# Spatial dependence of local density of states in semiconductor-
superconductor hybrids

Qingzhen Wang  QuTech and Kavli Institute of Nanoscience, Delft University of
Technology, Delft, 2600 GA, The Netherlands  Yining Zhang  QuTech and Kavli
Institute of Nanoscience, Delft University of Technology, Delft, 2600 GA, The
Netherlands  Saurabh Karwal  QuTech and Netherlands Organization for Applied
Scientific Research (TNO), Delft, 2628 CK, The Netherlands  Srijit Goswami
<EMAIL_ADDRESS>QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, Delft, 2600 GA, The Netherlands

###### Abstract

Majorana bound states are expected to appear in one-dimensional semiconductor-
superconductor hybrid systems, provided they are homogenous enough to host a
global topological phase. In order to experimentally investigate the
uniformity of the system, we study the spatial dependence of the local density
of states in multiprobe devices where several local tunnelling probes are
positioned along a gate-defined wire in a two-dimensional electron gas.
Spectroscopy at each probe reveals a hard induced gap, and an absence of
subgap states at zero magnetic field. However, subgap states emerging at
finite magnetic field are not always correlated between different probes.
Moreover, we find that the extracted critical field and effective $g$-factor
of the lowest energy subgap state varies significantly across the length of
the wire. Upon studying several such devices we do however find examples of
striking correlations in the local density of states measured at different
tunnel probes. We discuss possible sources of variations across devices.

##  I Introduction

Majorana bound states (MBSs) obey non-Abelian exchange statistics and are
potential building blocks of topological qubits [1, 2]. In this context, one-
dimensional (1D) semiconductor-superconductor hybrids have been widely
studied, where a topological phase transition is accompanied by the emergence
of MBSs at the system edges [3, 4]. together with a closing and reopening of
the superconducting gap in the hybrid bulk [5]. Tunnelling spectroscopy
provides information about the local density of states (LDOS), and is often
used to search for signatures of MBS [6]. However, it has been suggested that
some of these observations could arise due to trivial reasons such as disorder
or inhomogeneity of the chemical potential [7, 8]. Strong local perturbations
would effectively segment the wire and thus prevent the creation of a global
topological phase. It has therefore become clear that a prerequisite for
reliably creating MBSs is spatial uniformity of microscopic properties across
the length of the 1D hybrid system. These include the chemical potential, the
induced superconducting gap and the effective $\mathit{g}$-factor.

Information about the bulk density of states of the hybrid region can be
inferred by measuring the non-local conductance in a three-terminal geometry
[9, 10, 11]. However, these measurements are only sensitive to the minimum
energy scale of all the bulk states, and thus do not immediately reveal local
properties. An alternative method to probe the bulk, and therefore get
information about the wave function of subgap states, is to perform local
tunnelling spectroscopy along the hybrid. While such experiments have been
performed in hybrid nanowires, technical difficulties have led to soft
superconducting gaps [12] or additional tunnelling currents that obscure the
direct measurement of the LDOS in the hybrid [13]. Furthermore, the
transparency of these tunnel probes is not tunable, thus preventing a
systematic study of LDOS in the bulk. These issues can be mitigated by using a
two-dimensional electron gas (2DEG), which offers flexibility in device design
and fabrication, allowing one to pattern an arbitrary number of tunable tunnel
probes along the 1D hybrid, thus providing information about spatial variation
in the LDOS. It has also been proposed that a gate-defined hybrid wire with
multiple tunnel junctions is more resilient to inhomogeneous confinement
potential, thereby making this device geometry a promising way to probe the
LDOS [14].

Figure 1: The multiprobe device. (a) A false-colored scanning electron
microscope (SEM) image of a device with the Al strip and normal contacts. Nine
normal contacts are placed from the edge of the wire (”1”) to the bulk (up
till ”9”). In the circuit diagram, the applied bias voltages and measured
currents are only shown for the first four probes for simplicity. (b) SEM
image of a device after the global gate deposition. (c) SEM image of a device
after the tunnel gates deposition. The applied global gate voltages
$V_{\mathrm{GG}}$ and the applied voltages of the first four tunnel gates
$V_{\mathrm{T1}}$ to $V_{\mathrm{T4}}$ are labelled. The first and second
images are from lithographically similar devices. All scale bars here are
$200\text{\,}\mathrm{n}\mathrm{m}$.

