Ballistic superconductivity and tunable $\pi$-junctions in InSb quantum wells

TL;DR

This paper demonstrates ballistic superconductivity and tunable $$-junctions in InSb quantum wells, revealing new possibilities for studying topological states and quantum phenomena in semiconductor-superconductor hybrid systems.

Contribution

First experimental realization of superconducting Josephson junctions in high-quality InSb quantum wells with tunable $0$-$$ states and phase diagram mapping.

Findings

Supercurrent transport over several microns in InSb quantum wells.

Clear signatures of ballistic superconductivity observed.

Controlled transitions between $0$ and $$ states achieved.

Abstract

Two-dimensional electron gases (2DEGs) coupled to superconductors offer the opportunity to explore a variety of quantum phenomena. These include the study of novel Josephson effects, superconducting correlations in quantum (spin) Hall systems, hybrid superconducting qubits and emergent topological states in semiconductors with spin-orbit interaction (SOI). InSb is a well-known example of such a strong SOI semiconductor, however hybrid superconducting devices in InSb quantum wells remain unexplored. Here, we interface InSb 2DEGs with a superconductor (NbTiN) to create Josephson junctions (JJs), thus providing the first 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 Land\'{e} g-factor and gate tunability of the junctions to control the current-phase relation, and drive transitions between the $0$ and $\pi$-states. This control over the free energy landscape allows us to construct a phase diagram identifying these $0$ and $\pi$-regions, in agreement with theory. Our results establish InSb quantum wells as a promising new material platform to study the interplay between superconductivity, SOI and magnetism.

Tables

Table 1. Table S1: Electron density, mobility and mean free path for the high and low mobility wafer, obtained from quantum Hall measurements at T = 300 ​ mK 𝑇 300 mK T=300~{}\text{mK} .

	High mobility wafer	Low mobility wafer
$n\left({\text{cm}}^{-2}\right)$	$2.71\cdot {10}^{11}$	$2.71\cdot {10}^{11}$
$\mu \left({\text{cm}}^2/\text{Vs}\right)$	146,400	61,500
$l_{\text{e}}\left(\mu \text{m}\right)$	1.26	0.53

Equations

$$g{\mu }_{\text{B}}B=\pi \frac{{\hslash }^2\sqrt{2\pi n}}{m^{\ast }2L}.$$

$$\frac{\Delta \sigma (B)}{e^2/2\pi h}=$$

$$+\sum _{n=1}^{\infty }\left[\frac{3}{n}-\frac{3a_{\text{n}}^2+2a_{\text{n}}{H}_{\text{s}}-1-2(2n+1)H_{\text{s}}}{(a_{\text{n}}+H_{\text{s}})a_{\text{n-1}}{a}_{\text{n+1}}-2H_{\text{s}}[(2n+1)a_{\text{n}}-1]}\right]\text{,}$$

$$a_{\text{n}}=n+\frac{1}{2}+H_{\varphi }+H_{\text{s}}{H}_{\text{tr,ϕ,s}}=\frac{\hslash }{4eDB{\tau }_{\text{tr,ϕ,s}}}{\Delta }_{\text{SO}}=\sqrt{\frac{2{\hslash }^2}{{\tau }_{\text{tr}}{\tau }_{\text{s}}}},$$

$$\frac{\Delta {\rho }_{\text{xx}}(T)}{{\rho }_{\text{xx,0}}(T)}\propto \frac{\alpha T}{\text{sinh}(\alpha T)},$$

Full text

Ballistic superconductivity and tunable $\pi$-junctions in InSb quantum wells

Chung Ting Ke

Equal contributions

QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands

  

Christian M. Moehle

Equal contributions

QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands

  

Folkert K. de Vries

QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands

  

Candice Thomas

Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

  

Sara Metti

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA

  

Charles R. Guinn

Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

  

Ray Kallaher

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

Microsoft Quantum at Station Q Purdue, Purdue University, West Lafayette, Indiana 47907, USA

  

Mario Lodari

QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands

  

