# Parametric exploration of zero-energy modes in three-terminal InSb-Al
nanowire devices

Ji-Yin Wang<EMAIL_ADDRESS>QuTech and Kavli Institute of Nanoscience,
Delft University of Technology, 2600 GA Delft, The Netherlands  Nick van Loo
Grzegorz P. Mazur  Vukan Levajac  Filip K. Malinowski  Mathilde Lemang
Francesco Borsoi  QuTech and Kavli Institute of Nanoscience, Delft University
of Technology, 2600 GA Delft, The Netherlands  Ghada Badawy  Sasa Gazibegovic
Department of Applied Physics, Eindhoven University of Technology, 5600 MB
Eindhoven, The Netherlands  Erik P.A.M. Bakkers  Department of Applied
Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The
Netherlands  Marina Quintero-Pérez  QuTech and Kavli Institute of Nanoscience,
Delft University of Technology, 2600 GA Delft, The Netherlands  Netherlands
Organisation for Applied Scientific Research (TNO), Delft 2600 AD, Netherlands
Sebastian Heedt  QuTech and Kavli Institute of Nanoscience, Delft University
of Technology, 2600 GA Delft, The Netherlands  Leo P. Kouwenhoven  QuTech and
Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft,
The Netherlands

###### Abstract

We systematically study three-terminal InSb-Al nanowire devices by using
radio-frequency reflectometry. Tunneling spectroscopy measurements on both
ends of the hybrid nanowires are performed while systematically varying the
chemical potential, magnetic field and junction transparencies. Identifying
the lowest-energy state allows for the construction of lowest- and zero-energy
state diagrams, which show how the states evolve as a function of the
aforementioned parameters. Importantly, comparing the diagrams taken for each
end of the hybrids enables the identification of states which do not coexist
simultaneously, ruling out a significant amount of the parameter space as
candidates for a topological phase. Furthermore, altering junction
transparencies filters out zero-energy states sensitive to a local gate
potential. Such a measurement strategy significantly reduces the time
necessary to identify a potential topological phase and minimizes the risk of
falsely recognizing trivial bound states as Majorana zero modes.

##  I I. Introduction

Superconductor-semiconductor hybrids have attracted great interest in recent
years for their potential applications in creating Majorana zero modes (MZMs)
bib:Liang2008 ; bib:Roman2010 ; bib:Yuval2010 . Extensive experiments have
been carried out on such hybrid nanowires bib:Vincent2012 ; bib:Mingtang2012 ;
bib:Anindya2012 ; bib:Churchill2013 ; bib:Mingtang2016 ; bib:Chen2017 ;
bib:Hao2021 ; bib:Vaitiekenas2020  and hybrid two-dimensional electron gases
(2DEGs) bib:Fabriz2017 ; bib:Fornieri2019 ; bib:Ren2019 . Zero-bias peaks
(ZBPs), observed at the ends of such hybrids, were initially considered as
evidence for the existence of MZMs. However, such ZBPs could also originate
from alternative trivial mechanisms, such as quasi-Majoranas bib:Adriaan2019 ,
disorder effects bib:JieLiu2012 ; bib:HainingPan2020 ; bib:HainingPan2021_1 ;
bib:HainingPan2021_2 ; bib:HainingPan2021_3 , or a combination of Zeeman and
Little-Parks effects bib:Valentini2021 . On the other hand, end-to-end
correlations are a unique property of paired MZMs in a topological
superconductor, and could be used to distinguish MZMs from trivial Andreev
bound states in three-terminal architectures bib:Liu2013 ; bib:Lu2014 ;
bib:Lai2019 ; bib:Anselmetti2019 . Simulations taking into account the
physical details of experimental devices (i.e. superconductor-semiconductor
coupling, band offset at the interface, multiple subbands, and disorder
effects) predict a significantly reduced and complex topological phase space
bib:Antipov2018 . Therefore, finding such a phase in the large parameter space
requires the development of a detection method capable of scanning the entire
parameter space within a practical time bib:Pikulin2021 . Radio-frequency (rf)
techniques have been succesfully implemented on superconducting qubits
bib:Wallraff2004 , spin qubits bib:Hornibrook2014  and hybrid devices
bib:Razmadze2019 ; bib:Jasper2019 . Compared to traditional dc conductance
measurements, it enables a fast and high-resolution exploration of all
essential parameters in hybrid devices.

Three-terminal InSb-Al nanowire devices are systematically investigated using
rf reflectometry. Local tunneling spectroscopy is performed at two ends of the
hybrid nanowires, while exploring the chemical potential (controlled by the
so-called ‘super gate’) and external magnetic field. The lowest-energy states
(LESs) and zero-energy states (ZESs) are extracted as a function of super gate
voltage and magnetic field, forming LES or ZES diagrams. As MZMs in an
idealized model feature end-to-end correlations, the extracted diagrams of the
two sides are compared to filter out uncorrelated ZESs. Stability of ZESs to
transparency variation is studied by altering barrier gate settings, and zero-
energy Andreev states residing around junctions are successfully identified.
In addition, induced superconductivity on two ends of the hybrid nanowires is
extracted, helpful for quantifing superconductor-semiconductor coupling in the
hybrid nanowires. By applying the aforementioned experimental procedure,
typical patterns of ZBPs are indentified in the studied devices, but after a
closer inspection non-topological explanations are more likely. The approach
is able to significantly accelerate the idenfication on a potential
topological phase.