Here we study the LDOS of quasi-1D hybrid wires, defined by electrostatic
gating in an InSbAs 2DEG with epitaxial aluminium. Several tunnel probes
positioned along the wire enables a simultaneous measurement of the position-
dependent LDOS. At zero magnetic field, we measure a hard superconducting gap
without any subgap states, confirming a strong proximity effect and the
presence of clean tunnel junctions. As we increase the magnetic field, we in
general do not observe any obvious correlation between the emerging subgap
states at neighbouring probes, suggesting that these states are localized
within $250\text{\,}\mathrm{n}\mathrm{m}$ along the hybrid. Furthermore, we
find that the critical field ($B_{\mathrm{c}}$) and the effective $g$-factor
($g^{*}$) exhibit significant fluctuations along the wire. In contrast, some
devices show remarkably correlated subgap states with a spatial extension of
more than $1.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. We discuss
possible explanations of this inconsistency between different devices.

Figure 2: Hard superconducting gap at zero magnetic field. (a) Tunnelling
conductance $G_{\mathrm{i}}$ as a function of individual tunnel gate
$\mathrm{TG_{i}}$ and the corresponding applied bias voltage $V_{\mathrm{i}}$
(i $\in\mathrm{\\{1,2,3\\}}$). With a substantial change of the out-of-gap
state conductance, no discrete subgap states are observed within the gap,
indicating clean tunnel junctions. Probe numbers are labelled on the top-right
corner. (b) Exemplary line traces indicate that the presence of two sharp
coherence peaks and a hard induced superconducting gap (the lines are
laterally offsetted by 0.1$G_{0}$ for clarity). The measured lock-in signals
are higher than the noise floor due to the additional parasitic capacitance in
the circuit and a detailed comparison with the numerical derivative of the DC
current is made in the Fig. S1. $V_{\mathrm{GG}}$ is at
$-2.6\text{\,}\mathrm{V}$.

##  II Results

###  II.1 Device and basic characterization

Figure 3: Field evolution of LDOS for device A. Tunnelling conductance $G$ of
each tunnel probes with a schematic of the device. The measurements of probe
1234 and 6789 are obtained by sweeping the four biases at the same time and
recording the signals with four lock-in amplifiers. The spectrum of probe 5 is
obtained in a three-terminal measurement circuit. No obvious correlation of
subgap states between neighboring probes are observed. $V_{\mathrm{GG}}$ is at
$-2.6\text{\,}\mathrm{V}$.

The InSbAs 2DEG with epitaxial aluminium grown by molecular beam epitaxy has
been shown to have a good proximity effect, high g-factor and large spin-orbit
coupling [15, 16]. The structure of the multiprobe devices is illustrated in
Fig. 1, together with the circuit diagram. First a
$2.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$-long,
$130\text{\,}\mathrm{n}\mathrm{m}$-wide aluminium strip is defined by chemical
etching and nine Ti/Pd normal contacts are deposited along the strip with a
center-to-center separation of $250\text{\,}\mathrm{n}\mathrm{m}$. The
aluminium strip remains electrically grounded during measurement, and the bias
voltages applied on each contact $V_{\mathrm{i}}$ (i$\in$
$\mathrm{\\{1,2,...9\\}}$) can be varied independently. After depositing a
$20\text{\,}\mathrm{n}\mathrm{m}$ thick AlOx dielectric layer, a global gate
(GG) is deposited. Applying a negative voltage to GG depletes the 2DEG around
the Al strip, thereby defining the 1D hybrid wire. At the same time, the 2DEGs
between any two normal contacts are also depleted, ensuring that no current
flows between neighboring tunnel probes. After depositing an additional
$20\text{\,}\mathrm{n}\mathrm{m}$ layer of AlOx, nine tunnel gates are
deposited over the pinholes in the GG. The applied tunnel gate voltages
$V_{\mathrm{Ti}}$ (i$\in$ $\mathrm{\\{1,2,...9\\}}$) control the individual
tunnel barrier, allowing one to perform local spectroscopy along the wire. The
final image of one of the three measured devices (denoted Device A) is shown
in Fig. 1c. We also present measurements of two other devices (denoted device
B and C) with the same material but with only four tunnel probes (device
images shown in Fig. 5).

