Since the switching and retrapping current are different, the superconducting area is asymmetric with respect to the center of the current bias map. The second characteristic feature of a three-terminal junction is indicated by three lines in the bias map marked C10, C20, and C21, respectively. They each represent a particular combination of bias currents I10 and I20 for which either V10, V20, or V21 is zero (see Figure 3). This is a generic feature of a three-terminal Josephson junction that exhibits dissipationless transport in all of its junctions. They are the result of a current being able to flow to ground by two different paths in the multiterminal junction. For example, current I20 can not only flow to ground directly via the junction JJ20 formed between superconducting electrodes 2 and 0 but also take a detour through the other arms. Thus, part of the current will also flow from terminal 2, via terminal 1, to ground. For example, if I20 ≥ 0 and I10 ≤ 0, this will lead to a compensation of currents in junction JJ20. As a result, the superconducting region in the bias map is extended along the diagonal. The slope of the extension is determined by the ratio of the normal state resistances of the junctions.17 In Figure 3a, line C20 therefore represents a compensation of currents in the probed junction, whereas line C10 corresponds to a compensation of currents in junction JJ10 formed between electrode 1 and 0, visible as a reduced resistance. Line C21 represents the compensation in the coupling junction JJ21. Here, the condition sgn(I10) = sgn(I20) needs to be fulfilled so that the current components provided from terminal 1 and 2 flowing through JJ12 have opposite signs.

Figure 3.

Figure 3

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Differential resistance—current bias maps for different measurement configurations. (a) Current bias map of the three-terminal junction measured in the setup shown in Figure 1a. The bias map depicts the differential resistance of dV20/dI20 as a function of applied currents I10 and I20. The three different lines (dotted, dashed, and dashdot) in the graph indicate the three regimes of compensated currents. The compensated currents are an effect mediated by the nongrounded Josephson junction coupling the two others. In this case, the coupling is mediated by JJ12. (b) Current bias map of the measurement configuration shown in (c). The junction determining the coupling between the two grounded junction is switched in order to increase the effects mediated by the coupling. The coupling junction in this case is JJ20. (c) Schematic depiction of the electronic setup of the second three-terminal measurement configuration with ground on terminal 2 is also shown. (d) Current bias map generated with the solution of the RCSJ network model. The simulation is carried out with values extracted from the measurements. A direct comparison of the experimental results in (b) and the theoretically expected behavior in (d) gives a reasonable qualitative agreement.

In order to reveal the effect of the junction properties on the superconducting area, a second configuration shown in Figure 3c is investigated, where now JJ20 mediates the coupling between JJ10 and JJ21. Figure 3b shows the differential resistance dV20/dI20 as a function of the applied currents I20 and I21. Visible again is an extended superconducting region and the features due to compensating currents indicated by the lines C10, C20, and C21.

The third feature in the current bias maps is equipotential lines, which we attribute to MAR resonances in the junction according to the analysis of Pankratova et al.17Figure 4 shows the differential resistance dV20/dI20 as a function of the DC voltages V20 and V21 recorded in the measurement configuration shown in Figure 3c. The equipotential lines visible in the current bias color maps do indeed appear as lines of constant voltage for the respective junctions. We attribute them to MAR resonances in the sample. However, it is difficult to assign the respective lines to a fixed value of V = (2Δ)/(en), but it is reasonable to assume that the lines correspond to higher order MAR resonances.

Figure 4.

Figure 4

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Bias map shown in Figure 3c but now with the differential resistance dV20/dI20 plotted as a function of the DC voltage drops V20 and V21. The horizontal and vertical lines at zero voltage can be attributed to the superconducting state between the corresponding pairs of terminals. The vertical and horizontal lines marked in blue and red are attributed to MAR in junctions JJ20 and JJ21, respectively. The diagonal line at V20 = V21 corresponds to the superconducting state between the terminals 1 and 0.