Giordano Scappucci

QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands

  

Tiantian Wang

Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

  

Rosa E. Diaz

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

  

Geoffrey C. Gardner

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

Microsoft Quantum at Station Q Purdue, Purdue University, West Lafayette, Indiana 47907, USA

  

Michael J. Manfra

Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA

Microsoft Quantum at Station Q Purdue, Purdue University, West Lafayette, Indiana 47907, USA

School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907, USA

  

Srijit Goswami

S.Goswami@tudelft.nl

QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands

Two-dimensional electron gases (2DEGs) coupled to superconductors offer the opportunity to explore a variety of quantum phenomena. These include the study of novel Josephson effectsRiwar et al. (2016), superconducting correlations in quantum (spin) Hall systemsHart et al. (2014); Pribiag et al. (2015); Wan et al. (2015); Amet et al. (2016); Lee et al. (2017); Finocchiaro et al. (2018), hybrid superconducting qubitsCasparis et al. (2018); Wang et al. (2018) and emergent topological states in semiconductors with spin-orbit interaction (SOI)Rokhinson et al. (2012); Nichele et al. (2017); Ren et al. (2018); Fornieri et al. (2018). InSb is a well-known example of such a strong SOI semiconductor, however hybrid superconducting devices in InSb quantum wells remain unexplored. Here, we interface InSb 2DEGs with a superconductor (NbTiN) to create Josephson junctions (JJs), thus providing the first 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 Landé g-factor and gate tunability of the junctions to control the current-phase relation, and drive transitions between the [math] and $\pi$-states. This control over the free energy landscape allows us to construct a phase diagram identifying these [math] and $\pi$-regions, in agreement with theory. Our results establish InSb quantum wells as a promising new material platform to study the interplay between superconductivity, SOI and magnetism.

The combined effect of SOI and a Zeeman field is known to significantly alter the current-phase relation of JJsBezuglyi et al. (2002); Yokoyama et al. (2014); Szombati et al. (2016). In particular, one expects a complete reversal of the supercurrent (i.e., a $\pi$-JJ)Hart et al. (2017); Chen et al. (2018); Li et al. (2018) when the Zeeman and Thouless energy of the system become comparable. It was shown recently that such a $0\!-\!\pi$ transition in a 2D system is in fact accompanied by a topological phase transitionPientka et al. (2017); Hell et al. (2017); Fornieri et al. (2018); Ren et al. (2018). This, combined with the promise of creating scalable topological networksNayak et al. (2008); Karzig et al. (2017); Plugge et al. (2017), provides a strong motivation to study induced superconductivity in 2DEGs. Key requirements for the semiconductor include low disorder, large SOI and a sizable g-factor, combined with the ability to grow it on the wafer scale.

InSb satisfies all of these requirementsGilbertson et al. (2009); Kallaher et al. (2010); Nedniyom et al. (2009); Qu et al. (2016) and has emerged as a prime material candidate for engineering topological superconductivity, as evident from nanowire-based systemsMourik et al. (2012); Zhang et al. (2018). However, despite significant progress in the growth of InSb 2DEGsYi et al. (2015); Masuda et al. (2018), material challenges have prevented a systematic study of the superconducting proximity effect in these systems. We overcome these issues and reliably create JJs in high mobility InSb quantum wells, allowing us to realize ballistic $\pi$-junctions that can be controlled by magnetic and electric fields.

The JJs are fabricated in an InSb 2DEG wafer grown by molecular beam epitaxy, with a nominal electron density $n=2.7\cdot 10^{11}$ $\text{cm}^{-2}$ and mobility $\mu\approx~{}150,000~{}\text{cm}^{2}/\text{Vs}$, which corresponds to a mean free path $l_{\text{e}}\approx 1.3~{}\mu\text{m}$. Figure 1a shows a cross-sectional illustration and scanning electron micrograph of a typical JJ. Following a wet etch of the 2DEG in selected areas, NbTiN is deposited to create side-contacts to the 2DEG, thus defining a JJ of width $W$ and length $L$. Prior to sputtering NbTiN, an in-situ argon plasma cleaning of the exposed quantum well is performed in order to obtain good electrical contacts. A metal top-gate, 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 Supplementary Information (SI).