All measurements were conducted in a dilution refrigerator with
$20\text{\,}\mathrm{m}\mathrm{K}$ base temperature with standard lock-in
techniques. More details about the measurement scheme can be found in the
measurements methods in the supplementary information.

We begin the device characterization through tunnelling spectroscopy
measurements as a function of tunnel gates. Three examples of the measured
spectrum are illustrated in Fig. 2(a-c). In the tunnelling regime (Fig. 2d),
all three probes show sharp superconducting coherence peaks at approximately
$\pm~{}$$0.26\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ and a suppression of the
in-gap conductance. The tunnel gate voltages $V_{\mathrm{Ti}}$ affect the
transparency of the tunnel junctions. While the the out-of-gap conductance
varies between around half of the $G_{\mathrm{0}}$ to nearly zero, the
coherence peaks remain at the same energies, as shown in Fig. 2a-c.
Importantly, we note that there are no obvious charging effects, and no
additional subgap states appear over this range of transparency. These
spurious states are often observed in hybrid devices and are attributed to a
nonuniform confinement potential in the semiconductor junctions [17, 14, 18].
The absence of these subgap states at zero magnetic field allows us to extract
information about the LDOS in the hybrid wire.

###  II.2 Spatial dependence of microscopic parameters

The precursor of MBSs in a 1D hybrid system is an extended Andreev bound state
(ABS) across the entire wire. By applying a large enough magnetic field, a
topological phase may arise, where the ABS evolves into spatially separated
MBSs localized at the ends of the wire. A persisting ZBP is then expected to
appear at the edges, along with a closing and reopening of the gap in the bulk
of the wire. If the spatial separation of a series of tunnel probes is
sufficiently small, it should then be possible to map the wave function of the
MBSs, which in theory decays exponentially from the wire edge into the bulk
[14].

We measure the tunnelling conductance as a function of the individual applied
bias $V_{\mathrm{i}}$ and a global magnetic field $B$ parallel to the
aluminium strip, as shown in Fig. 3. The tunnel gates voltages are adjusted
such that all probes have out-of-gap conductance well below G0, and are
therefore in the tunnelling regime. The most clear observation from Fig. 3 is
that there is no systematic correlation in the field evolution of the subgap
states moving from the edge to the bulk, indicating the absence of an extended
ABS in the wire. For example, the field value where the lowest subgap states
cross zero energy differs by about $400\text{\,}\mathrm{m}\mathrm{T}$ between
probe 1 and 9. Furthermore, even when we compare the subgap states from
neighboring probes (separated by $250\text{\,}\mathrm{n}\mathrm{m}$), their
evolution with magnetic field seems uncorrelated. For example, the subgap
states from probe 2 reach zero energy at around $1.1\text{\,}\mathrm{T}$ while
this occurs at $1.3\text{\,}\mathrm{T}$ for probe 3. The measured spectra of
probe 3, 5 and 7 look qualitatively similar, but a more detailed comparison
shows that the extracted microscopic parameters are different, and thus these
subgap states are also not actually correlated (detailed in Fig. 4). The
measured tunnel spectra at individual probes can also depend on the chemical
potential in the wire. Thus we also performed the measurements at
$V_{\mathrm{GG}}=$-1.8\text{\,}\mathrm{V}$$, just below the threshold voltage
required to deplete the bare 2DEG. Similarly uncorrelated subgap states are
observed for this set of measurements (Fig. S2).

Figure 4: Spatial dependence of superconducting gap $\Delta$, critical field
$B_{\mathrm{c}}$ and effective g-factor $g^{*}$. (a) A sketch of the field
dependence of the lowest subgap states. (b) $\Delta,B_{\mathrm{c}}$ and
$g^{*}$ are plotted as a function of distance $x$ to the edge of the wire
(bottom axis) and the corresponding probe number (top axis).

