# InSbAs two-dimensional electron gases as a platform for topological
superconductivity

Christian M. Moehle∗ QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GA Delft, The Netherlands  Chung Ting Ke∗
QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
2600 GA Delft, The Netherlands  Qingzhen Wang  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  Di Xiao  Department of Physics and
Astronomy, Purdue University, West Lafayette, Indiana 47907, USA  Birck
Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA
Saurabh Karwal  QuTech and Netherlands Organization for Applied Scientific
Research (TNO), 2628 CK Delft, The Netherlands  Mario Lodari  QuTech and Kavli
Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The
Netherlands  Vincent van de Kerkhof  QuTech and Kavli Institute of
Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
Ruben Termaat  QuTech and Kavli Institute of Nanoscience, Delft University of
Technology, 2600 GA Delft, The Netherlands  Geoffrey C. Gardner  Microsoft
Quantum Purdue, Purdue University, West Lafayette, Indiana 47907, USA
Giordano Scappucci  QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GA Delft, The Netherlands  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
Purdue, Purdue University, West Lafayette, Indiana 47907, USA  School of
Materials Engineering, Purdue University, West Lafayette, Indiana 47907, USA
Srijit Goswami+ QuTech and Kavli Institute of Nanoscience, Delft University of
Technology, 2600 GA Delft, The Netherlands

###### Abstract

+ E-mail<EMAIL_ADDRESS>

Topological superconductivity can be engineered in semiconductors with strong
spin-orbit interaction coupled to a superconductor. Experimental advances in
this field have often been triggered by the development of new hybrid material
systems. Among these, two-dimensional electron gases (2DEGs) are of particular
interest due to their inherent design flexibility and scalability. Here we
discuss results on a 2D platform based on a ternary 2DEG (InSbAs) coupled to
in-situ grown Aluminum. The spin-orbit coupling in these 2DEGs can be tuned
with the As concentration, reaching values up to $400$ $\mathrm{meV\AA}$, thus
exceeding typical values measured in its binary constituents. In addition to a
large Landé g-factor $\sim$ 55 (comparable to InSb), we show that the clean
superconductor-semiconductor interface leads to a hard induced superconducting
gap. Using this new platform we demonstrate the basic operation of phase-
controllable Josephson junctions, superconducting islands and quasi-1D
systems, prototypical device geometries used to study Majorana zero modes.

Topological phases of matter are currently a subject of intense research.
Following early theoretical proposals [1, 2], materials with large spin-orbit
interaction (such as InAs and InSb) coupled to superconductors have emerged as
a promising platform to engineer topological superconductivity in the form of
Majorana zero modes (MZMs). In this context, one-dimensional nanowires have
been studied extensively over the years [3, 4, 5]. More recently, several
efforts have been focused on engineering MZMs in two-dimensional electron
gases (2DEGs). Not only do 2DEGs provide a scalable platform for future
development of topological qubits, but their inherent flexibility allows for
the realization of more complex devices. The versatility of the 2DEG platform
can be seen in the variety of experiments performed on quasi-1D structures
[6], superconducting islands [7], multi-terminal Josephson junctions (JJs)
[8], and phase-biased JJs [9, 10], all of which are promising architectures to
create topological systems. Many of these studies have been performed on InAs
2DEGs where it is possible to create a pristine interface between the
superconductor Aluminum (Al) and the 2DEG allowing for a strong
superconducting proximity effect [11, 12].

The InSb 2DEG is another appealing platform, primarily due to its
significantly larger g-factor and spin-orbit coupling. Whereas the former
allows the hybrid system to enter the topological regime at a lower magnetic
field, the latter is crucial in determining the topological gap that protects
the MZMs. These 2DEGs have recently been proximitized by ex-situ
superconductors [13], however there exist no reports of InSb-based hybrid
systems with in-situ grown superconductors. This could be related to the band
offset at the InSb-Al interface, which (unlike InAs) prevents an efficient
accumulation of charge carriers and hence induced superconductivity [14]. It
would thus be ideal to have a material system with the desirable properties of
both InAs and InSb.

In this work we explore such a new hybrid material: ternary (InSbAs) 2DEGs
coupled to in-situ grown Al. Using magneto-transport experiments we
demonstrate a large g-factor ($\sim$ 55) and exceptionally strong spin-orbit
coupling exceeding the values of either InAs or InSb. In addition, the
pristine semiconductor-superconductor interface leads to a hard induced
superconducting gap that is revealed by spectroscopy measurements.
Furthermore, using these ternary 2DEGs we demonstrate the stable operation of
prototypical devices studied in the context of MZMs: phase-controllable JJs,
superconducting islands, and quasi-1D structures. Our results show that
InSbAs/Al 2DEGs offer the combined advantages of their binary constituents and
are therefore a promising platform to realize topological superconductivity.

