Abstract
Planar Josephson junctions (JJs) made in semiconductor quantum wells with large spinorbit coupling are capable of hosting topological superconductivity. Indium antimonide (InSb) twodimensional electron gases (2DEGs) are particularly suited for this due to their large Landé gfactor and high carrier mobility, however superconducting hybrids in these 2DEGs remain unexplored. Here we create JJs in high quality InSb 2DEGs and provide evidence of ballistic superconductivity over micronscale lengths. A Zeeman field produces distinct revivals of the supercurrent in the junction, associated with a 0−π transition. We show that these transitions can be controlled by device design, and tuned insitu using gates. A comparison between experiments and the theory of ballistic πJosephson junctions gives excellent quantitative agreement. Our results therefore establish InSb quantum wells as a promising new material platform to study the interplay between superconductivity, spinorbit interaction and magnetism.
Introduction
Twodimensional electron gases (2DEGs) coupled to superconductors offer the opportunity to explore a variety of quantum phenomena. These include the study of novel Josephson effects^{1}, superconducting correlations in quantum (spin) Hall systems^{2,3,4,5,6,7}, hybrid superconducting qubits^{8,9}, and emergent topological states in semiconductors with strong spinorbit interaction (SOI)^{10,11,12,13}. Topological superconductivity in such 2DEGs can be realized using planar Josephson junctions (JJs), where the combined effect of SOI and a Zeeman field is known to significantly alter the currentphase relation^{14,15,16}. In particular, one expects a complete reversal of the supercurrent (i.e., a π–JJ)^{17,18,19} when the Zeeman and Thouless energy of the system become comparable. It was shown recently that such a 0–π transition in a 2D system is in fact accompanied by a topological phase transition^{12,13,20,21}. This, combined with the promise of creating scalable topological networks^{22,23,24}, provides a strong motivation to study induced superconductivity in 2DEGs.
Key requirements for the semiconductor include low disorder, large SOI and a sizable Landé gfactor, combined with the ability to grow it on the wafer scale. InSb satisfies all of these requirements^{25,26,27,28} and has emerged as a prime material candidate for engineering topological superconductivity, as evident from nanowirebased systems^{29,30}. However, despite significant progress in the growth of InSb 2DEGs^{31,32}, material challenges have prevented a systematic study of the superconducting proximity effect in these systems.
Here, we overcome these issues and reliably create JJs, thus providing evidence of induced superconductivity in high quality InSb quantum wells. The JJs support supercurrent transport over several microns and display clear signatures of ballistic superconductivity. Furthermore, we exploit the large gfactor and gate tunability of the junctions to control the currentphase relation, and drive transitions between the 0 and πstates. This control over the free energy landscape allows us to construct a phase diagram identifying these 0 and πregions, in agreement with theory.
Results
Induced superconductivity in InSb 2DEGs
The JJs are fabricated in an InSb 2DEG wafer grown by molecular beam epitaxy, with a nominal electron density n = 2.7 × 10^{11} cm^{−2} and mobility μ ≈ 150,000 cm^{2}V^{−1}s^{−1}, which corresponds to a mean free path l_{e} ≈ 1.3 μm. Figure 1a shows a crosssectional illustration and scanning electron micrograph of a typical JJ. Following a wet etch of the 2DEG in selected areas, NbTiN is deposited to create sidecontacts to the 2DEG, thus defining a JJ of width W and length L. Prior to sputtering NbTiN, an insitu argon plasma cleaning of the exposed quantum well is performed in order to obtain good electrical contacts. A metal topgate, deposited on a thin dielectric layer is used to modify the electron density in the JJ. Details of the device fabrication and wafer growth can be found in the Methods section.
The junctions are measured using a quasifour terminal currentbiased circuit (Fig. 1a) at a temperature of 50 mK. We observe a clear supercurrent branch with zero differential resistance, dV/dI, followed by a jump to the resistive branch at switching current, I_{s}. In small perpendicular magnetic fields, B_{z}, Fraunhoferlike interference patterns are observed, as seen in Fig. 1b. The magnitude of supercurrent is controlled using the gate (Fig. 1c). Lowering the gate voltage, V_{g}, leads to a reduction of the electron density in the 2DEG and therefore to a suppression of I_{s} and an increase in the normal state resistance, R_{n}. In addition, we observe multiple Andreev reflections indicating an induced superconducting gap of 0.9 meV, and excess current measurements allow us to estimate transparencies in the range of 0.6–0.7 (representative data are provided in the Supplementary Note 2).