The junctions are measured using a quasi-four terminal current-biased circuit (Fig. 1a) at a temperature of 50 mK. We observe a clear supercurrent branch with zero differential resistance, $\text{d}V/\text{d}I$, followed by a jump to the resistive branch at switching current, $I_{\text{s}}$. In small perpendicular magnetic fields, $B_{\text{z}}$, Fraunhofer-like interference patterns are observed, as seen in Fig. 1b. The magnitude of supercurrent is controlled using the gate (Fig. 1c). Lowering the gate voltage, $V_{\text{g}}$, leads to a reduction of the electron density in the 2DEG and therefore to a suppression of $I_{\text{s}}$ and an increase in the normal state resistance, $R_{\text{n}}$. 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 is provided in the SI).

Studying JJs of varying lengths ($L=0.7-4.7~{}\mu\text{m}$), 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 $500~{}\text{nm}$ (see SI). In this limit the product of the critical current, $I_{\text{c}}$, and $R_{\text{n}}$ is proportional to the Thouless energyAltshuler and Spivak (1987), $E_{\text{Th}}=\hbar v_{\text{F}}l_{\text{e}}/2L^{2}$, where $v_{\text{F}}$ is the Fermi velocity in the 2DEG. Thus, for ballistic (diffusive) transport where $l_{\text{e}}=L$ ($l_{\text{e}}<L$), we expect $I_{\text{c}}R_{\text{n}}$ to scale as $1/L$ ($1/L^{2}$). Figure 1d shows $I_{\text{s}}R_{\text{n}}$ref (a) for a set of JJs. We find a $1/L$ scaling (black dots) indicative of ballistic superconductivity, with deviations only for the longer ($L\geq 2.7~{}\mu\text{m}$) junctions. Such a $1/L$ dependence up to micron-scale lengths has only recently been experimentally observed in clean graphene-based JJs Ben Shalom et al. (2016); Borzenets et al. (2016). To confirm the scaling arguments we also include data from a lower mobility wafer with $l_{\text{e}}\approx 0.5~{}\mu\text{m}$ (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.

Using these ballistic junctions, we now explore their response to a Zeeman field. 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, $E_{\text{Z}}\,=\,g\mu_{\text{B}}B$, leading to a shift in the Fermi surfaces ref (b) by $\pm\delta k/2$. Therefore, the Cooper pairs (electrons with opposite momentum and spin) possess a finite momentum, given by $\bm{k}_{\text{F}}\cdot\delta\bm{k}=E_{\text{Z}}(m^{*}/\hbar^{2})$, where $\bm{k}_{\text{F}}$ 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, $\Psi(\bm{r})\propto\cos(\delta\bm{k}\cdot\bm{r})$Bulaevskii et al. (1977); Demler et al. (1997); Buzdin (2005). Depending on the length of the Cooper pair trajectories, $|\bm{r}|$, the order parameter is either positive or negative, corresponding to the ground state of the JJ being at 0 or $\pi$ superconducting phase difference, respectively. This oscillation of the order parameter results in a modulation of the critical current $I_{\text{c}}\propto|\Psi|$, where a minimum of $I_{\text{c}}$ is expected whenever the order parameter switches signBezuglyi et al. (2002); Yokoyama et al. (2014). Taking only trajectories perpendicular to the contacts ($\delta\bm{k}\,=\,\delta k\hat{x}$, $\bm{k}_{\text{F}}=k_{\text{F}}\hat{x}$), a JJ with length $L$ will display minima in $I_{\text{c}}$ when $L\delta k\,=\,(2\text{N}+1)\pi/2$, 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 $E_{\text{Z}}=\pi E_{\text{Th}}$ giving:

[TABLE]

The $0\!-\!\pi$ 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.