$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ 2DEGs with varying As concentration, $x$,
are grown by molecular beam epitaxy (MBE) on undoped, semi-insulating GaAs
(100) substrates (see Fig. 1a for a schematic of the layer stack). The growth
starts with a $100\text{\,}\mathrm{n}\mathrm{m}$ GaAs buffer layer, directly
followed by a $1\text{\,}\mu\mathrm{m}$ thick AlSb nucleation layer [15] and a
$4\text{\,}\mu\mathrm{m}$ thick $\mathrm{Al_{0.1}In_{0.9}Sb}$ layer. The
latter forms a closely matched pseudo-substrate for the
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ growth and the bottom barrier of the
quantum well [16]. The As concentration in the
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ is controlled by the growth temperature
and the As flux. In this study, heterostructures with $x=0$, 0.053, 0.080,
0.130, 0.140 and 0.240 are grown. The semiconductor growth is terminated by
the deposition of 2 monolayers (ML) InAs, serving as a screening layer to
prevent intermixing between the semiconductor structure and the
superconducting Al layer [17]. After the semiconductor growth, the
heterostructures are transferred under ultra-high vacuum to a second MBE
chamber to deposit $7\text{\,}\mathrm{n}\mathrm{m}$ of Al, using methods
described in [17]. Figure 1b displays a bright field scanning transmission
electron micrograph focusing on the Al/$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$
interface for $x=0.130$. The interface appears sharp with a slight change of
atomic contrast that is attributed to the relaxed InAs screening layer [17].
Further details about the growth process can be found in the Supplementary
Information (SI).

We characterize the semiconducting properties of the InSbAs 2DEGs by removing
the Al in the active device area to fabricate Hall bars. After the Al removal,
the 2DEG is etched in unwanted areas, followed by the deposition of a
$\mathrm{SiN}_{x}$ dielectric layer. Lastly, a Ti/Au top-gate is evaporated
and used to control the electron density in the 2DEG (see Fig. 1c for a
schematic). We find peak mobilities of
$20\,000\text{\,}\mathrm{-}$$28\,000\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$
(see SI for mobility-density curves, and further details about the device
fabrication).

Figure 1: Hybrid Al/InSbAs heterostructures. a, Layer stack of the
Al-$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ hybrid heterostructures. b, Bright-
field scanning transmission electron micrograph of the
Al-$\mathrm{InSb_{0.870}As_{0.130}}$ interface along the [110] zone axis of
the semiconductor. Red lines indicate the boundaries of the Aluminum. c,
Schematic of a Hall bar that is used to extract the 2DEG properties.

To study the spin-orbit coupling in these 2DEGs, we measure the longitudinal
conductivity, $\sigma_{\mathrm{xx}}$, in perpendicular magnetic fields,
$B_{\perp}$, at $300\text{\,}\mathrm{m}\mathrm{K}$ using standard lock-in
techniques. The simultaneously measured transversal Hall resistance allows us
to deduce the density, $n$, in the 2DEG at every gate voltage,
$V_{\mathrm{g}}$. Figure 2a shows the magneto-conductivity correction,
$\Delta\sigma_{\mathrm{xx}}(B_{\perp})=\sigma_{\mathrm{xx}}(B_{\perp})-\sigma_{\mathrm{xx}}(0)$,
at $V_{\mathrm{g}}=$0\text{\,}\mathrm{V}$$ for $\mathrm{\mathrm{x}=0}$, 0.053,
0.130 and 0.240, where the individual curves are offsetted for clarity. We
observe clear weak anti-localization (WAL) peaks that are caused by the
suppression of coherent backscattering due to spin-orbit coupling. Since we
expect the linear Rashba term to be the dominating spin-orbit contribution in
these asymmetric quantum wells, we fit the WAL peaks with the Rashba-dominated
Iordanskii, Lyanda-Geller, and Pikus (ILP) model [18, 19] (grey curves in Fig.
2a). This allows us to extract the spin-orbit length,
$l_{\mathrm{so}}=\sqrt{D\tau_{\mathrm{so}}}$, where
$D=v_{\mathrm{F}}l_{\mathrm{e}}/2$ is the diffusion constant. Here,
$\tau_{\mathrm{so}}$ is the spin-orbit scattering time, $v_{\mathrm{F}}$ the
Fermi velocity and $l_{\mathrm{e}}$ the mean free path. The linear Rashba
parameter is given by $\alpha=\Delta_{\mathrm{so}}/2k_{\mathrm{F}}$, where
$\Delta_{\mathrm{so}}=\sqrt{2\hbar^{2}/\tau_{\mathrm{so}}\tau_{\mathrm{e}}}$
is the spin-split energy, $k_{\mathrm{F}}$ the Fermi wave vector, and
$\tau_{\mathrm{e}}$ the elastic scattering time.