Ballistic superconductivity
Studying JJs of varying lengths (L = 0.7–4.7 μm), we gain insight into the transport regime. These devices fall in the long junction limit, since their lengths exceed the induced superconducting coherence length of around 500 nm (see Supplementary Note 2). In this limit the product of the critical current, I_{c}, and R_{n} is proportional to the Thouless energy^{33}, E_{Th} = ℏv_{F}l_{e}/2L^{2}, where v_{F} is the Fermi velocity in the 2DEG. Thus, for ballistic (diffusive) transport where l_{e} = L (l_{e} < L), we expect I_{c}R_{n} to scale as 1/L (1/L^{2}). In our experiments we measure I_{s}, but expect it to be close to I_{c}, since the Josephson energy (≈20 K) is significantly larger than the fridge temperature (≈50 mK). Figure 1d shows I_{s}R_{n} for a set of JJs. We find a 1/L scaling (black dots) indicative of ballistic superconductivity, with deviations only for the longer (L ≥ 2.7 μm) junctions. Such a 1/L dependence was predicted decades ago^{34} but has only recently been experimentally observed over micronscale lengths in clean graphenebased JJs^{35,36}. To confirm the scaling arguments we also include data from a lower mobility wafer (see Supplementary Note 1) with l_{e} ≈ 0.5 μm (red dots) and find a 1/L^{2} scaling, consistent with diffusive behavior. In the remainder of this work we focus on JJs fabricated on the high mobility wafer.
0–π transitions in Josephson junctions
Using these ballistic junctions, we now explore their response to a Zeeman field. The theory of JJs with large SOI subjected to a magnetic field has been discussed extensively^{14,17,20}. Below we briefly describe the essential elements of the physical picture. At zero B the Fermi surfaces are split due to the Rashba SOI (solid lines of Fig. 2a inset). The magnetic field then splits the bands by the Zeeman energy, E_{Z} = gμ_{B}B, leading to a shift in the Fermi surfaces by ±δk/2. The depicted shift of the Fermi surfaces assumes that the spinorbit energy dominates over the Zeeman energy, which is indeed the case for the measured JJs (see Supplementary Note 3 for a detailed discussion). Therefore, Cooper pairs (electrons with opposite momentum and spin) now possess a finite momentum, given by k_{F} ⋅ δk = E_{Z}(m^{*}/ℏ^{2}), where k_{F} is the Fermi momentum and m^{*} the effective mass. This translates to a phase acquired by the superconducting order parameter along the direction of current flow, Ψ(r) ∝ cos(δk ⋅ r)^{37,38,39}. Depending on the length of the Cooper pair trajectories, r, the order parameter is either positive or negative, corresponding to the ground state of the JJ being at 0 or π superconducting phase difference, respectively. This oscillation of the order parameter results in a modulation of the critical current I_{c} ∝ Ψ, where a minimum of I_{c} is expected whenever the order parameter switches sign^{14,15}. Taking only trajectories perpendicular to the contacts \(\left( {\delta {\mathbf{k}} = \delta {{k}}\widehat {\mathbf{x}},{\mathbf{k}}_{\mathrm{F}} = {{k}}_{\mathrm{F}}\widehat {\mathbf{x}}} \right)\), a JJ with length L will display minima in I_{c} when Lδk = (2N + 1)π/2, with N = 0, 1, 2... The condition for the first minimum (N = 0) can be expressed as a resonance condition in terms of the Zeeman and ballistic Thouless energy as E_{Z} = πE_{Th} giving:
The 0–π transition therefore depends on three experimentally accessible parameters: (1) applied magnetic field, (2) length of the JJ, and (3) carrier density. In the following, we demonstrate independent control of each of these parameters, allowing for a complete study of the free energy landscape of the junctions.