Simulation

Neglecting multiple couplings, our three-terminal Josephson junction can be described by a model that connects three Josephson junctions, i.e., JJ10, JJ20, and JJ21, in a triangular network. A schematic of the network is provided in the Supporting Information. We employed the simulation approach presented in the work of Gupta et al.25 to determine the relevant junction parameters. The IV characteristics in the two-terminal measurements presented above are used to extract first estimates of the critical currents and normal state resistances. In the second step, these values are adjusted in a semiqualitative manner until a satisfactory agreement with the experimental data is obtained. The resulting bias map of the simulated three-terminal Josephson junction, corresponding to the measurements shown in Figure 3c, is shown in Figure 3d. An approximate matching of the critical current area as well as the slope and position of the arms is achieved. The corresponding parameters, i.e., critical currents and normal state resistances for each individual junction are presented in the Supporting Information. It turns out that a good matching is obtained by assuming a much higher critical current for junction JJ10 compared to the other two junctions. In general, the critical currents of the individual junctions in the network are smaller than the values obtained from the two-terminal measurements because the bypass currents are not included. Since the junction capacitances are negligible and Joule heating is not implemented, the simulation fails to reproduce the hysteresis of the device, which leads to a slight shift from zero in the experimental data. Since in the simulation effects are not included, the experimentally observed tapering of the three arms/slope for the higher current regimes cannot be reproduced. The tapering is a result of the dynamic reduction of the critical current due to Joule heating.

Diode Effect

Due to the breaking of inversion symmetry by the device layout, three-terminal Josephson junctions are expected to exhibit a diode effect when exposed to a perpendicular magnetic field.25,53Figure 5a shows the current–voltage characteristics of JJeff10 with terminal 2 left floating. The current from terminal 1 to terminal 0 was increased from zero in either the positive or negative direction to investigate the switching current for both current directions. By applying a magnetic field of 50 mT, it is found that the switching current for positive bias currents (I+) is smaller than that for negative currents (I–). As can be seen in Figure 5b, reversing the magnetic field to −50 mT causes the polarity of the diode effect to switch, with I+ now being larger than I–.

Figure 5.

Figure 5

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Diode effect in a three-terminal Josephson junction. (a) Current–voltage characteristics of the effective junction JJeff10 for negative and positive bias currents at an applied out of plane magnetic field of 50 mT. In this case, the critical current is larger for positive bias currents. (b) Corresponding set of IV characteristics after inversing the magnetic field to −50 mT with the critical current for negative bias currents being larger. (c) Switching currents I+ and I– as a function of magnetic field for positive and negative bias currents, respectively. (d) Diode efficiency η as a function of magnetic field. (e) Periodic switching of current I10 while linearly increasing the control terminal bias I20. (f) Response of JJeff10 to an alternating square-wave current I10 and a linearly increasing current I20 shown in (e). A field of 50 mT is applied.

In order to quantify the performance of the Josephson diode, its efficiency is defined as follows: η = δIc/(I+ + |I–|), where δIc = (I+ – |I–|). At ±50 mT, we get a diode efficiency η of about 0.04 for JJeff10. For a fully symmetric three-terminal Josephson junction, the diode efficiency should follow a Φ0/2 periodicity, where Φ0 = h/(2e) is the magnetic flux quantum and h is the Planck constant.25 A maximum diode efficiency ηmax of 0.28 is expected at an applied external flux of Φ0/4. For our device, the magnitude of the magnetic field corresponding to Φ0/4 can be calculated from the junction size estimated from the SEM micrograph shown in Figure 1a. Assuming a junction area of 100 × 100 nm2, one obtains a magnetic field of ≈50 mT, corresponding to the field applied in our experiment. The experimentally determined value at ±50 mT of η ≈ 0.04 is considerably smaller than the expected ηmax. We attribute the smaller value to the asymmetric layout of our multiterminal Josephson junctions and to the unbalanced switching currents. Figure 5c displays the critical currents I– and I+ as a function of magnetic field with the evolution of the diode efficiency η shown in Figure 5d. It can be seen that around zero magnetic field both currents I– and I+ are equal yielding η = 0%. When increasing the magnetic field into the respective direction, the difference between I– and I+ becomes larger, yielding increased efficiency. The existence of the superconducting diode effect is a good measure of the quality of the device and can be seen as a proof-of-principle experiment for the quality and coupling of the three-terminal Josephson junction. In our case, the diode effect is effectively a demonstration that our description of the device as a coupling of three Josephson junctions is adequate.

The diode characteristics of the single junction under application of an alternating current are depicted in the Supporting Information. Here, we focus on the characteristics determined by the three-terminal layout. By biasing the second terminal with a current I02, it is possible to control the diode characteristics.25 This is illustrated in Figure 5e,f, where the Josephson bias current I10 is periodically switched between ±5 μA while the voltage drop V10 is recorded. At the same time, the control bias current I20 is ramped from 0 to 15 μA. The voltage drop of JJeff10 recorded in Figure 5f shows that at zero control bias current, there is already a diode effect, with a superconducting state at I10 = −5 μA and a voltage drop of 0.075 mV at +5 μA. As long as the junction JJ20 is in the superconducting state, i.e., up to I20 = 1 μA, the diode characteristics do not change. Beyond this value, however, the voltage drop of the diode makes a sudden jump and increases approximately linearly. For negative diode current biases, the diode remains in the superconducting state up to the maximum control current of 15 μA. Closer inspection reveals that at a current I20 of about 2.5 μA, the slope of the linear increase in V10 decreases slightly. We attribute this to the fact that the two other junctions, JJ20 and JJ21, are in the resistive state at the respective bias current.