Figure 2: Large and tuneable spin-orbit coupling. a, Magneto-conductivity
correction at $V_{\mathrm{g}}=$0\text{\,}\mathrm{V}$$ for the different
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ 2DEGs. The $x=0$ curve is measured at
$V_{\mathrm{g}}=$0.2\text{\,}\mathrm{V}$$ due to a high resistance at
$V_{\mathrm{g}}=$0\text{\,}\mathrm{V}$$. The gray lines are ILP fits to the
weak anti-localization data. In the inset the extracted linear Rashba
coefficient is plotted for the four As concentrations, showing a monotonic
increase with increasing As concentration. For the higher As concentrations,
$\alpha$ is 3-4 times larger as compared to the value for pure InSb ($x=0$).
b, Spin-orbit length plotted against $V_{\mathrm{g}}$. c, $\alpha$ as a
function of $V_{\mathrm{g}}$. $l_{\mathrm{so}}$ ($\alpha$) decreases
(increases) with increasing As concentration when comparing at a fixed gate
voltage.

As shown in Fig. 2a, the ILP model fits the experimental data well. The
resulting linear Rashba parameter for the four different As concentrations is
plotted in the inset. It is striking that $\alpha$ increases monotonically
with increasing As concentration. Compared to the value for pure InSb
($\alpha\approx$100\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\AA$$), it is
noteworthy that for the higher As concentrations, the linear Rashba parameter
is 3-4 times larger ($300-400$ $\mathrm{meV\AA}$). We proceed by measuring WAL
as a function of gate voltage ($V_{\mathrm{g}}$) for the different As
concentrations. In Fig. 2b and 2c we show $l_{\mathrm{so}}$ and $\alpha$
plotted against $V_{\mathrm{g}}$ (see SI for the same plots as a function of
electron density). The trend of decreasing (increasing) $l_{\mathrm{so}}$
($\alpha$) with increasing As concentration persists also when comparing at
other gate voltages. We note, however, that specifically for $x=0.240$, the
spin-orbit coupling becomes so large that $l_{\mathrm{so}}$ is smaller than
$l_{\mathrm{e}}$. This might lead to inaccuracies in the extracted fit
parameters as the ILP model is valid when $l_{\mathrm{e}}$ is the smallest
length scale.

The systematic increase in spin-orbit coupling with As concentration can arise
from a combination of several effects. Firstly, bandstructure calculations of
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ show that the Rashba parameter is
strongly influenced by the As concentration [20, 21], which has been observed
in experiments on ternary nanowires [22]. Secondly, electric fields across the
2DEG can also influence the spin-orbit interaction. We note that even at
$V_{\mathrm{g}}=$0\text{\,}\mathrm{V}$$, $\alpha$ increases monotonically with
$x$, suggesting that the external electric field from the applied gate voltage
is not the primary source of the enhancement. However the internal field
(generated at the 2DEG-gate dielectric interface) could be a strong function
of the As concentration. We indeed observe that the nominal density in the
2DEG (density at $V_{\mathrm{g}}=$0\text{\,}\mathrm{V}$$) increases
systematically with As concentration (see SI), indicating a stronger downward
band bending at the dielectric-semiconductor interface resulting in an
increased electric field. While from these studies it is difficult to
disentangle the effects of the bulk semiconductor from the interfaces, similar
experiments on deep InSbAs quantum wells would shed more light on the origins
of the enhanced spin-orbit interaction.

Having established strong spin-orbit coupling in
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ 2DEGs, we proceed by measuring the
perpendicular g-factor, $g^{*}$. By comparing the temperature dependence of
Shubnikov-de Haas oscillations for an odd-even filling factor couple, an
expression for $g^{*}$ can be obtained [23]. For $x=0$ we find $g^{*}=47.8\pm
2.8$. The same analysis is done for $x=0.130$, where we obtain $g^{*}=54.6\pm
3.1$ (see SI for details about the data analysis, as well as effective mass
measurements for the two As concentrations). This shows that besides strong
spin-orbit coupling, InSbAs 2DEGs also possess a large g-factor that is
comparable to the one of pure InSb.

Given that the semiconducting properties of InSbAs 2DEGs are favorable to
realize topological phases, we now demonstrate that these 2DEGs also have
excellent coupling to Al. To do so, we use devices as shown in the false-
colored scanning electron micrograph (SEM) in Fig. 3a. This device can either
be operated as a gate-tunable JJ (gray circuit) or a spectroscopy device
(black circuit) to measure the local density of states. All devices are
measured at a temperature of $30\text{\,}\mathrm{m}\mathrm{K}$.