Magnetic fielddriven 0–π transitions
We start by varying B_{y}, while n (controllable by V_{g}) and L remain fixed. The orientation of the magnetic field reflects the Fermi surfaces described, and avoids unwanted geometric effects^{40}. Figure 2a shows the expected oscillation of I_{s} with increasing B_{y}, displaying two distinct minima at B_{y} = 470 mT and B_{y} = 1250 mT (see Supplementary Note 4 for details about magnetic field alignment). This behavior is consistent with a magnetic fielddriven 0–π transition, as discussed above, where the first (second) minimum corresponds to a transition of the JJ state from 0 to π (π–0). This interpretation is corroborated by the occurrence of the second minimum at a field value, which is approximately three times larger than the first. Note that this is incompatible with a Fraunhofer interference pattern that might arise from the finite thickness of the 2DEG. Furthermore, taking into account the gate dependence of the transition and other geometric considerations (discussed in detail in the Supplementary Note 5) allows us to conclusively rule out such a mechanism for the supercurrent modulation.
Next, we investigate how the length of the JJ influences B_{0π}, the magnetic field at which the transition occurs. Figure 2b presents the I_{s} oscillation for JJs with four different lengths, showing that B_{0π} is systematically reduced for increasing L. Plotting B_{0π} with respect to 1/L (inset of Fig. 2b), we find a linear dependence as expected from Eq. (1). The transition points are therefore determined by the ballistic E_{Th}, consistent with the conclusions from Fig. 1d. Finally, we check the dependence of the transition on the electron density. In Fig. 2c, we plot I_{s} versus B_{y} for different gate voltages using a JJ with L = 1.1 μm. As V_{g} is lowered, B_{0π} shifts to smaller values, again in qualitative agreement with Eq. (1). Interestingly, above a certain magnetic field the state of the JJ (0 or π) becomes gatedependent. For example at B_{y} = 400 mT, the junction changes from a 0JJ (V_{g} = 0 V) to a πJJ (V_{g} = −0.4 V), with a transition at V_{g} = −0.2 V. This indicates the feasibility of tuning the JJ into the πstate using gate voltages, while the magnetic field remains fixed.
Gatedriven 0–π transitions
These gatedriven transitions are demonstrated in Fig. 3a–d, which show a sequence of I–V_{g} plots for increasing inplane magnetic fields. At B_{y} = 250 mT, I_{s} displays a monotonic reduction with decreasing V_{g}. At a higher magnetic field, B_{y} = 325 mT, I_{s} reveals a markedly different behavior, whereby the supercurrent first decreases and then (at V_{g} = −0.32 V) shows a clear revival, indicative of a gatedriven 0–π transition, where the resonance condition (E_{Z} = πE_{Th}) is achieved by tuning the electron density. Increasing B_{y} further, continuously moves the transition point to higher gate voltages (larger density), perfectly in line with expectations for a 0–π transition. Figure 3e shows two linecuts from Fig. 3d. At zero current bias, dV/dI shows a clear peak, indicative of a reentrance of the supercurrent due to the the 0–π transition. However, at high bias, dV/dI increases monotonically, similar to the response at zero magnetic field. This eliminates trivial interference effects as an explanation for the supercurrent modulation, where one would expect a correlation between the two curves^{35,41,42}.
Construction of the 0–π phase diagram
In contrast to the fielddriven measurements (Fig. 2), controlling the transition with a gate avoids the need for timeconsuming field alignment procedures, thus allowing us to efficiently explore a large parameter space in magnetic field and gate voltage. We now combine these results to construct a 0–π phase diagram of the JJ. The combination of a high quality 2DEG and relatively long devices results in well defined magnetoresistance oscillations, allowing us to directly extract the electron density in the junction. Figure 4a shows the Landau fan diagram in perpendicular magnetic fields, B_{z}, from which we identify the filling factors, ν = nh/eB_{z} (Fig. 4b), and thereby obtain the n vs. V_{g} curve (Fig. 4c). We then plot all the transition points in Fig. 4d. The axes represent the two important energy scales in the system (B_{y} ∝ E_{Z} and \(\sqrt n \propto E_{{\mathrm{Th}}}\)), thereby highlighting the 0 and π regions in the phase space. Finally, we compare our results with the theory of ballistic JJs represented by Eq. (1). To do so, we independently extract the effective mass (see Supplementary Note 7), m* = (0.022 ± 0.002)m_{e}, and fit the data to a single free parameter, g_{y} (the inplane gfactor), giving g_{y} = 25 ± 3 in good agreement with previous measurements on similar InSb quantum wells^{28}.