We start by varying $B$, while $n$ (controllable by $V_{\text{g}}$) and $L$ remain fixed. Figure 2a shows the expected oscillation of $I_{\text{s}}$ with increasing $B_{\text{y}}$, displaying two distinct minima at $B_{\text{y}}=470~{}\text{mT}$ and $B_{\text{y}}=1250~{}\text{mT}$ (see SI for details about magnetic field alignment). This behavior is consistent with a magnetic field driven $0\!-\!\pi$ transition, as discussed above, where the first (second) minimum corresponds to a transition of the JJ state from [math] to $\pi$ ($\pi$ to [math]). 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 SI) allows us to conclusively rule out such a mechanism for the supercurrent modulation.

Next, we investigate how the length of the JJ influences $B_{0\!-\!\pi}$, the magnetic field at which the transition occurs. Figure 2b presents the $I_{\text{s}}$ oscillation for JJs with four different lengths, showing that $B_{0\!-\!\pi}$ is systematically reduced for increasing $L$. Plotting $B_{0\!-\!\pi}$ 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 $E_{\text{Th}}$, consistent with the conclusions from Fig. 1d. Finally, we check the dependence of the transition on the electron density. In Fig. 2c, we plot $I_{\text{s}}$ versus $B_{\text{y}}$ for different gate voltages using a JJ with $L=1.1~{}\mu\text{m}$. As $V_{\text{g}}$ is lowered, $B_{0\!-\!\pi}$ shifts to smaller values, again in qualitative agreement with Eq. 1. Interestingly, above a certain magnetic field the state of the JJ ([math] or $\pi$) becomes gate-dependent. For example at $B_{\text{y}}=400~{}\text{mT}$, the junction changes from a 0-JJ ($V_{\text{g}}=0~{}\text{V}$) to a $\pi$-JJ ($V_{\text{g}}=-0.4~{}\text{V}$), with a transition at $V_{\text{g}}=-0.2~{}\text{V}$. This indicates the feasibility of tuning the JJ into the $\pi$-state using gate voltages, while the magnetic field remains fixed.

These gate-driven transitions are demonstrated in Fig. 3a-d, which show a sequence of $I-V_{\text{g}}$ plots for increasing in-plane magnetic fields. At $B_{\text{y}}=250~{}\text{mT}$, $I_{\text{s}}$ displays a monotonic reduction with decreasing $V_{\text{g}}$. At a higher magnetic field, $B_{\text{y}}=325$ mT, $I_{\text{s}}$ reveals a markedly different behavior, whereby the supercurrent first decreases and then (at $V_{\text{g}}=-0.32~{}\text{V}$) shows a clear revival, indicative of a gate-driven $0\!-\!\pi$ transition, where the resonance condition ($E_{\text{Z}}=\pi E_{\text{Th}}$) is achieved by tuning the electron density. Increasing $B_{\text{y}}$ further, continuously moves the transition point to higher gate voltages (larger density), perfectly in line with expectations for a $0\!-\!\pi$ transition. Figure 3e shows two line-cuts from Fig. 3d. At zero current bias, $\text{d}V/\text{d}I$ shows a clear peak, indicative of a re-entrance of the supercurrent due to the the $0\!-\!\pi$ transition. However, at high bias, $\text{d}V/\text{d}I$ 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 Calado et al. (2015); Ben Shalom et al. (2016); Allen et al. (2017).

In contrast to the field-driven measurements (Fig. 2), controlling the transition with a gate avoids the need for time-consuming 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 $0\!-\!\pi$ phase diagram of the JJ. The combination of a high quality 2DEG and relatively long devices results in well defined magneto-resistance oscillations, allowing us to directly extract the electron density in the junction. Figure 4a shows the Landau fan diagram in perpendicular magnetic fields, $B_{\text{z}}$, from which we identify the filling factors, $\nu=nh/eB_{\text{z}}$ (Fig. 4b), and thereby obtain the $n$ vs. $V_{\text{g}}$ curve (Fig. 4c). We then plot all the transition points in Fig. 4d. The axes represent the two important energy scales in the system ($B_{\text{y}}\propto E_{\text{Z}}$ and $\sqrt{n}\propto E_{\text{Th}}$), thereby highlighting the [math] and $\pi$ 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 SI), $m^{*}=(0.022\pm 0.002)m_{\text{e}}$, and fit the data to a single free parameter, $g_{y}$ (the in-plane g-factor), giving $g_{y}=25\pm 3$ in good agreement with previous measurements on similar InSb quantum wells Qu et al. (2016).