Figure 3: Hard induced superconducting gap. a, False-colored SEM of a combined
JJ and tunneling spectroscopy device. A cross-sectional schematic the JJ part
is shown in the bottom. All devices have a fixed JJ length of
$L\approx$150\text{\,}\mathrm{n}\mathrm{m}$$. b, Differential resistance as a
function of applied current bias and perpendicular magnetic field for the
$x=0.053$ JJ. The Fraunhofer interference pattern signifies a uniform current
distribution in the JJ. c, Differential resistance as a function of applied
current bias and gate voltage for the same JJ, showing that the switching
current, $I_{\mathrm{s}}$, can be fully suppressed. d, Differential
conductance as a function of splitgate voltage and applied voltage bias for
$x=0.053$ and $x=0.130$. The color scale has been saturated to increase the
visibility of the tunneling regime. For both As concentrations, the induced
superconducting gap is visible (region of suppressed conductance), stable over
a large range in $V_{\mathrm{g,S}}$. e, Linecuts at the indicated positions
(gray markers) on linear and logarithmic scale. The size of the induced gap is
similar for both As concentrations. The in-gap conductance is suppressed by
2-3 orders of magnitude as compared to the out-of gap conductance, indicating
a hard induced gap.

It is important to note that we do not see any induced superconductivity for
pure InSb, presumably due to an unfavorable band alignment at the Al-InSb
interface. In stark contrast, all the InSbAs 2DEGs (irrespective of As
concentration) have excellent coupling to the superconductor, where all JJs
display supercurrents and pronounced multiple Andreev reflections (see SI). In
Fig. 3b we show a representative Fraunhofer interference pattern for the JJ
with $x=0.053$, where the black regions correspond to zero resistance. The
size of the switching current, $I_{\mathrm{s}}$, can be controlled by the gate
voltage, as demonstrated in Fig. 3c for the same As concentration. Upon
lowering $V_{\mathrm{g}}$, $I_{\mathrm{s}}$ shrinks and correspondingly, the
normal-state resistance of the JJ, $R_{\mathrm{n}}$, increases.

Clean and transparent interfaces are crucial for the realization of MZMs as
they allow the proximitized semiconductor to obtain a hard induced
superconducting gap with a vanishing in-gap density of states. It is therefore
important to measure the density of states in the proximitized 2DEG directly.
To this end, we operate the devices shown in Fig. 3a as spectroscopy devices.
We apply a voltage bias to the left normal contact, and measure the current
flowing through the Al lead, while energizing the splitgates with a negative
voltage, $V_{\mathrm{g,S}}$, to create a barrier. In the tunneling regime the
measured conductance is proportional to the density of states in the
proximitized 2DEG. In Fig. 3d we present tunneling spectroscopy maps for
$x=0.053$ and $x=0.130$. In both measurements we note the emergence of a
region with suppressed conductance, reflecting the induced superconducting
gap. The gaps persist over an extended range in $V_{\mathrm{g,S}}$ with a
slight dependence of the gap size on the out-of-gap conductance. In Fig. 3e we
show two representative linecuts in the tunneling regime. We find that the
size of the induced gap is similar for both As concentrations
($\Delta^{*}\approx$220\text{\,}\mu\mathrm{e}\mathrm{V}$$). Turning to the
linecuts on the logarithmic scale and comparing the out-of-gap conductance
with the in-gap conductance, we see that the in-gap conductance is suppressed
by 2-3 orders of magnitude for both As concentrations. This confirms the
excellent Al-2DEG interface.

Figure 4: Prototypical Majorana devices. a, False-colored SEMs of a phase-
biased JJ before (left) and after (right) gate deposition. The split gates at
the top and bottom edge of the JJ are used to perform tunneling spectroscopy
at both ends of the JJ. This device is fabricated on a 2DEG with $x=0.080$. b,
Differential conductance as a function of voltage bias applied to the top
contact and perpendicular magnetic field while the bottom tunneling contact is
floating (top panel). The bottom panel shows the tunneling spectroscopy map as
a function of bottom voltage bias and perpendicular magnetic field (top
tunneling contact floating). For both measurements the splitgates are fixed at
$V_{\mathrm{g,S1}}=$-2.36\text{\,}\mathrm{V}$$,
$V_{\mathrm{g,S2}}=$-1.36\text{\,}\mathrm{V}$$ and
$V_{\mathrm{g,S3}}=V_{\mathrm{g,S4}}=$-1.48\text{\,}\mathrm{V}$$. c, Linecuts
from b at the indicated positions. d, SEMs of a superconducting island before
(top) and after (bottom) gate deposition, fabricated on a 2DEG with $x=0.140$.
The two tunnel gates are used to tune the transmission between the leads and
the island. The central gate depletes the surrounding 2DEG and changes the
charge occupancy of the island. e, Differential conductance at fixed tunnel
gate voltages ($V_{\mathrm{g,T1}}=$-2.157\text{\,}\mathrm{V}$$,
$V_{\mathrm{g,T2}}=$-2.052\text{\,}\mathrm{V}$$) as a function of applied
voltage bias and central gate voltage. 2e-periodic Coulomb oscillations are
visible at $V=$0\text{\,}\mathrm{m}\mathrm{V}$$ and 1e-periodic Coulomb
oscillations at high biases. f, Linecuts from e at the indicated positions. g,
SEMs of a quasi-1D grounded superconducting strip before (top) and after
(bottom) gate deposition. The left gate is used to create a tunnel barrier
between the bulk Al contact and the superconducting strip, and the central
gate depletes the surrounding 2DEG. This device is fabricated on a 2DEG with
$x=0.140$. h, Differential conductance at fixed gate voltages
($V_{\mathrm{g,T}}=$-3.96\text{\,}\mathrm{V}$$,
$V_{\mathrm{g,C}}=$-4.40\text{\,}\mathrm{V}$$) as a function of applied
voltage bias and parallel magnetic field. i, Linecuts from h at the indicated
positions.