We use the the measurement presented in Fig. 3 and Fig. S2 to extract the
spatial dependence of three microscopic parameters in the hybrid: the induced
superconducting gap $\Delta$, the critical field $B_{\mathrm{c}}$ and the
effective $g$-factor $g^{*}$ of the lowest-energy subgap states. They are
labelled in an exemplar field evolution of the lowest subgap states, as shown
in Fig. 4a. The size of the induced gap $\Delta$, $B_{\mathrm{c}}$ and $g^{*}$
characterize the degree of hybridization of the wavefunction across the
superconductor-semiconductor interface. It has been shown that this coupling
between the two materials in hybrid nanowires can be modulated by the use of
the electric field [19, 20, 10]. $\Delta$ is determined by locating the
applied bias voltages corresponding to the coherence peaks maxmima at $B=$0\.
$B_{\mathrm{c}}$ is defined here as the field value at which the lowest states
reach zero energy and is extracted by locating the first local maximum in the
zero-bias conductance traces as a function of magnetic field. $g^{*}$ is
defined by $g_{eff}=\frac{2}{\mu_{B}}|\frac{\Delta E}{\Delta B}|$ [21], where
$\mu_{B}$ is Bohr magneton and $|{\frac{\Delta E}{\Delta B}}|$ the absolute
average of the slope from the linear fitting of the lowest subgap states at
positive and negative biases.

Figure 5: Field evolution of LDOS on device B and C. (a) The false-colored SEM
image of device Bwith a similar shape of Al strip, but only four normal
contacts that are separated by around $500\text{\,}\mathrm{n}\mathrm{m}$. The
scale bar here is $200\text{\,}\mathrm{n}\mathrm{m}$. (b)) The field
dependence of the leftmost three tunnel probes. The lowest three subgap states
spectra have almost the same field dependence between probe 1 and probe 3, but
only the second lowest subgap states are present in probe 2. (c) The false-
colored SEM image of another four-probe device C and (d) the field evolution.
The lowest subgap states are almost perfectly correlated within all three
probes. Details on peak-matching for confirming these correlations are shown
in Fig. S6.))

As seen in Fig. 4b, the induced gap $\Delta$ in our devices varies between
$0.25\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ and
$0.28\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ along the wire, with an average
value of $0.26\text{\,}\mathrm{m}\mathrm{V}$ (GG = $-2.6\text{\,}\mathrm{V}$)
and $0.27\text{\,}\mathrm{m}\mathrm{V}$ (GG = $-1.8\text{\,}\mathrm{V}$).
Additionally, the similar magnitude at two different $V_{\mathrm{GG}}$ is
probably due to the weak gating effect of the hybrid sections, which is
achieved purely by the fringing field of the applied global gate voltages. The
spread of the data points can be captured by the calculated coefficient of
variation (CV), which is the ratio of the standard deviation to the mean.
$\mathrm{CV}_{\Delta}$ about 3.1% for GG = $-2.6\text{\,}\mathrm{V}$ and 3.4%
for GG = $-1.8\text{\,}\mathrm{V}$. This variation may be due to mesoscopic
variations in the wire or the different tunnel broadening at each probe. For
the critical field $B_{\mathrm{c}}$, however, we find a much stronger
variation of the extracted values across different probes. The averaged value
is at $1.16\text{\,}\mathrm{T}$ for GG = $-2.6\text{\,}\mathrm{V}$ and
$1.18\text{\,}\mathrm{T}$ for GG = $-1.8\text{\,}\mathrm{V}$, with the CV
reaching about 9.4% in both cases. This significant spread could arise from a
nonuniform electro-chemical potential in the wire, which is undesirable in
realizing a global topological phase transition. The effective g-factor is
indicative of the extent of hybridization of wavefunction throughout the
cross-sectional interface of the hybrid [21, 19, 20], and eventually
determines the required critical field for a topological phase transition. The
extracted data here shows a large amount of fluctuation, ranging from about 8
to 11. These values are significantly smaller than the g-factor of bare InSbAs
2DEGs [15, 22], indicative of hybridization with the superconductor. The error
bars originate from the process of linear fitting. For $V_{\mathrm{GG}}$ =
$-2.6\text{\,}\mathrm{V}$, the mean is 9.4 with a CV of 6.7%, and for
$V_{\mathrm{GG}}$ = $1.8\text{\,}\mathrm{V}$, the mean is 9.6 with a CV of
7.5%. The significant spread of $B_{\mathrm{c}}$ and $g^{*}$, together with
the uncorrelated LDOS shown in Fig. 3, indicates a non-uniform chemical
potential along the wire, which is nonideal for creating Majoranas.