##  II II. Experimental Setup

Figure 1: Rf reflectometry measurement setup and its basic characteristics.
(a) Measurement circuit together with a false-color SEM image of a three-
terminal InSb-Al device. The superconducting lead (blue) is made from Al,
connecting from the substrate to the nanowire. The superconductor-
semiconductor hybrid is $\sim$2 $\mu$m long. Two probe leads (yellow), made
from Ti/Au, are bonded to two superconducting inductors $L_{\mathrm{1}}$,
$L_{\mathrm{2}}$. Voltages $V_{\mathrm{LB}}$ and $V_{\mathrm{RB}}$ are applied
to bottom gates (orange) for tuning the left and right tunneling barriers,
respectively. The chemical potential of the hybrid nanowire is tuned by the
super gate (turquoise) with voltage $V_{\mathrm{SG}}$. (b) and (c) Dc
conductance of left and right junction ($g_{\mathrm{L}}$, $g_{\mathrm{R}}$)
versus corresponding barrier gate voltage ($V_{\mathrm{LB}}$,
$V_{\mathrm{RB}}$). (d) and (e) Corresponding rf response as a function of
barrier gates in the same range as (b) and (c). (f) and (g) Line cuts at
specific gate voltages from (d) and (e). In (d)-(g), $V_{\mathrm{rf}}$ is the
amplitude of the reflected rf signal with $\sim$30 dB amplification at 4 K and
$\sim$65 dB amplification at room temperature. (h) Bias-voltage waveforms
applied at two probe leads as a function of time. In the shaded time period,
voltage-bias spectroscopy of either side is performed. T is the time period
for measuring bias-spectroscopy traces on both sides.  Figure 2: (a) and (e)
Selected segments of tunneling spectroscopy at the left junction at
$B_{\mathrm{||}}$ = 0 T and $B_{\mathrm{||}}$ = 0.48 T, respectively. Line
cuts are taken at colored bars and shown in (b) and (f). (c) and (g) are
similar as (a) and (e), but for the right junction. (d) and (h) show line cuts
from (c) and (g). Line cuts at zero magnetic field show a hard gap on both
sides. Results at $B_{\mathrm{||}}$ = 0.48 T illustrate the coexistence of
zero-bias peaks on two sides. Note that the curves shown in (b), (d), (f) and
(h) are shifted vertically for better visibility.

Fig. 1(a) shows a circuit diagram of the measurement setup together with a
false-color scanning electron microscope (SEM) image of Device 1. Three-
terminal devices are fabricated from InSb nanowires bib:Gada2019  using the
recently developed shadow-wall lithography technique, enabling high-quality
semiconductor-superconductor quantum devices bib:Sebastian2020 ;
bib:Francesco2020 . In the SEM image, an Al film (blue) is connected from the
substrate to the nanowire to serve as a superconducting drain lead, while
Ti/Au contacts (yellow) are fabricated on both ends of the hybrid nanowire to
serve as probe leads. Voltages $V_{\mathrm{LB}}$, $V_{\mathrm{RB}}$ are
applied onto barrier gates (orange) to tune the transparency of the tunneling
junctions. The voltage on the super gate (turquoise), $V_{\mathrm{SG}}$, is
changing the chemical potential of the hybrid nanowire. In Fig. S1, we show
SEM pictures of two additional measured devices and a schematic of the cross-
section of the device.

In order to accelerate tunneling spectroscopy at both junctions, an rf-
conductance measurement scheme is employed bib:Schoelkopf1998 ; bib:Reilly2007
; bib:Razmadze2019 . The left and right probe lead of the device are connected
to two superconducting spiral inductors ($L_{\mathrm{1}}$, $L_{\mathrm{2}}$)
bib:Hornibrook2014 . Together with parasitic capacitors ($C_{\mathrm{P1}}$,
$C_{\mathrm{P2}}$) to ground, the inductors form two rf resonant circuits with
typical resonance frequencies 250-450 MHz (see optical images of inductor
chips with devices in Fig. S2). Each resonator acts as an impedance
transformer for the corresponding tunneling junction. On resonance, the
typical junction impedance ($\sim$150 k$\Omega$) is converted towards 50
$\Omega$, which is the characteristic impedance of the transmission lines in
the cryostat. Consequently, the reflection of the rf circuits at resonance
displays a sensitive dependence on the differential conductance of the
tunneling junctions. Fig. 1(b) and Fig. 1(c) present the pinch-off curves of
the junctions at the two ends of the hybrid nanowire, at 10 mV dc bias, while
Fig. 1(d)-(g) show corresponding response of the resonator circuits. The rf
reflection has a sensitive response to conductance changing from 0.005
$G_{\mathrm{0}}$ to 0.6 $G_{\mathrm{0}}$ ($G_{\mathrm{0}}$=2$e^{2}/h$). Such a
broad conductance response allows sensitive rf detection at different
tunneling transparencies.

The integration time per data point is about 1 ms, approximately two orders of
magnitude less than the integration time of a conventional lock-in conductance
measurement. To take advantage of the reduced integration time, we employ a
rastering scheme bib:Reilly2007 ; bib:Stehlik2015  to rapidly sweep the dc
voltage bias applied at the tunneling junctions. Fig. 1(h) shows the waveforms
generated by an arbitrary waveform generator (AWG). To perform the tunneling
spectroscopy measurements at the two ends of the nanowire, the AWG generates a
pair of triangular pulses, each sweeping the dc voltage bias at one of the
junctions. The waveforms are accompanied by triggers, synchronizing the data
aquisition with the voltage sweeps. Throughout the experiment, we perform
pairs of tunneling spectroscopy measurement (typically 200 data points for
each side with a total duration of $\sim$0.4 s), which we repeat while varying
gate voltages and the external magnetic field.