Using the InSbAs/Al hybrid platform we realize three prototypical device
architectures used to study MZMs: phase-controllable JJs, superconducting
islands and quasi-1D superconducting strips. Figure 4a shows false-colored
SEMs of a JJ embedded in a superconducting loop. A perpendicular magnetic
field, $B_{\perp}$, penetrating through the loop can be used to tune the phase
difference across the JJ. Split gates are positioned at the top and bottom
edge of the JJ to perform tunneling spectroscopy at both ends of the phase-
controllable JJ. In Fig. 4b we show spectroscopy maps at the top (top panel)
and bottom (bottom panel) of the JJ, with representative linecuts shown in
Fig. 4c. In both cases we observe a clear flux-dependent modulation of the
gap, consistent with the phase modulation expected for Andreev bound states.
Such three-terminal devices are important to check for correlations between
the two ends of the JJ, when tuned into the topological regime.

A second approach to create MZMs is based on a floating narrow strip of
superconductor, a superconducting island (see Fig. 4d). Here two tunnel gates
are used to tune the transmission between the bulk Al contacts and the island.
The central gate depletes the 2DEG in areas that are not covered with Al, and
changes the charge occupancy on the island. In Fig. 4e we show a differential
conductance map at fixed tunnel gate voltages, $V_{\mathrm{g,T}}$, varying the
applied voltage bias, $V$, and the central gate voltage, $V_{\mathrm{g,C}}$.
At $V=$0\text{\,}\mathrm{m}\mathrm{V}$$ (see also gray linecut in Fig. 4f) we
observe Coulomb peaks that have twice the spacing as compared to the Coulomb
peaks at high biases (black linecut in Fig. 4f). The Coulomb peaks at
$V=$0\text{\,}\mathrm{m}\mathrm{V}$$ reflect 2e-periodic Cooper pair transport
through the superconducting island, while at high biases, quasiparticles with
1e charge are allowed to tunnel through the island.

Another strategy to study MZMs employs a grounded narrow strip of
superconductor (see Fig. 4g). The narrow gate is used to define a tunnel
barrier between the bulk Al contact and the quasi-1D strip, and the central
gate depletes the remaining exposed 2DEG. In Fig. 4h we present a tunneling
spectroscopy map at fixed gate voltages as a function of applied voltage bias,
$V$, and magnetic field, $B_{\parallel}$. $B_{\parallel}$ is oriented along
the superconducting finger. Two representative linecuts are shown in Fig. 4i.
Whereas at $B_{\parallel}=$0\text{\,}\mathrm{T}$$ we observe a hard induced
superconducting gap (note that the gap size is doubled due to the
superconducting contact), a zero energy state emerges around
$B_{\parallel}=$0.7\text{\,}\mathrm{T}$$. While these results are promising,
further experiments are required to comment on the origin of these states.

In conclusion, we have shown that InSbAs 2DEGs offer the combined advantages
of the more commonly studied binary materials. In addition to a large
g-factor, they have excellent coupling to in-situ grown Aluminum. Furthermore,
the spin-orbit coupling in these ternary 2DEGs is significantly stronger than
in either InAs or InSb. Using this hybrid system, we realize distinct device
architectures that can be used to study topological superconductivity.

Data availability
Raw data and analysis scripts for all presented figures are available at the
4TU.ResearchData repository, DOI: https://doi.org/10.4121/14565228

Author information
Corresponding author
+E-mail<EMAIL_ADDRESS>

Author contributions
∗C. M. M. and C. T. K contributed equally, fabricated and measured the devices
together with Q. W., S. K. V. vd K. and R. T. The MBE growth of the
semiconductor heterostructures and the characterization of the materials was
performed by C. T., D. X. and G. C. G. under the supervision of M. J. M. The
effective mass and g-factor measurements were provided by M. L. and G. S. The
manuscript was written by C. M. M., C. T. K and S. G., with inputs from all
co-authors. S. G. supervised the experimental work in Delft.