Conclusions

In conclusion, we have fabricated a three-terminal Josephson junction in situ, where the weak link material consists of a selectively grown topological insulator. The superconducting Nb electrodes are defined by shadow evaporation. This approach allows the fabrication of arbitrarily shaped junctions for future quantum computing applications, in particular, structures for braiding Majorana zero modes. The single junctions that make up the device are of very high quality, with high transparency and evidence of MAR. The three-terminal junction exhibits all the characteristics of a fully coupled multiterminal Josephson junction. The observed slight asymmetries in the switching currents are attributed to the intrinsic asymmetry of the T-shaped junction and to variations in the interface transparency. The bias current maps of the differential resistance show all features expected for multiterminal Josephson junctions where all junctions interact in the superconducting state. The proof of principle now allows for more sophisticated experiments to detect signatures of the topological superconductivity. By leaving one terminal floating and applying an external magnetic field, a diode effect is observed, where the switching current depends on the direction of the bias current. As a next step, connecting superconducting electrodes by a superconducting loop to achieve phase bias would allow for an efficient control of the diode characteristics, offering a great potential for superconducting electronic circuits.26

Methods

Fabrication

The samples are fabricated using a combination of selective-area growth and shadow mask evaporation.37,38 This gives the possibility to prepare samples with arbitrary geometry and exceptional interface transparency between the topological insulator and the parent superconductor. 10 nm of SiO2 and 25 nm of Si3N4 are deposited by thermal oxidation and plasma-enhanced chemical vapor deposition (PECVD), respectively, on a Si(111) wafer. Trenches in the shape of a T and a width of 100 nm are etched into the stack by means of a resist process using electron beam lithography and reactive-ion etching (RIE). A second stack of 300 nm SiO2 and 100 nm Si3N4 is deposited using PECVD. This stack is subsequently used to define the bridge for shadow evaporation. To do so, the second Si3N4 layer is patterned into the shape of the bridge using a negative resist process and RIE. By etching the sample with hydrofluoric acid in the second etching step, the bridge is under-etched creating a suspended shadow mask above the trench. The TI growth takes place under rotation of the sample around its normal axis. This ensures homogeneous growth under the shadow mask. The Nb contacts are deposited in situ. For this purpose, 50 nm of Nb is deposited from an angle without rotation of the sample. The shadow mask then patterns the Josephson junction itself, without the need for etching. Finally, the sample is capped by using a 5 nm layer of Al2O3 to prevent oxidation. The coarse shapes of the electrodes are defined ex situ using an SF6 RIE process without damaging the junction area or the nanoribbon at any point.

Measurements

The sample characteristics were measured in a dilution refrigerator with a base temperature of T ≈ 10 mK. Figure 1a shows a typical measurements configuration, where two current sources supply currents I10 and I20 from the left and right terminal, i.e., terminals 1 and 2, respectively, to the bottom electrode, i.e., terminal 0, respectively. Voltages V10 and V20 are measured accordingly. The voltages are measured in a quasi-four-point measurement scheme. The differential resistance of the sample is measured using a lock-in amplifier by the addition of a 10 nA AC current to the applied DC current. For the measurements of the diode effect, an out-of-plane magnetic field is applied.

Acknowledgments

We thank H. Kertz for technical assistance and F. Lentz and Stefan Trellenkamp for electron beam lithography. We thank M. Gupta for providing the code for the solution of the RCSJ network model. We are grateful for fruitful discussions with K. Moors, R. Riwar, and P. Schüffelgen. This work was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1-390534769 as well as financially supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy within Bavaria’s High-Tech Agenda Project “Bausteine für das Quantencomputing auf Basis topologischer Materialien mit experimentellen und theoretischen Ansätzen” (grant no. 07 02/686 58/1/21 1/22 2/23).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnano.4c15893.

  • Corresponding counterparts of the bias maps, simulation and differential resistances for all of the junction in the three-terminal junctions; and behavior of the device under application of an magnetic field and an extension of the diode characteristic measurements (PDF)

Author Present Address

§ II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, D-50937 Köln, Germany

The authors declare no competing financial interest.

Supplementary Material

nn4c15893_si_001.pdf (798.6KB, pdf)

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