##  III III. Results

###  III.1 A. Tunneling spectroscopy

Initially, basic characterization of devices and resonators is performed
before moving to tunneling spectroscopy with rf. First, cross-talk between the
super gate and barrier gates is measured at a fixed dc voltage bias. While
sweeping $V_{\mathrm{SG}}$, barrier gate voltages are changed accordingly to
maintain a constant junction conductance (see Fig. S3(a) and Fig. S3(b)).
Next, the magnetic field is aligned along the nanowire axis. Furthermore, the
resonator frequency shift in an external magnetic field is characterized (see
Fig. S3(c) and Fig. S3(d)). As the external magnetic field is swept, the
probing frequencies are adjusted to maintain a high sensitivity of the rf
conductance measurement. Finally, tunneling spectroscopy is performed on both
sides of the device by applying the dc bias waveform illustrated in Fig. 1(h),
while stepping $V_{\mathrm{SG}}$ and the parallel magnetic field,
$B_{\mathrm{||}}$.

Figure 3: ZES diagrams as a function of $V_{\mathrm{SG}}$ and
$B_{\mathrm{||}}$ for the left junction (a) and right junction (b). Black and
white pixels indicate the presence and absence of ZESs, respectively. (c)
Diagram with black pixels indicating the coexistence of ZESs on both ends. In
each panel, there are 2523 data points along $V_{\mathrm{SG}}$ and 61 along
$B_{\mathrm{||}}$ (i.e. total number of pixels is 2523$\times$61). The full
data set is acquired in $\sim$142 hours. Magenta rectangles mark the region
with a relatively large density of coexistent ZESs.

Fig. 2 shows segments of tunneling spectroscopy measurements on two nanowire
ends at different $B_{\mathrm{||}}$. Fig. 2(a) and Fig. 2(c) show the results
as a function of $V_{\mathrm{SG}}$ at $B_{\mathrm{||}}$=0. Line cuts at
different $V_{\mathrm{SG}}$ are presented in Fig. 2(b) and Fig. 2(d).
Suppressed conductance in between two pronounced coherence peaks suggests a
hard superconducting gap. The superconducting gap is $\sim$260 $\mu$eV,
consistent with previous report based on the same fabrication platform
bib:Sebastian2020 ; bib:Francesco2020 . Fig. 2(e) and Fig. 2(g) present an
example of sub-gap features at $B_{\mathrm{||}}$=0.48 T for the same
$V_{\mathrm{SG}}$ range as Fig. 2(a) and Fig. 2(c). ZBPs are formed at both
ends between $V_{\mathrm{SG}}$ = $-0.23$ V and $V_{\mathrm{SG}}$ = $-0.15$ V
(see Fig. 2(f) and Fig. 2(h) for line cuts). ZBPs, peaks in differential
conductance at zero energy, indicate the existence of ZESs, which are states
with zero energy. Having established our setup for tunneling spectroscopy at
both nanowire ends, we start mapping ZESs as a function of multiple parameters
over a large range with high resolution.

###  III.2 B. Diagrams of lowest- and zero-energy states

For a given combination of $B_{\mathrm{||}}$ and $V_{\mathrm{SG}}$, the
presence of ZBPs is validated by analyzing one $V_{\rm rf}$-$V_{\mathrm{L/R}}$
trace (see Fig. S5 for details). This is repeated for all measured parameter
values and presented as ZES diagrams, as shown in Fig. 3. Fig. 3(c) shows
coexisting ZESs on both ends of the hybrid nanowire. In these diagrams, three
distinct regimes can be observed. (1) For negative super gate voltages
($V_{\mathrm{SG}}$ $<$ $-0.5$ V), ZESs only appear at high magnetic field (on
the order of $B_{\mathrm{||}}$ = 0.8 T) and there are no coexistent ZESs. (2)
For positive super gate voltages ($V_{\mathrm{SG}}$ $>$ 0.5 V), ZESs are
ubiquitous at fields as low as $B_{\mathrm{||}}$ = 0.2 T. Here, coexistent
ZESs are sparsely distributed in parameter space. (3) In an intermediate
regime ($-0.5$ V $<$ $V_{\mathrm{SG}}$ $<$ $0.5$ V), ZESs emerge at moderate
magnetic fields compared to the other two regimes. Notably, ZESs form regular
shapes in parameter space and there is a significant amount of coexisting
ZESs. This behavior is reproduced for two other InSb-Al hybrid nanowires
presented in this work (see Fig. S11 and Fig. S13). A recent work on InSb-Al
hybrid islands reports three similar regimes in $V_{\mathrm{SG}}$ bib:Jie2020
. It is explained by a tunable superconductor-semiconductor coupling with
$V_{\mathrm{SG}}$ bib:Antipov2018 ; bib:Mikkelsen2018 . The intermediate
regime is identified to be the most promising region to search for a
topological superconducting phase. We focus the subsequent measurements on
this intermediate super gate regime (marked by magenta rectangles in Fig. 3).

Fig. 4 presents high-resolution diagrams obtained in the intermediate super
gate regime. Fig. 4(a) and Fig. 4(b) show the energy of LESs probed at the two
junctions, E${}_{0}^{L}$ and E${}_{0}^{R}$, versus $V_{\mathrm{SG}}$ and
$B_{\mathrm{||}}$ (LES extraction method is shown in Fig. S5). At zero
magnetic field, $E_{\mathrm{0}}^{\mathrm{L/R}}$ is close to the
superconducting gap ($\sim$260 $\mu$eV). As $B_{\mathrm{||}}$ increases,
$E_{\mathrm{0}}^{\mathrm{L/R}}$ starts to drop due to the emergence of sub-gap
states. In order to illustrate the dependence of
$E_{\mathrm{0}}^{\mathrm{L/R}}$ on $B_{\mathrm{||}}$, examples of two vertical
line cuts are shown in Fig. 4(d) and Fig. 4(e). These are picked to illustrate
two types of behavior: For the blue line cuts ($V_{\mathrm{SG}}$ = $-0.18$ V),
the behavior on both sides of the nanowire is similar. Sub-gap states emerge
and drop to zero energy, with a comparable effective g-factor (solid green
lines are fitting traces to the linear part of the data). On the other hand,
the magenta line cuts ($V_{\mathrm{SG}}$ = $-0.6$ V) show an example where on
one junction, a sub-gap state drops to zero energy while on the other side no
sub-gap states emerge. In order to identity LESs that may extend between the
two ends of the hybrid nanowire, the energy difference between LESs,
$|{E_{\mathrm{0}}^{\mathrm{L}}}-E_{\mathrm{0}}^{\mathrm{R}}|$, is calculated
and shown in Fig. 4(c). Fig. 4(f) shows the line cuts at the same
$V_{\mathrm{SG}}$ as in Fig. 4(d) and Fig. 4(e). Notably, for the blue line
cut the energy difference of the LESs on both ends is close to zero within a
large range of field, indicating a potential correlation. In contrast, the
magenta line cut shows a large energy difference for almost all field values,
which signifies uncorrelated behavior.