Notes
The authors declare no competing interests.

Acknowledgments
We thank Michael Wimmer and Michal P. Nowak for useful discussions. Moreover
we would like to thank J.H. Dycus, M.E. Salmon and R.E. Daniel from Eurofins
EAG Materials Science for the TEM studies. The research at Delft was supported
by the Dutch National Science Foundation (NWO), the Early Research Programme
of the Netherlands Organisation for Applied Scientific Research (TNO) and a
TKI grant of the Dutch Topsectoren Program. The work at Purdue was funded by
Microsoft Quantum.

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## Supplementary Information

##  S1 Wafer growth and characterization

Figure S1: a, Asymmetric (115) reciprocal space map for the
$\mathrm{InSb_{0.760}As_{0.240}}$ quantum well heterostructure. b, High-
resolution x-ray diffraction $2\theta-\omega$ scans obtained along (004) for
the $\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ quantum well heterostructures with
different As concentrations.

InSbAs quantum well heterostructures are grown in a Veeco Gen 930 molecular
beam epitaxy (MBE) system equipped with valved crackers for As and Sb, and
effusion cells for Ga, In, and Al. The growths are performed on GaAs (100)
substrates with a $0.5\text{\,}\mathrm{\SIUnitSymbolDegree}$ miscut toward the
(111)B crystalline direction to minimize surface roughness [1]. The growth is
initiated with a $100\text{\,}\mathrm{n}\mathrm{m}$ thick GaAs buffer layer,
directly followed by a $1\text{\,}\mu\mathrm{m}$ thick AlSb layer and a
$4\text{\,}\mu\mathrm{m}$ thick $\mathrm{Al_{0.1}In_{0.9}Sb}$ layer that also
serves as bottom barrier for the InSbAs quantum wells. The AlSb and AlInSb
layers help accommodate the large lattice mismatch between the GaAs substrate
and InSbAs by forming a pseudo-substrate on which InSbAs can be coherently
strained.

The growth rate for all the semiconducting layers is kept at roughly
$0.5\text{\,}\mathrm{M}\mathrm{L}\mathrm{/}\mathrm{s}$, under group V element
rich conditions. The incorporation of As and the resulting alloy content of
the InSbAs quantum well is controlled by the growth temperature and the Sb/As
flux ratio. The growth temperature is monitored by a BandiT spectrometer
through blackbody radiation fitting. For the $x=0.053$, 0.130 and 0.240
samples, the growth temperatures were
$445\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$477\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$483\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. While the Sb flux
was kept around
$4.2\text{\times}{10}^{-7}\text{\,}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$
for all three samples to maintain group V overpressure, the As flux changed
from
$4.2\text{\times}{10}^{-6}\text{\,}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$
($x=0.053$ and 0.130) to
$1.1\text{\times}{10}^{-5}\text{\,}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$
($x=0.240$). The semiconductor growth terminates under As rich conditions with
the epitaxy of 2 ML InAs to prevent interdiffusion between the quantum well
and the superconducting layer on top [2].

The heterostructures are then transferred under ultra-high vacuum to a Veeco
620 MBE system equipped with an Al effusion cell, and a quartz crystal monitor
to determine the growth rate. Inside this MBE chamber, a moveable cryocooler
is used to contact and cool the wafers down to liquid nitrogen temperature
within a few hours [2]. After 6 hours of cooling, a 7 nm thin layer of Al is
deposited on the semiconductor heterostructure with a typical growth rate of
$0.3\text{\,}\AA\mathrm{/}\mathrm{s}$. Immediately after the Al deposition,
the samples are transferred into another chamber where they are oxidized for
15 min under an $\mathrm{O_{2}}$ pressure of
$5\text{\times}{10}^{-5}\text{\,}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$ to
stabilize the Al films [3].

High-resolution X-ray diffraction (HRXRD) measurements were performed to
evaluate the As concentration, $x$, of the
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ quantum wells and reciprocal space maps
were acquired to assess the strain of these layers. These measurements were
performed using a X’pert PANalytical diffractometer with a copper X-ray tube
operating at a wavelength $\lambda=$1.5406\text{\,}\AA$$. Asymmetric (115)
reciprocal space maps (averaged at 2 opposite azimuth angles
$\phi=$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$) confirm that the
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ layers are coherently strained for the
largest investigated composition $x=0.240$, as shown in Fig. S1a, where the
InSbAs peak is along the same red dashed vertical line as the
$100\text{\,}\mathrm{\char 37\relax}$ relaxed AlInSb peak. Similar reciprocal
space maps were obtained for the heterostructures with $x=0.053$ and 0.130,
confirming that InSbAs quantum wells are coherently and fully strained in the
investigated composition range. $2\theta-\omega$ scans are presented in S1b.
The peak positions of the coherently strained
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ layers give the As concentration for each
sample, using the 4 micron fully relaxed $\mathrm{Al_{0.1}In_{0.9}Sb}$ layer
as the substrate in the analysis.