Our work provides the first evidence of induced superconductivity in high quality InSb 2DEGs and demonstrates the creation of robust, gate-tunable $\pi$-Josephson junctions. We show that the $0\!-\!\pi$ transition can be driven both by magnetic fields and gate voltages. The significant region of phase space where the $\pi$-JJ is stable could prove advantageous in the study of topological superconductivity in planar JJs Pientka et al. (2017); Hell et al. (2017); Fornieri et al. (2018); Ren et al. (2018). 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 regimeFinocchiaro et al. (2018). Finally, the ability to control the ground state between [math] and $\pi$ states using gates is analogous to recent experimental results in ferromagnetic JJsGingrich et al. (2016), and could possibly serve as a semiconductor-based platform for novel superconducting logic applications Terzioglu and Beasley (1998). We therefore establish InSb 2DEGs as a new, scalable platform for developing hybrid superconductor-semiconductor technologies.

Acknowledgments: 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.

Author contributions: 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 co-authors. S. G. supervised the project.

Supplementary Information

Wafer growth and characterization

InSb-based 2DEGs were grown on semi-insulating GaAs (100) substrates by molecular beam epitaxy in a Veeco Gen 930 using ultra-high purity techniques and methods as described in Ref. 1. The layer stack of the heterostructure is shown in Fig. S1a. The growth has been initiated with a $100\,\text{nm}$ thick GaAs buffer followed by a $1\,\mu\text{m}$ thick AlSb nucleation layer. The metamorphic buffer is composed of a superlattice of $300\,\text{nm}$ thick In0.91Al0.09Sb and $200\,\text{nm}$ thick In0.75Al0.25Sb layers, repeated 3 times, and directly followed by a $2\,\mu\text{m}$ thick In0.91Al0.09Sb layer. The active region consists of a $30\,\text{nm}$ thick InSb quantum well and a $40\,\text{nm}$ thick In0.91Al0.09Sb top barrier. The Si $\delta$-doping layer has been introduced at $20\,\text{nm}$ from the quantum well and the surface. The In${}_{\text{x}}$Al${}_{1-\text{x}}$Sb buffer, the InSb quantum well and the In${}_{\text{x}}$Al${}_{1-\text{x}}$Sb setback were grown at a temperature of $440$ ℃ under a p(1x3) surface reconstruction. The growth temperature was lowered to $340$ ℃, where the surface reconstruction changed to c(4x4), just before the $\delta$-doping layer, to facilitate Si incorporation Liu et al. (1998). The scanning transmission electron micrograph of Fig. S1b reveals the efficiency of the metamorphic buffer to filter the dislocations.

The wafer is characterized by measuring the (quantum) Hall effect in a Hall bar geometry at $T=300~{}\text{mK}$. From a linear fit to the transversal resistance in a magnetic field range up to 1 T, we extract an electron density $n=2.71\cdot 10^{11}~{}\text{cm}^{-2}$, and by using the longitudinal resistivity at zero field, we obtain a mobility $\mu=146,400~{}\text{cm}^{2}/\text{Vs}$ (see Tab. S1). We calculate the corresponding mean free path to be $l_{\text{e}}=1.26~{}\mu\text{m}$. In Tab. S1, we also include $n$, $\mu$ and $l_{e}$ for the low mobility wafer, obtained from a quantum Hall measurement on this wafer. Data from the low mobility wafer is shown in Fig. 1d in the main text.