###  II.3 Extended subgap states in other multiprobe devices

We repeat similar measurements in two additional multiprobe devices with a
similar design. The SEM images of devices B and C are shown in Fig. 5a and
Fig. 5c, respectively. These devices are fabricated with the same 2DEG
heterostructures and have the identical shape of Al strip. However, four
normal probes are now arranged with a larger separtion of around
$500\text{\,}\mathrm{n}\mathrm{m}$. Basic characterization in Fig. S4 confirms
similar hard gap and absence of subgap states in tunnelling spectroscopy, as
the behaviour observed in device A. Field dependence measurements are
conducted in a comparable tunnelling regime as depicted in Fig. 3 for the
first three probes from the edge for both devices. Remarkably, the subgap
states of probe 1 and probe 3 now have a remarkably similar dependence on the
magnetic field, which we attribute to extended states over
$1.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. However, spectroscopy at
probe 2 looks different. While some states evolve similarly at all three
probes (Fig. S6), others do not. This suggests that the wavefunction of these
states is not uniform across the width of the hybrid region. The measurements
for device C shows that the lowest subgap states from all probes have the same
dependence on the magnetic field, confirming their spatial correlation over
$1.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (Fig. S6).

##  III Discussion

The variations in the extent of the ABS wavefunction across different devices
warrants a further discussion. We propose a few explanations for this observed
discrepancy. First of all, we know that the semiconductor 2DEGs used in this
study have a typical peak mobility of about
$28\,000\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$
(which corresponds to a mean-free path of about
$280\text{\,}\mathrm{n}\mathrm{m}$) [15]. Thus, intrinsic disorder could be a
factor responsible for the device to device variations. Secondly, the system
could be sensitive to potential inhomogeneities arising from electrostatic
gating. The tuning of chemical potential of a 1D hybrid wire is determined in
part by the fringing field of the global gate. The roughness in the edge
profile of the aluminium could therefore result in local variations of
chemical potential along the wire. This may explain the observations in device
B (Fig. 5b) where the spectra are only correlated on one side of the wire.

Additionally, while the device geometry of device A looks nominally similar to
that of device B/C (apart from the number of probes), they actually have
different dimensions of the pinholes and dielectric thicknesses (Fig. S5),
which could potentially lead to different electric fields at the hybrid
region. In fact, we observe this experimentally while measuring the tunnelling
spectra as a function of tunnel gate voltages at finite field (Fig. S4). The
lowest energy subgap states in device A can be affected upon changing the the
corresponding tunnel gate voltages, in contrast with device B/C, where they
remain unaffected. To qualitatively understand this difference, we performed
electrostatic simulations in COMSOL, based on the realistic device geometry
(Fig. S5). We find that in device A, the tunnel gate voltages can create
stronger fringing fields in the hybrid region (Fig. S3) and thereby
effectively lead to the formation of invasive tunnel probes. On the other
hand, as a result of the narrower pinholes and the thicker dielectric layers,
the tunnel gates in device B/C have a significantly weaker effect on the
hybrid region. This is in accordance with the experimental observations
whereby device B/C show stronger correlations between probes as compared to
device A. Therefore, it is important to take these electrostatic effects into
consideration while designing devices to study the LDOS in hybrid systems.

##  IV Conclusions

In conclusion, we have used tunnelling spectroscopy to investigate the local
density of states in gate-defined wires based on a 2DEG semiconductor-
superconductor hybrid structure. This is achieved by implementing a multiprobe
device geometry, with up to nine side probes placed at different positions
along the wire.  At zero magnetic field, we observed hard superconducting gaps
and clean tunnel junctions, indicating uniform proximity over
$2.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. As the magnetic field
increases, subgap states appear and eventually cross zero energy. However,
these states are generally not correlated between neighboring probes. The
critical field $B_{\mathrm{c}}$ and effective $g$-factor $g^{*}$ are extracted
at two different global gate voltages $V_{\mathrm{GG}}$ and exhibit
significant spatial fluctuations. Measurements from comparable devices show a
completely different behaviour, where the subgap states evolve identically as
a function of magnetic field, suggesting correlations over
$1.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. In particular, even in
the case of perfect probe-to-probe correlation, we find no clear evidence of a
gap reopening, suggesting that the non-uniformity in our devices may be more
than what is required to host a global topological phase [23].