##  S2 Device fabrication

The processing of Sb-based 2DEGs in proximity to Aluminum is more challenging
than InAs/Al systems. This is due to the potential intermixing of Al and Sb,
which becomes more severe at elevated temperatures. We therefore need to
ensure that all the processing steps (as described below) are performed at as
low a temperature as possible. This includes room temperature “baking” of the
resist for every lithography step, and low temperature atomic layer
deposition.

The devices presented in the main text are fabricated using electron beam
lithography. First, the Al and the 2DEG is etched in unwanted areas. The Al
etch is performed in Transene D etchant at a temperature of
$48.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $9\text{\,}\mathrm{s}$.
Subsequently, using the same PMMA mask, the 2DEG is etched in a wet etch
solution consisting of $560\text{\,}\mathrm{m}\mathrm{l}$ deionized water,
$9.6\text{\,}\mathrm{g}$ citric acid powder, $5\text{\,}\mathrm{m}\mathrm{l}$
$\text{H}_{2}\text{O}_{2}$ and $4\text{\,}\mathrm{m}\mathrm{l}$
$\text{H}_{3}\text{PO}_{4}$. We etch for
$2\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$, resulting in an etch depth around
$100\text{\,}\mathrm{n}\mathrm{m}$. A second Al etch is performed to define
the normal regions using Transene D etchant at a temperature of
$38.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $17\text{\,}\mathrm{s}$.
Next, a $60\text{\,}\mathrm{n}\mathrm{m}$ thick $\mathrm{SiN}_{x}$ dielectric
layer is sputtered for the Hall bars and the Josephson junction/spectroscopy
devices. For the superconducting island and the quasi-1D superconducting strip
we use a $40\text{\,}\mathrm{n}\mathrm{m}$ thick $\mathrm{AlO}_{x}$ dielectric
deposited by atomic layer deposition at
$40\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Finally, top-gates are deposited
by evaporating
$10\text{\,}\mathrm{n}\mathrm{m}$/$190\text{\,}\mathrm{n}\mathrm{m}$ of Ti/Au.

##  S3 Mobility and density

Figure S2: a, Electron density plotted against the gate voltage for all As
concentrations. At a fixed gate voltage, the electron density increases
monotonically with $x$. b, Electron mobility as a function of $n$.

The $\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ 2DEGs are characterized by measuring
the Hall effect in a gated Hall bar geometry at
$T=$300\text{\,}\mathrm{m}\mathrm{K}$$. From a linear fit to the transversal
resistance in a magnetic field range up to $\pm$0.5\text{\,}\mathrm{T}$$, we
extract the electron density, $n$, at every gate voltage, $V_{\mathrm{g}}$.
Figure S2a shows $n$ plotted against $V_{\mathrm{g}}$ for all As
concentrations, $x$. At a fixed gate voltage, the electron density increases
systematically with $x$. The nominal density,
$n(V_{\mathrm{g}}=$0\text{\,}\mathrm{V}$)$, increases from
$0.15\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$ for $x=0$, to
$1.35\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$ for $x=0.240$.
This suggests that the incorporation of As causes an increasing negative band
offset at the dielectric-2DEG interface.

Using the longitudinal resistivity, we calculate the mobility, $\mu$, at every
gate voltage. Figure S2b shows a plot of $\mu$ as a function of $n$ for all As
concentrations. We observe peak mobilities between
$20\,000\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$ and
$28\,000\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$ (the
peak mobility for $x=0.240$ could not be reached because of gate leakage). The
peak mobilities occur around
$n=$0.35\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$, with a
slight trend of moving towards higher densities for higher As concentrations.
We attribute the peak mobilities to the de-population of the second subband.

##  S4 Effect of electron density on spin-orbit coupling

Figure S3: a, Spin-orbit length plotted against the electron density for all
As concentrations. b, Linear Rashba parameter plotted against $n$. When
comparing at the same density, $l_{\mathrm{so}}$ ($\alpha$) decreases
(increases) for increasing As concentration.