Figure 4: High-resolution LES and ZES diagrams. (a) Energy of LESs,
$E_{\mathrm{0}}^{\mathrm{L}}$, probed on the left junction as a function of
$V_{\mathrm{SG}}$ and $B_{||}$. (b) Energy of LESs,
$E_{\mathrm{0}}^{\mathrm{R}}$, probed on the right junction versus
$V_{\mathrm{SG}}$ and $B_{\mathrm{||}}$. (c) Energy difference between the
LESs probed on the two ends. (d)-(f) Line cuts taken at different $V_{SG}$
from (a)-(c). In (d) and (e), solid green lines are linear fits. (g) and (h)
ZES diagrams for the left and right junction, respectively. (i) Diagram of
coexisting ZESs on both ends. Gates are swept by following the cyan dashed
lines ($g_{\mathrm{L}}$ $\sim$0.035 G0, $g_{\mathrm{R}}$ $\sim$0.064
$G_{\mathrm{0}}$) in Fig. S4. In (a)-(c) and (g)-(i), each panel includes
522$\times$101 pixels. The dataset is acquired in $\sim$35 hours.

From the LES diagrams, states with zero energy are identified. These states
are presented in ZES diagrams, shown in Fig. 4(g)-Fig. 4(i) (see the example
of the extraction process in Fig. S5). Similar to Fig. 3, black pixels in Fig.
4(g) and Fig. 4(h) represent the presence of ZESs, while white pixels indicate
the absence of ZESs. Regular features are observed in these diagrams,
including parabolic and oscillatory shapes, as well as clusters of black
pixels. The intersection of the two diagrams yields a diagram (Fig. 4(i))
consisting of ZESs which coexist on both sides. Around $V_{\mathrm{SG}}$
$\sim$ $-0.2$ V and $B_{\mathrm{||}}$ $>$ 0.5 T, a high density of coexistent
ZESs is observed which indicates a potential candidate region for a
topological phase. In the following section, this region and several of the
regular patterns will be further analyzed.

###  III.3 C. Detailed analysis of ZES and LES diagrams

####  III.3.1 C1. Coexisting ZBP clusters

Figure 5: Clusters of ZBPs with respect to super gate and magnetic field.
(a)-(c) ZES diagrams for the left junction, the right junction, and
coexistence in both junctions (same as Fig. 4(g)-Fig. 4(i), but with reduced
super gate range). In (a), the green mark indicates the position of
$V_{\mathrm{SG}}$ = $-0.18$ V and corresponding tunneling spectroscopy data is
shown in (d). The results at $B_{\mathrm{||}}$ = 0.82 T (yellow mark) is shown
in (f). In (b), tunneling spectroscopy data of the vertical (pink mark) and
horizontal (red mark) line cut is shown in (e) and (g), respectively. In (c),
the horizontal and vertical dashed line indicates $B_{\mathrm{||}}$ = $0.82$ T
and $V_{\mathrm{SG}}$ = $-0.18$ V, respectively.

ZES diagrams constructed in the previous section identify a region with
coexisting ZESs on both ends of the nanowire hybrid. Fig. 5(a)-Fig. 5(c) show
a zoom-in of Fig. 4(g)-Fig. 4(i). At fixed $V_{\mathrm{SG}}$ = $-0.18$ V,
tunneling spectroscopy on both ends of the hybrid nanowire shows a sub-gap
state reaching zero energy around 0.5 T (Fig. 5(d) and Fig. 5(e)). After a
single oscillation, it sticks to zero energy for $\sim$300 mT. On the other
hand, when the magnetic field is fixed at 0.82 T, a stable ZES is observed on
the left side (Fig. 5(f)). Remarkably, it persists for more than 800 mV in
$V_{\mathrm{SG}}$. Considering that the dielectric layer is made from 20 nm
HfOx, chemical potential is changed by a significant amount in this super gate
range. Such stable ZBPs as a function of super gate voltage and magnetic
field, together with similar behavior on both sides of the device as a
function of magnetic field, have been previously interpreted as evidence of
MZMs. However, tunneling spectroscopy along super gate on the right side of
the device (Fig. 5(g)) reveals a strikingly different behavior, with only two
crossings through zero energy. The different behaviors on the two sides of the
device can be recognized as well from ZES diagrams in Fig. 5(a)-Fig. 5(c).
Such clearly distinct behavior with respect to changes in the chemical
potential implies that the ZESs on the two sides of the device do not
originate from an unbroken topological superconducting phase.