##  V Reference

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##  VI Acknowledgments

We would like to express our gratitude to Nick van Loo and Greg Mazur for
their constant support and productive, in-depth discussions at various stages
of the work. We also want to thank Ji-Yin Wang, Vukan Levajac, Bas ten Haaf
and Christian Prosko for providing valuable feedback on the manuscript. We
thank Di Xiao, Candice Thomas and Michael J. Manfra for providing the
semiconductor heterostructures used in this work. The experimental research at
Delft was supported by the Dutch National Science Foundation (NWO) and a TKI
grant of the Dutch Topsectoren Program.

##  VII Author contributions

Q.W. and S.K. fabricated the devices. Measurements were performed by Y.Z. and
Q.W. The manuscript was written by Q.W., Y.Z. and S.G., with inputs from all
coauthors. S.G. supervised the experimental work.

##  VIII Data availability

Raw data and analysis scripts for all presented figures are available at
zenodo.org/doi/10.5281/zenodo.11203149

## Supplementary information

##  IX Fabrication

Three devices were measured to obtain the data presented in the main text and
Supplementary (device A, B and C). To prevents the potential intermixing of Al
and Sb at high temperature, all the fabrication processes are performed at
room temperature unless otherwise specified. We use electron beam lithography
to define the required nano-structures and use unexposed PMMA as etch masks.
The device fabrication starts by etching the Al and the 2DEG in undesired
areas. The Al etch is performed in Transene D etchant at a temperature of
$48.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $9\text{\,}\mathrm{s}$.
Afterwards, using the same PMMA mask, the 2DEG is etched in a III-V etchant
($560\text{\,}\mathrm{m}\mathrm{L}$ deionized water,
$5\text{\,}\mathrm{m}\mathrm{L}$ H2O2 and $4\text{\,}\mathrm{m}\mathrm{L}$ 85%
H3PO4 and $9.6\text{\,}\mathrm{g}$ citric acid powder) for
$70\text{\,}\mathrm{s}$. This leads to an etching depth of about
$75\text{\,}\mathrm{n}\mathrm{m}$ and produces a series of un-etched units
(called ”mesa”) which are electrically isolated. Then a sets of fine markers
made of Ti/Au are evaporated around each mesa. The purpose of the fine markers
is to improve the alignment accuracy between the Al strip, normal contacts and
two gate layers to ensure the functionality of the final device. The next step
is to define the Al strip, where we carry out a second Al etch in
$38.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ Transene D for
$10\text{\,}\mathrm{s}$. Multiple fine normal contacts made of
$5\text{\,}\mathrm{n}\mathrm{m}$/$15\text{\,}\mathrm{n}\mathrm{m}$ Ti/Pd are
then deposited around the Al strip. Subsequently, a thicker
$5\text{\,}\mathrm{n}\mathrm{m}$/$115\text{\,}\mathrm{n}\mathrm{m}$ Ti/Au
evaporation defines the contact leads. Afterwards, we deposit a
$20\text{\,}\mathrm{n}\mathrm{m}$-thick global AlOx dielectric at
$40\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The first layer of gate electrodes
(global gate) are formed with a
$5\text{\,}\mathrm{n}\mathrm{m}$/$25\text{\,}\mathrm{n}\mathrm{m}$ Ti/Pd
evaporation for the fine structures, and then a
$5\text{\,}\mathrm{n}\mathrm{m}$ Ti and $115\text{\,}\mathrm{n}\mathrm{m}$ Au
evaporation for the coarse gate leads. After depositing the second layer of
$20\text{\,}\mathrm{n}\mathrm{m}$-thick global AlOx dielectric, we deposit a
$5\text{\,}\mathrm{n}\mathrm{m}$/$35\text{\,}\mathrm{n}\mathrm{m}$ Ti/Au for
the fine structure of the second gate layer and an additional
$5\text{\,}\mathrm{n}\mathrm{m}$ Ti/$115\text{\,}\mathrm{n}\mathrm{m}$ Ti/Au
evaporation for the coarse gate leads.