Device fabrication

The devices are fabricated using electron beam lithography. First, mesa structures of width $W$ and length $L$ 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 $\text{H}_{2}\text{O}_{2}$ and 7 ml $\text{H}_{3}\text{PO}_{4}$, and etch for 5 min, which results in an etch depth around $150~{}\text{nm}$. This is followed by the deposition of superconducting contacts in an ATC 1800-V 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 $200~{}\text{nm}$. Next, a $45~{}\text{nm}$ thick layer of $\text{AlO}_{\text{x}}$ dielectric is added by atomic layer deposition at 105 ℃, followed by a top-gate consisting of 10 nm/170 nm of Ti/Au.

Multiple Andreev reflections and excess current

To further characterize the superconductivity in our JJs, we study multiple Andreev reflections (MAR) in a representative JJ, by measuring its differential resistance, $\text{d}V/\text{d}I$, as a function of applied bias voltage, $V$. In Fig. S2a, we observe three dips in $\text{d}V/\text{d}I$, the first, at $2\Delta$, corresponding to the coherence peaks of the superconducting density of states, and two MAR peaks at $2\Delta/2$ and $2\Delta/3$. From these peaks we extract an induced superconducting gap $\Delta=0.9~{}\text{meV}$. In addition, we estimate the transparancy of the same JJ by measuring its excess current, $I_{\text{exc}}$, and normal state resistance, $R_{\text{n}}$. This measurement is shown in Fig. S2b, where we perform a linear fit in the high bias region of the $I-V$ curve ($V>2\Delta$) and obtain $I_{\text{exc}}=9~{}\mu\text{A}$ and $R_{\text{n}}=50~{}\Omega$. Using the OBTK model Flensberg et al. (1988), we find a value of 0.62 for the transparency of the JJ.

Weak anti-localization and spin-orbit interaction energy

To obtain an estimate of the typical energy scale associated with the spin-orbit interaction, we performed weak anti-localization (WAL) measurements. We use a Hall bar device (inset Fig. S3) fabricated on the high mobility wafer (Fig. S1), and apply magnetic field perpendicular to the Hall bar. The measurement in Fig. S3 reveals the typical WAL peak around zero field. This peak is caused by suppression of coherent backscattering at small magnetic fields due to the spin-orbit interaction. As we expect the Dyakonov Perel scattering mechanism to be dominating in our high mobility wafer, we use the theory developed by Iordanskii, Lyanda-Geller and Pikus Iordanskii et al. (1994) to fit the data:

[TABLE]

where $\Psi$ is the Digamma function, $C$ the Euler constant, and

[TABLE]

with $D=v_{\text{F}}l_{e}/2$, and \tau_{\text{tr,\phi,s}} the scattering times for elastic, inelastic and spin-orbit scattering, respectively. We find a spin-orbit energy splitting at the Fermi level ($\Delta_{\text{SO}}$) of 0.93 meV. The Rashba spin-orbit parameter of $\alpha=36~{}\text{meV}\text{\AA}$ is calculated following $\alpha=\Delta_{\text{SO}}/k_{\text{F}}$, where $k_{F}$ is deduced from a classical Hall measurement. Finally, we compare $\Delta_{\text{SO}}$ to the Zeeman energy. For a Landé g-factor of 25, $\Delta_{\text{SO}}\,>\,E_{\text{Z}}$ up to 640 mT. We are therefore in the spin-orbit dominated regime for the $0\!-\!\pi$ transition.

Magnetic field alignment

To ensure we are sweeping the magnetic field in the plane of the JJs only, we characterize the misalignment of our vector magnet axes, $B^{\prime}_{\text{y}}$ and $B^{\prime}_{\text{z}}$, used to apply the magnetic field in-plane and out-of-plane of the JJ, $B_{\text{y}}$ and $B_{\text{z}}$. In Fig. S4a we present a systematic measurement of the Fraunhofer interference pattern induced by $B^{\prime}_{\text{z}}$ with increasing $B^{\prime}_{\text{y}}$. We track the magnetic field at which the central lobe reaches its maximum $I_{\text{s}}$, $B^{\prime}_{\text{z,max}}$ and plot this for all $B^{\prime}_{\text{y}}$ in Fig. S4b. The linear dependence observed, represents a small misalignment angle of $\theta=1.4^{\circ}$. We take this angle into account when sweeping the in-plane field, $B_{\text{y}}=\cos(\theta)B^{\prime}_{\text{y}}+\sin(\theta)B^{\prime}_{\text{z}}$, and disregard it for the out-of-plane direction, $B_{\text{z}}=B^{\prime}_{\text{z}}$.