####  III.3.2 C2. Parabolic patterns in ZES diagrams

In Fig. 4, ZESs form parabolic patterns in the
$V_{\mathrm{SG}}$-$B_{\mathrm{||}}$ space. Such parabolic patterns can
represent the onset of a topological phase when Majorana zero modes at two
ends of a short hybrid nanowire strongly interact bib:Sarma2012 ; bib:Chen2017
. In Fig. 6, an example of such a parabola is shown and its tunneling
spectroscopy data is presented. In Fig. 6(a), an orange rectangle marks the
region with such a parabolic pattern. This region is re-plotted in Fig. 6(b).
In order to understand the pattern, three line cuts are made at different
magnetic fields and corresponding spectroscopic results are plotted in Fig.
6(c)-Fig. 6(e). At $B_{\mathrm{||}}$ = 0.2 T (Fig. 6(c)), there is a pair of
levels emerging below the superconducting gap, marked by white dashed lines.
Once the magnetic field is increased to 0.3 T (Fig. 6(d)), the pair of sub-gap
levels merges at zero energy, forming ZBPs. This field corresponds to the
onset of ZESs in Fig. 6(b). At higher $B_{\mathrm{||}}$, for example
$B_{\mathrm{||}}$ = 0.5 T (Fig. 6(e)), the sub-gap levels form two crossings
at zero energy. The evolution of these sub-gap levels in $B_{\mathrm{||}}$ at
$V_{\mathrm{SG}}$ = 0.35 V is shown in Fig. 6(f). In Fig. S6, another
parabolic pattern which has similar properties as Fig. 6 is presented. This
behavior is fully explained by Zeeman-driven Andreev level splitting in a
quantum dot proximitized by a superconducting lead bib:Lee2014 .

Figure 6: Parabolic patterns in ZES diagrams. (a) ZES diagram for the left
junction (same as Fig. 4(g)). The region marked by the orange rectangle is re-
plotted in (b). Three line cuts are made at different $B_{\mathrm{||}}$ and
corresponding tunneling spectroscopy data is displayed in (c)-(e). White
dashed lines serve as guide to the eye. (f) Evolution of sub-gap levels in
magnetic field at $V_{\mathrm{SG}}$ = 0.35 V (red vertical line cut from (b)).

####  III.3.3 C3. Oscillatory patterns in LES diagrams

The oscillation of LESs in magnetic field with an increasing amplitude and
period is consistent with the prediction of smoking gun evidence for MZMs
bib:Sarma2012 . This type of behavior would result in oscillatory patterns in
LES and ZES diagrams. An example of such patterns is shown in Fig. 7(a) and
Fig. 7(b). A line cut through such a circle, shown in Fig. 7(c)
($V_{\mathrm{SG}}$ = $-0.08$ V), shows the energy of a LES dropping to zero
before oscillating with increasing amplitude and period, thus matching the
smoking gun predictions. The tunneling spectroscopy data as a function of
$B_{\mathrm{||}}$ corresponding to this particular line cut is shown in Fig.
7(e). In this map, three discrete sub-gap states are observed and marked with
an orange, white, and black lines. The LES (orange) first comes down in energy
and crosses through zero energy, before interacting with another state
(white). Interaction between these states results in an anti-crossing, which
can be attributed to the mixing of different spin species of two states via
spin-orbit interaction bib:Moor2018, and is represented by a gradient color.
The LES crosses through zero once more, and subsequently interacts with
another state (black). In addition, two spectroscopy results at a lower
($V_{\mathrm{SG}}$ = $-0.12$ V) and higher ($V_{\mathrm{SG}}$ = $-0.05$ V)
super gate voltage are presented in Fig. 7(d) and Fig. 7(f), respectively. By
changing the super gate voltage, the magnitude of the interaction between the
states can be tuned. Consequently, the interactions can become negligible
which results in states crossing rather than anti-crossing. This indicates
that the results in Fig. 7(e) or Fig. 7(c) arise from several sub-gap states
interacting with each other. Thus, while the behavior of LESs in Fig. 7(e) and
Fig. 7(c) is consistent with the predicted evidence of MZMs, such oscillations
can originate from anti-crossings between the LES and other states. The
analysis made above suggests that although oscillatory patterns in LES and ZES
diagrams are expected for interacting MZMs, they may also originate from
interactions between topologically trivial Andreev bound states.

Figure 7: Oscillatory patterns in LES diagrams. (a) LES diagram for the right
junction (same as Fig. 4(b)). The region marked by the orange rectangle is
plotted in (b). (c) Example of an oscillating LES, showing
$E_{\mathrm{0}}^{\mathrm{R}}$ versus $B_{\mathrm{||}}$ for the line cut taken
at $V_{\mathrm{SG}}$ = $-0.08$ V (marked by the magenta bar in (b)). Tunneling
spectroscopy data taken at $V_{\mathrm{SG}}$ = $-0.12$ V (black bar),
$V_{\mathrm{SG}}$ = $-0.08$ V (magenta bar), and $V_{\mathrm{SG}}$ = $-0.05$ V
(blue bar) is shown in (d), (e) and (f), respectively. In (d)-(f), three sub-
gap states are marked by orange, white, and black lines which serve as a guide
to the eye. Gradient colors indicate interaction between states.  Figure 8:
Gate-dependent induced superconductivity. (a) Superconducting gap as a
function of super gate voltage VSG extracted from the left (yellow) and right
(green) junction. (b) Effective g-factor versus VSG for the left (yellow) and
right (green) end of the hybrid nanowire. Magenta and blue points correspond
to the data extracted from the traces in Fig. 4(d) and Fig. 4(e).