##  X Measurement Methods

For all the measurements, the alignment of the magnetic field with respect to
the gate-define wire is expected to be accurate within $\pm$
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$ and calibrated through performing
tunnelling spectroscopy of the hybrid section as a function of field angle.
During the transport measurement, the aluminium is always electrically
grounded. Each available Ohmic lead is biased with both DC and AC voltages and
also connected to a current-to-voltage converter, transmitting the outcome to
Keithley multimeter DM6500 and lock-in amplifer SR830. Two types of circuits
are implemented: for the results presented in Fig. 2, probe 5 in Fig. 3 and
Fig. 5b, a common three-terminal circuit as presented in [24] is used. The
rest of measurements, in Fig. 3 and Fig. 5d, are acquired with a special
multi-terminal circuit. For the case of Fig. 3, the same DC voltage biases are
applied on four different probes simultaneously, but each with an AC
excitation of distinct frequencies. Thus, we can register four lock-in
responses concurrently and perform more efficient measurements. In particular,
we have compared the results obtained in the multi-terminal circuit
configuration with those obtained with the simplistic two-terminal circuit,
and found no qualitative difference in determining the energies of the subgap
states Fig. S7.

In the three-terminal circuit, when a DC voltage and a lock-in AC excitation
is applied to one Ohmic contact, the another one is kept grounded and vice
versa. The amplitudes of AC excitations are always $5\text{\,}\upmu\mathrm{V}$
and frequencies are $19.99\text{\,}\mathrm{H}\mathrm{z}$ (lockin-1) and
$29.99\text{\,}\mathrm{H}\mathrm{z}$ (lockin-2). In this way, we can measure
the full conductance matrix $G$ in two measurement runs . In the multi-
terminal circuits, if two DC biases are swept simultaneously, the frequencies
of the AC excitations are $19.99\text{\,}\mathrm{H}\mathrm{z}$ (lockin-1) and
$27.77\text{\,}\mathrm{H}\mathrm{z}$ (lockin-2). If four DC biases are swept,
the frequencies of the AC excitations are $23.33\text{\,}\mathrm{H}\mathrm{z}$
(lockin-1), $27.77\text{\,}\mathrm{H}\mathrm{z}$
(lockin-2),$13.33\text{\,}\mathrm{H}\mathrm{z}$ (lockin-3) and
$19.99\text{\,}\mathrm{H}\mathrm{z}$ (lockin-4). Therefore, the tunnelling
signals at each tunnel probe can only be demodulated by the single lock-in
amplifier whose AC excitation has the same frequency. The voltage-divider
effect is accounted for by correcting the real DC voltage drop on each contact
with the known fridge line resistances and the resistance from the current-to-
voltage converter module. In this way, the evolution of the energies of the
subgap states as a function of gate voltage or magnetic field can be correctly
resolved. Offsets of the applied voltage biases on each contacts are corrected
via averaging the coherence peaks in the conductance line traces.

##  XI COMSOL Simulations

We use AC/DC module in COMSOL Multiphysics 6.1 to simulate the electrostatics
in a simplified but realistic geometry. To be more precise, the following
specifications are taken directly from the multiprobe devices shown in the
main text: the thickness of the 2DEG, the width and thickness of the aluminium
strip, the thickness of the AlOx dielectric, the relative position and the
size of the pinhole, the thickness of the global gate and tunnel gate layer.
Note that the 2DEG is treated as a semiconductor material with a relative
permittivity of 17.7. Aluminium is kept electrically ground and the voltages
are applied to the global gate and the tunnel gate. The only variance between
device A and device B/C are the different AlOx thickness and the width of the
pinhole. The plots shown in Fig. S5 are obtained by performing a parametric
sweep of the $V_{\mathrm{GG}}$ and $V_{\mathrm{TG}}$ with
$1\text{\,}\mathrm{V}$ step.