In-plane interference considerations

We observe a switching current, $I_{\text{s}}$ modulation in a JJ with $L=1.1$ $\mu$m, with minima at 470 mT and 1250 mT, which are attributed to Zeeman induced $0\!-\!\pi$ transitions. One might be inclined to believe that this modulation is caused by an in-plane Fraunhofer interference effect, due to the finite thickness ($d=30\,\text{nm}$) of the InSb quantum well. The $I_{s}$ minima of such a Fraunhofer pattern are expected to occur at $B_{\text{node}}=\text{N}\Phi_{0}/A$, where $\Phi_{0}$ is the magnetic flux quantum, $A=d\cdot L$ is the cross-sectional 2DEG area and $\text{N}=1,2,3,...$. The second minimum should thus occur at twice the value of the first, which is not the case here. Moreover, based on the estimated cross-sectional area of the JJ (see Fig. S5), one would expect the first node to be at 60 mT, inconsistent with the observation. In fact, it has been shown Monaco et al. (2009); Chiodi et al. (2012) that a oscillatory interference pattern is not expected at all in such an SNS junction with $L\gg d$. Finally, for an in-plane interference effect one expects the $B$ value at which the $I_{s}$ minima occur to increase for more negative gate voltages, since the wavefunction is then squeezed and $d$ effectively reduced. However, we observe the opposite behavior (i.e., the minima move to lower $B$), as expected for Zeeman-induced $0\!-\!\pi$ transitions. To conclude, we rule out an in-plane interference effect as a possible explanation for the supercurrent modulation.

Additional gate-driven $0\!-\!\pi$ transitions

Here, we present additional data of the gate-driven $0\!-\!\pi$ transitions in the JJ with $L=1.1~{}\mu\text{m}$. The gate voltages of the $0\!-\!\pi$ transitions presented in the phase diagram are extracted from the plots in Fig. S6.

Error analysis for gate-driven $0\!-\!\pi$ transitions

To systematically extract the value where gate-driven $0\!-\!\pi$ transition occurs and its error, we use a fit of the linetraces from Fig. S6, at zero $I$. At the transition point, a peak in $\text{d}V/\text{d}I$ indicates the $0\!-\!\pi$ transition. As an example, we show a single linetrace at $310$ mT in Fig. S7a, and extract the standard deviation, $\sigma$, based on a Gaussian fit of the peak. Subsequently, we used the gate to density mapping to convert $\sigma$ to the error bar shown in the phase diagram. This fitting procedure is used for all magnetic fields (Fig. S7b).

Effective mass measurement

To extract the effective mass of the electrons in the InSb 2DEG, the temperature dependence of the Shubnikov-de Haas (SdH) oscillation amplitude is measured in a Hall bar geometry. Figure S8a shows the magnetoresistance oscillations after the subtraction of a polynomial background, $\Delta\rho_{\text{xx}}$, as a function of filling factor, $\nu$, for temperatures ranging from $T=1.73~{}\text{K}$ to $T=10~{}\text{K}$. At a fixed filling factor, the effective mass, $m^{*}$, can be obtained from a fit to the damping of the SdH oscillation amplitude with increasing temperature, using the expression

[TABLE]

where $\rho_{\text{xx,0}}(T)$ is the temperature-dependent low-field resistivity and $\alpha=\pi k_{\text{B}}m^{*}\nu/(\hbar^{2}n)$. Figure S8b shows such fits to the oscillation minima and maxima of $\nu=10$ and $\nu=12$, resulting in a mean effective mass of $m^{*}=(0.022\pm 0.002)\cdot m_{e}$, with $m_{e}$ being the free electron mass.