####  III.3.4 C4. Induced superconductivity

In addition to extracting of the energy of various sub-gap states in the
system, two other important parameters can be determined from the data: the
superconducting gap size $\Delta$ and the effective g-factor $g^{*}$. From the
local tunneling spectroscopy measurements, the superconducting gap can be
estimated by fitting zero-field gL/R-VL/R traces with the BCS$-$Dynes formula
bib:Dynes1978 , where gL/R is obtained from Vrf-gL/R correspondence in Fig. 1.
On the other hand, the effective g-factor is determined by making a linear fit
to the energy dependence of the LES as a function of magnetic field. In Fig.
8, the evolution of these two parameters is shown as a function of super gate
voltage. At both ends of the nanowire, the estimated gap size behaves
similarly and remains largely unaffected by the super gate. It only shows a
small dip in the vicinity of VSG=$-0.25$ V, and from the spectra shown in Fig.
2 it can be seen that these dips correspond to the energy minima of two LESs.
However, if we instead look at the field evolution it becomes apparent that
the extracted g-factors do not behave in a similar way. This indicates that
the LESs at two ends are uncorrelated, which offers an alternative perspective
to the energy evolution which we investigated in previous sections. In
addition, the right side of the device shows an absence of sub-gap states
below VSG=$-0.4$ V (see an example in Fig. S7). The corresponding g-factor is
estimated from the gap edge and remains close to 2, as the measured properties
are dominated by the Al film. It is worth to note that with local tunneling
spectroscopy, only the superconducting properties in the vicinity of the
junctions are detected while leaving the bulk properties inaccessible. Thus,
these measurements could be complemented by non-local measurements in order to
investigate the induced superconductivity in the bulk of the hybrids
bib:Pikulin2021 .

###  III.4 D. Influence of barrier gates

Figure 9: Overlapping ZES diagrams taken at different barrier gate settings
for the left junction (a), right junction (b), and coexistence in both
junctions (c). The panels are obtained by overlapping ZES diagrams from Fig.
S8 ($g_{\mathrm{L}}\sim$ 0.027 $G_{\mathrm{0}}$, $g_{\mathrm{R}}\sim$ 0.025
$G_{\mathrm{0}}$), Fig. S9 ($g_{\mathrm{L}}\sim$ 0.029 $G_{\mathrm{0}}$,
$g_{\mathrm{R}}\sim$ 0.004 $G_{\mathrm{0}}$), Fig. 4 ($g_{\mathrm{L}}\sim$
0.03 $G_{\mathrm{0}}$, $g_{\mathrm{R}}\sim$ 0.064 $G_{\mathrm{0}}$), and Fig.
S10 ($g_{\mathrm{L}}\sim$ 0.047 $G_{\mathrm{0}}$, $g_{\mathrm{R}}\sim$ 0.129
$G_{\mathrm{0}}$).

ZESs formed in the vicinity of the junction can mimic MZM behavior with
respect to magnetic field and chemical potential variations bib:ChunXiao2017 .
Tunnel transparency is an important experimental parameter which can be used
for distinguishing MZMs from trivial Andreev bound states. To investigate the
stability of ZESs shown in Fig. 4, similar measurements are performed with
three different barrier gate settings. The corresponding processed diagrams
are plotted in Fig. S8, Fig. S9, and Fig. S10 in the supplementary material.
ZES diagrams obtained for four different barrier gate settings are overlapped
to form ZES histograms in Fig. 9. As shown in Fig. 9(a), the left end of the
hybrid exhibits clusters of ZESs in the space of chemical potential and
magnetic field. As those ZESs are stable against variation of all
experimentally accessible parameters, they are compatible with MZMs. However,
as shown in Fig. 9(b), the right side of the studied hybrid does not show
similar stability and thus indicates that the device does not exhibit an
unbroken topological superconducting phase. As expected, the uncorrelated
behavior between left and right end of the hybrid is recognized as well in the
histogram of coexisting ZESs (Fig. 9(c)). Similar studies have been performed
on the other two devices and details are presented in Fig. S11-Fig. S14 in the
supplementary material. The results do not yield evidence for correlated MZMs
either.

##  IV IV. Discussion

In this work, three-terminal InSb-Al hybrid nanowire devices have been
systematically studied using rf reflectometry. This approach introduces a
critical experimental technique to quickly map out a large phase space at both
ends of superconductor-semiconductor hybrid nanowires, which is crucial for
searching for candidate regions of a topological phase. A wide range of
chemical potential can be mapped out in high resolution by varying the super
gate voltage, and consequently the chance of missing topologically non-trivial
regions is minimized. Tunneling spectroscopy depending on super gate voltage
and magnetic field enables the extraction of LESs and ZESs, as well as induced
superconducting gap and effective g-factor. Constructed diagrams of ZESs and
LESs indicate the most promising regions for searching for MZMs
bib:Pikulin2021 . Combined with the tunneling-spectroscopy data, clusters of
ZPBs, parabolic and oscillatory patterns in ZES and LES diagrams are analyzed
for the first time in large scale and high resolution. While such patterns
mimic the predicted behavior of MZMs, the systematic exploration of all
accessible experimental parameters suggest a non-topological origin. Further
analysis by altering barrier gates indicate that the ZESs and LESs observed in
this work are likely localized in the vicinity of the tunnel junctions.
Additionally, the simultaneous detection of LESs at both ends of the hybrid
enables the possibility to look for correlated ZESs, which is essential to
prove the existence of paired MZMs. From the comparison of the LES and ZES
diagrams at two ends of the hybrid, no indication of correlated behavior is
observed in this work.

The absence of an unbroken topological superconducting phase in these samples
can be attributed to several physical origins. Possible reasons include
disorder, inhomogeneous interface band bending, local chemical potential
fluctuations and a non-perfect detection method. Here, four possible
shortcomings are elaborated. (1) In addition to forming trivial sub-gap levels
which can mimic MZMs, disorder effects could push the Majorana wavefunctions
away from the ends of the hybrid nanowires bib:Liu2013 ; bib:HainingPan2021_3
. In this case, even if a topological phase is formed in the bulk, the MZM
wavefunction cannot be probed by tunneling probes at the edge of the hybrid
nanowire. (2) Local chemical potential fluctuations with an energy above the
helical gap can break a single topological phase into segments. In this case,
the two ends of a hybrid nanowire are not necessarily correlated. In real
devices, such fluctuations can originate from non-uniform gating effects,
grain boundaries in aluminium, and disorder from impurities. (3) Recently, the
interface band bending between the superconductor and semiconductor was
recognized as an important ingredient for tuning the properties of the hybrid
bib:Antipov2018 ; bib:Mikkelsen2018 . In particular, band bending can lead to
the occupation of multiple subbands, and for the InSb-Al system the
experimental implications are still unknown. This makes it difficult to
predict the experimental conditions to achieve a topological superconducting
phase. (4) The experimental protocol in this work is specifically designed to
search for paired MZMs at two ends of an extended topological phase. If for
any reason, a topological phase would not be continuous across the entire
hybrid, it is possible that MZMs can be probed on one end but not on the
other. In this case, the protocol applied in this work would miss a potential
topological phase in the parameter space. Such a false negative cannot be
ruled out by these measurements.