In Fig. 2 of the main text we show the spin-orbit length, $l_{\mathrm{so}}$,
and the linear Rashba paramter, $\alpha$, plotted against the gate voltage for
all As concentrations, $x$. In Fig. S3a and b we show $l_{\mathrm{so}}$ and
$\alpha$ plotted against the electron density, $n$, respectively. When
comparing at the same density, the overall trend of decreasing (increasing)
$l_{\mathrm{so}}$ ($\alpha$) with increasing As concentration persists.

##  S5 Effective mass and g-factor

Figure S4: a, Shubnikov-de Haas oscillations for temperatures ranging from
$4\text{\,}\mathrm{K}$ to $40\text{\,}\mathrm{K}$ for the $x=0.130$ sample at
a fixed density of
$n=$3.41\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$ close to
peak mobility. b, Temperature dependence of the oscillation amplitude for the
$\nu=6$ minima (circles) and maxima (rhombs) for $x=0$ and $x=0.130$. The
solid lines are fits to the data in order to obtain the effective mass. c,
g-factor, $g^{*}$, for $x=0$ and $x=0.130$ considering the even-odd filling
factor couple $\nu=4-3$ and $\nu=6-3$. The fits are performed in a high and
low temperature range.

We extract the effective mass, $m^{*}$, of the electrons in the
$\mathrm{InSb}_{1-x}\mathrm{As}_{x}$ 2DEGs by measuring the temperature
dependence of the Shubnikov-de Haas (SdH) oscillations amplitude in gated Hall
bars. Figure S4a shows the longitudinal resistivity, $\rho_{\mathrm{xx}}$, as
a function of perpendicular magnetic field, $B$, for $x=0.130$ at a fixed
density of $n=$3.41\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$,
close to peak mobility. The temperature ranges from $T=$
$4\text{\,}\mathrm{K}40\text{\,}\mathrm{K}$. The same measurement is also done
for $x=0$ at
$n=$2.93\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$ (for $T=$
$1.7\text{\,}\mathrm{K}17.3\text{\,}\mathrm{K}$). After a polynomial
background subtraction, the effective mass can be obtained from a fit to the
thermal damping of the SdH oscillation amplitude, $\Delta\rho_{\mathrm{xx}}$,
normalized to the zero-field magnetoresistance value, $\rho_{xx,0}(T)$, at a
fixed filling factor, $\nu$, using [4]:

$\frac{\Delta\rho_{\mathrm{xx}}(T)}{\rho_{\mathrm{xx,0}}(T)}\propto\frac{\alpha
T}{\sinh(\alpha T)},$

Here, $\alpha=\pi k_{\mathrm{B}}m^{*}\nu/(\hbar^{2}n)$, where $k_{\mathrm{B}}$
is the Boltzmann constant and $\hbar$ is the Planck constant. Figure S4b shows
such fits for the $\nu=6$ minima (circles) and maxima (rhombs) for $x=0$ and
$x=0.130$. The resulting effective mass is lower for the ternary sample, with
a weighted mean value of $m^{*}=(0.0162\pm 0.0004)m_{0}$, where $m_{0}$ is the
free electron mass. The $x=0$ sample shows a heavier effective mass of
$m^{*}=(0.0180\pm 0.0002)m_{0}$.

We proceed by extracting the g-factor, $g^{*}$, from the temperature
dependence of the SdH oscillations, extending the temperature range from $T=$
$1.7\text{\,}\mathrm{K}44.7\text{\,}\mathrm{K}$. Using the method reported in
[5], an expression for $g^{*}$ can be obtained by combining the equations for
the thermal activation energy of an even-odd filling factor couple. We
consider the filling factor couples $\nu=4-3$ and $\nu=6-3$. Since we find
different activation energies from the linear fit of $\ln(\rho_{xx})$ vs.
$1/T$ depending on the temperature range, we report in Fig. S4c the extracted
values for the low- and high-temperature ranges. The $m^{*}$ values we found
earlier are used in the $g^{*}$ calculation. We find an average g-factor of
$g^{*}=47.8\pm 2.8$ for $x=0$, and $g^{*}=54.6\pm 3.1$ for $x=0.130$.

##  S6 Multiple Andreev reflections

Figure S5: Differential resistance plotted against the voltage drop between
the two Al leads for the JJs with $x=0.053$, 0.130 and 0.240. All measurements
are obtained at a JJ gate voltage of $V_{\mathrm{g}}=$0\text{\,}\mathrm{V}$$.
The curve for $x=0.130$ is offsetted for clarity. MAR up to several orders are
observed for all As concentrations.

In Fig. 3b and c of the main text we show a representative Fraunhofer
interference pattern and a gate dependence of the switching current for a
Josephson junction with $x=0.053$. As demonstrated in Fig. S5, we observe
pronounced subgap conductance modulations that are due to multiple Andreev
reflections (MAR) for all As concentrations. The observation of higher order
MAR shows that transport is phase-coherent across a length scale of several
times the junction length.

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