Figure S1: Comparison of the AC and DC signals In this example, the
conductance is either measured directly with lock-in amplifier (a, Figure 2a
in the main text), or calculated with a numerical derivative of the DC current
(b). The line-cut taken at the same gate voltage (c) shows that the in-gap
conductance in the DC derivative is about an order of magnitude smaller that
of the lock-in signal. This is because the additional parasitic components in
the circuit induce an additional phase shift, thus making the amplitude of the
measured lock-in signal higher. A Savitzky–Golay filter of window length of 5
is applied for the DC current before taking the numerical derivation. Figure
S2: Field evolution of LDOS at $V_{\mathrm{GG}}$ = $-1.8\text{\,}\mathrm{V}$:
Device A. Similar to the behavior observed in main text FIG. 3, no obvious
correlation of subgap states between neighboring probes are observed. Note
that the field evolution has only been recorded for the first seven probes
because of difficulties in adjusting the the tunnel gates for Probe 8 and
Probe 9. Figure S3: Effect of tunnel gates on spectroscopy: Device A. The
probe numbers are indicated in the top left of each figure. Overall, the
subgap states do not stay at constant energies upon changing the corresponding
tunnel gate voltages, indicating that the tunnel gates can have an effect on
the hybrid region. This observation also indicates that the electrostatics can
lead to false-positive observation of correlated subgap states, as a result of
fine-tuned gate voltages. Figure S4: Effect of tunnel gates on spectroscopy:
Device B and C. Hard gaps without the subgap states ((a-b) and (e-g)) are
observed are observed for both devices. At a finite field, the subgap states
stay at almost the same energies as a function of tunnel gate voltages over a
large range of transparency ((c-d) and (h-j)). Note that for device B, the
tunnel gate T1 has almost no tunability of the junction transparency, and
therefore only the T2 and T3-dependence are shown here. Figure S5: COMSOL
simulations of devices. (a) Top view of the mulitprobe device, with only one
representative pinhole on the global gate and only one tunnel gate. The
simulated electric field strength at the red cross is chosen to represent the
effect of the applied global gate voltages $V_{\mathrm{GG}}$ and tunnel gate
voltages $V_{\mathrm{TG}}$. (b) A cross-sectional view of the device stack,
taken along the dashed line in (a). The length scales that are relevant for
the electrostatics are the thickness of the first ALD layer
($t_{\mathrm{1}}$), the thickness of the second ALD layer ($t_{\mathrm{2}}$)
and the width of the pinhole ($w_{\mathrm{1}}$). The table summarizes these
parameters. (c) The simulated electric field norm for device A as a function
of $V_{\mathrm{GG}}$ and $V_{\mathrm{TG}}$. The change in the field amplitude
as a function of only TG or GG of the same range is very similar. (d)The
simulated electric field norm for device B/C. In contrast to (c), now the
global gate has a much stronger effect on the 2DEGs than the tunnel gate. This
is consistent with the overall thicker dielectric thickness ($t_{\mathrm{1}}$
+$t_{\mathrm{2}}$) and the smaller pinhole width ($w_{\mathrm{1}}$) in device
B/C, compared to device A.This may explain the weaker tunnel gate effect on
the measured spectrum Fig. S4, and the relatively stronger gating effect in
device A Fig. S3. More details about the COMSOL simulation can be found in the
methods section.  Figure S6: Identification of subgap states in Figure. 5:
Device B and C. At each field value, we find the local maxima in the
tunnelling conductance and overlay these states from different probes in the
last plot of each row. For device B ((a),top row), the lowest subgap states of
probe 1 and probe 3 show identical dependence on magnetic field, but are
absent in the spectra in probe 2. However, the evolution of the second-lowest
states of probe 1 and 3 matches well with that of the lowest states of probe
2. For device C((b),bottom row), the lowest energy subgap states from the
three probes show identical dependence. Figure S7: Spatial dependence of
superconducting gap $\Delta$, critical field $B_{\mathrm{c}}$ and effective
g-factor $g^{*}$ of the lowest subgap states: Device B and C. For device B,
the value of $B_{\mathrm{c}}$ and $g^{*}$ at probe 2 is clearly different from
those at probe 1 and 3. For Device C, however, these parameters are nearly
identical at various probes. Figure S8: Comparison of the measured signal with
different circuit configurations. Measurements here are performed either with
a two-terminal circuit diagram (with the voltage bias applied to either the
probe 3, as shown in (a) or the probe 7) or a five-terminal circuit (with
voltage biases applied simultaneously to four probes, shown in (c)). The
measured conductance via two kinds of circuits are then plotted together for
probe 3 (b) and probe 7 (d). There is no significant difference regarding the
positions and heights of the coherence peaks between these two measurements,
confirming the validity of the five-terminal circuit configuration for
investigating the subgap states.