For the aforementioned problems, several solutions can be proposed. (1) The
InSb nanowires in this work have transport mobilities of $\sim$ 4 $\times$ 104
cm2/V$\cdot$s bib:Gada2019 . The introduction of capping layers on top of the
semiconductor could alleviate disorder and improve transport mobilities
bib:Shabani2016 . (2) In order to improve resilience against local chemical
potential fluctuations, alternative superconductors with larger
superconducting gaps can be considered. Recent reports have successfully
realized growth of Sn ($\Delta\sim$ 700 $\mu$eV) bib:Pendharkar2021  and Pb
($\Delta\sim$ 1.25 meV) bib:Kanne2021  on semiconductor nanowires. These
hybrids may have a higher chance to achieve an uninterrupted topological phase
in spite of the challenges in fabrication. (3) Band bending at the interface
between superconductors and semiconductors can be engineered by inserting
modulation layers bib:Shabani2016 . Simultaneously, proper engineering of
these layers can also be used as a tool to influence the magnitude of the
induced superconducting gap. (4) Non-local measurements, proposed to probe the
superconducting gap on three-terminal devices bib:Tomas2018 , can complement
the fast rf reflectometry used in this work bib:Pikulin2021 . Corresponding
experimental work established its capability of detecting bulk properties
beyond the local characteristics bib:Gerbold2020 ; bib:D2020 , though the
measurement speed was slow. Combining our present protocol with non-local
measurement would strike a balance between measurement speed and detection
reliability.
The experimental protocol developed in this work, together with possible
improvements discussed above, will pave the way for unambiguously detecting
MZMs in superconductor-semiconductor hybrid systems in the future.

##  V Acknowledgement.

We thank Michael Wimmer and Anton Akhmerov for valuable comments on our
manuscript. We also thank John M. Hornibrook and David J. Reilly providing the
frequency multiplexing chips. We are grateful to Raymond Schouten, Olaf
Benningshof and J. Mensingh for valuable technical support. This work has been
financially supported by the Dutch Organization for Scientific Research (NWO)
and Microsoft Corporation Station Q.

##  VI Author contribution.

J.-Y.W., S.H. and L.P.K. conceived the experiment. J.-Y.W., G.M., N.v.L.,
M.L., F.B. and M.Q.P. fabricated the devices. J.-Y.W., V.L. and F.M. performed
the measurements. S.G., G.B. and E.P.A.M.B. carried out the material growth.
J.-Y.W. analyzed the data. J.-Y.W., N.v.L., G.P.M, V.L., and L.P.K wrote
manuscript with comments from all authors. J.-Y.W., S.H. and L.P.K. supervised
the project.