Our work provides the first evidence of induced superconductivity in high quality InSb 2DEGs and demonstrates the creation of robust, gatetunable πJosephson junctions. We show that the 0–π transition can be driven both by magnetic fields and gate voltages. The significant region of phase space where the π–JJ is stable could prove advantageous in the study of topological superconductivity in planar JJs^{12,13,20,21}. Moreover, these large SOI 2DEGs, in conjunction with our magnetic field compatible superconducting electrodes and clear Landau quantization, would also be excellent candidates to realize topological junctions in the quantum Hall regime^{7}. Finally, the ability to control the ground state between 0 and π states using gates is analogous to recent experimental results in ferromagnetic JJs^{43}, and could possibly serve as a semiconductorbased platform for novel superconducting logic applications^{44}. We therefore establish InSb 2DEGs as a new, scalable platform for developing hybrid superconductorsemiconductor technologies.
Methods
Wafer growth
InSbbased 2DEGs were grown on semiinsulating GaAs (100) substrates by molecular beam epitaxy in a Veeco Gen 930 using ultrahigh purity techniques and methods as described in ref. ^{45}. The layer stack of the heterostructure is shown in Supplementary Fig. 1a. The growth has been initiated with a 100 nm thick GaAs buffer followed by a 1 μm thick AlSb nucleation layer. The metamorphic buffer is composed of a superlattice of 300 nm thick In_{0.91}Al_{0.09}Sb and 200 nm thick In_{0.75}Al_{0.25}Sb layers, repeated three times, and directly followed by a 2 μm thick In_{0.91}Al_{0.09}Sb layer. The active region consists of a 30 nm thick InSb quantum well and a 40 nm thick In_{0.91}Al_{0.09}Sb top barrier. The Si δdoping layer has been introduced at 20 nm from the quantum well and the surface. The In_{x}Al_{1−x}Sb buffer, the InSb quantum well and the In_{x}Al_{1−x}Sb setback were grown at a temperature of 440 °C under a p(1 × 3) surface reconstruction. The growth temperature was lowered to 340 °C, where the surface reconstruction changed to c(4 × 4), just before the δdoping layer, to facilitate Si incorporation^{46}. The scanning transmission electron micrograph of Supplementary Fig. 1b reveals the efficiency of the metamorphic buffer to filter the dislocations.
Device fabrication
The devices are fabricated using electron beam lithography. First, mesa structures are defined by etching the InSb 2DEG in selected areas. We use a wet etch solution consisting of 560 ml deionized water, 9.6 g citric acid powder, 5 ml H_{2}O_{2} and 7 ml H_{3}PO_{4}, and etch for 5 min, which results in an etch depth around 150 nm. This is followed by the deposition of superconducting contacts in an ATC 1800V sputtering system. Before the deposition, we clean the InSb interfaces in an Ar plasma for 3 min (using a power of 100 W and a pressure of 5 mTorr). Subsequently, without breaking the vacuum, we sputter NbTi (30 s) and NbTiN (330 s) at a pressure of 2.5 mTorr, resulting in a layer thickness of approximately 200 nm. Next, a 45 nm thick layer of AlO_{x} dielectric is added by atomic layer deposition at 105 °C, followed by a topgate consisting of 10 nm/170 nm of Ti/Au.
Data availability
All data files are available at 4TU.ResearchData repository, https://doi.org/10.4121/uuid:5fab827387944cd796d4ba8ec00a62cf
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Acknowledgements
We thank Ady Stern, Attila Geresdi, and Michiel de Moor for useful discussions. The research at Delft was supported by the Dutch National Science Foundation (NWO) and a TKI grant of the Dutch topsectoren program. The work at Purdue was funded by Microsoft Quantum.
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C.T.K. and C.M.M. fabricated and measured the devices. C.T., G.C.G. and M.J.M. designed and grew the semiconductor heterostructures. C.T., S.M., C.R.G., R.K., T.W., R.E.D., G.C.G. and M.J.M. characterized the materials. M.L. and G.S. provided the effective mass measurements. C.T.K., C.M.M., F.K.d.V. and S.G. performed the data analysis. The manuscript was written by C.T.K., F.K.d.V., C.M.M. and S.G., with input from all coauthors. S.G. supervised the project.
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Ke, C.T., Moehle, C.M., de Vries, F.K. et al. Ballistic superconductivity and tunable π–junctions in InSb quantum wells. Nat Commun 10, 3764 (2019). https://doi.org/10.1038/s41467019117424
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