## References

  * (1) L. Fu and C. L. Kane, Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator, Phys. Rev. Lett. 100, 096407 (2008).
  * (2) R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures, Phys. Rev. Lett. 105, 077001 (2010).
  * (3) Y. Oreg, G. Refael, and F. von Oppen, Helical Liquids and Majorana Bound States in Quantum Wires, Phys. Rev. Lett. 105, 177002 (2010).
  * (4) V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers and L. P. Kouwenhoven, Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices, Science 336,1003–1007 (2012).
  * (5) M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu, Anomalous Zero-Bias Conductance Peak in a Nb-InSb Nanowire-Nb Hybrid Device, Nano Lett. 12, 6414–6419 (2012).
  * (6) A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum and H. Shtrikman, Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions, Nat. Phys. 8, 887–895 (2012).
  * (7) H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover, Phys. Rev. B 87, 241401(R) (2013)
  * (8) M. T. Deng, S. Vaitiekėnas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, C. M. Marcus, Majorana bound state in a coupled quantum-dot hybrid-nanowire system, Science 354, 1557-1562 (2016).
  * (9) J. Chen, P. Yu, J. Stenger, M. Hocevar, D. Car, S. R. Plissard, E. P. A. M. Bakkers, T. D. Stanescu, S. M. Frolov, Experimental phase diagram of zero-bias conductance peaks in superconductor/semiconductor nanowire devices, Sci. Adv. 3, 1701476 (2017).
  * (10) H. Zhang, M. W.A. de Moor, J. D.S. Bommer, D. Xu, G. Wang, N. van Loo, C.-X. Liu, S. Gazibegovic, J. A. Logan, D. Car, R. L. M. Op het Veld, P. J. van Veldhoven, S. Koelling, M. A. Verheijen, M. Pendharkar, D. J. Pennachio, B. Shojaei, J. Sue Lee, C. J. Palmstrøm, E. P.A.M. Bakkers, S. Das Sarma and L. P. Kouwenhoven, Large zero-bias peaks in InSb-Al hybrid semiconductor-superconductor nanowire devices, arXiv:2101.11456 (2021)
  * (11) S. Vaitiekėnas, G. W. Winkler, B. van Heck, T. Karzig, M.-T. Deng, K. Flensberg, L. I. Glazman, C. Nayak,P. Krogstrup, R. M. Lutchyn and C. M. Marcus, Flux-induced topological superconductivity in full-shell nanowires. Science 367, 1442 (2020)
  * (12) F. Nichele, A. C. C. Drachmann, A. M. Whiticar, E. C. T. O’Farrell, H. J. Suominen, A. Fornieri, T. Wang, G. C. Gardner, C. Thomas, A. T. Hatke, P. Krogstrup, M. J. Manfra, K. Flensberg and C. M. Marcus, Scaling of Majorana Zero-Bias Conductance Peaks, Phys. Rev. Lett. 119, 136803 (2017).
  * (13) A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portolés, A. C. C. Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. Kallaher, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern, C. M. Marcus, F. Nichele, Evidence of topological superconductivity in planar Josephson junctions, Nature 569, 89 (2019).
  * (14) H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp, B. I. Halperin, A. Yacoby, Topological superconductivity in a phase-controlled Josephson junction, Nature 569, 93 (2019).
  * (15) A. Vuik, B. Nijholt, A. R. Akhmerov, M. Wimmer, Reproducing topological properties with quasi-Majorana states, SciPost Phys. 7, 061 (2019).
  * (16) J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Zero-Bias Peaks in the Tunneling Conductance of Spin-Orbit-Coupled Superconducting Wires with and without Majorana End-States, Phys. Rev. Lett. 109, 267002 (2012).
  * (17) H. Pan and S. Das Sarma, Disorder effects on Majorana zero modes: Kitaev chain versus semiconductor nanowire, arXiv: 2012.12904 (2020).
  * (18) H. Pan, C.-X. Liu, M. Wimmer and S. Das Sarma, Quantized and unquantized zero-bias tunneling conductance peaks in Majorana nanowires: Conductance below and above 2$e^{2}/h$, arXiv: 2102.02218 (2021).
  * (19) H. Pan and S. Das Sarma, Crossover between trivial zero modes in Majorana nanowires, arXiv: 2102.07296 (2021).
  * (20) S. Das Sarma and H. Pan, Disorder-induced zero-bias peaks in Majorana nanowires, arXiv: 2103.05628 (2021).
  * (21) M. Valentini, F. Peñaranda, A. Hofmann, M. Brauns, R. Hauschild, P. Krogstrup, P. San-Jose, E. Prada, R. Aguado and G. Katsaros, Nontopological zero-bias peaks in full-shell nanowires induced by flux-tunable Andreev states. Science 373, 82-88 (2021)
  * (22) J. Liu, F.-C. Zhang, and K. T. Law, Majorana fermion induced nonlocal current correlations in spin-orbit coupled superconducting wires, Phys. Rev. B 88, 064509 (2013).
  * (23) H.-F. Lü, H.-Z. Lu, and S.-Q. Shen, Current noise cross correlation mediated by Majorana bound states, Phys. Rev. B 90, 195404 (2014).
  * (24) Y.-H. Lai, J. D. Sau, and S. Das Sarma, Presence versus absence of end-to-end nonlocal conductance correlations in Majorana nanowires: Majorana bound states versus Andreev bound states, Phys. Rev. B 100, 045302 (2019).
  * (25) G. L. R. Anselmetti, E. A. Martinez, G. C. M enard, D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm, C. M. Marcus, L. Casparis, and A. P. Higginbotham, End-to-end correlated subgap states in hybrid nanowires. Phys. Rev. B 100, 205412 (2019)
  * (26) A. E. Antipov, A. Bargerbos, G. W. Winkler, B. Bauer, E. Rossi and R. M. Lutchyn, Effects of Gate-Induced Electric Fields on Semiconductor Majorana Nanowires, Phys. Rev. X 8, 031041 (2018).
  * (27) D. I. Pikulin, B. van Heck, T. Karzig, E. A. Martinez, B. Nijholt, T. Laeven, G. W. Winkler, J. D. Watson, S. Heedt, M. Temurhan, V. Svidenko, R. M. Lutchyn, M. Thomas, G. de Lange, L. Casparis, and C. Nayak, Protocol to identify a topological superconducting phase in a three-terminal device, arXiv: 2103.12217 (2021).
  * (28) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.- S. Huang, J. Majer, S. Kumar, S. M. Girvin and R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature 431, 162-167 (2004).
  * (29) J. M. Hornibrook, J. I. Colless, A. C. Mahoney, X. G. Croot, S. Blanvillain, H. Lu, A. C. Gossard and D. J. Reilly, Frequency multiplexing for readout of spin qubits. Appl. Phys. Lett. 104, 103108 (2014).
  * (30) D. Razmadze, D. Sabonis, F. K. Malinowski, G. C. Ménard, S. Pauka, H. Nguyen, D. M. T. van Zanten, E. C.T. O’Farrell, J. Suter, P. Krogstrup, F. Kuemmeth and C. M. Marcus, Radio-Frequency Methods for Majorana-Based Quantum Devices: Fast Charge Sensing and Phase-Diagram Mapping, Phys. Rev. Applied 11, 064011 (2019).
  * (31) J. van Veen, D. de Jong, L. Han, C. Prosko, P. Krogstrup, J. D. Watson, L. P. Kouwenhoven and W. Pfaff, Revealing charge-tunneling processes between a quantum dot and a superconducting island through gate sensing, Phys. Rev. B 100, 174508 (2019).
  * (32) G. Badawy, S. Gazibegovic, F. Borsoi, S. Heedt, C.-A.Wang, S. Koelling, M. A. Verheijen, L. P. Kouwenhoven and E. P. A. M. Bakkers. High mobility stemless InSb nanowires, Nano Lett. 2019, 19, 3575-3582 (2019).
  * (33) S. Heedt, M. Quintero-Pérez, F. Borsoi, A. Fursina, N. van Loo, G. P. Mazur, M. P. Nowak, M. Ammerlaan, K. Li, S. Korneychuk, J. Shen, M. An Y. van de Poll, G. Badawy, S. Gazibegovic, N. de Jong, P. Aseev, K. van Hoogdalem, E. P. A. M. Bakkers, L. P. Kouwenhoven, Shadow-wall lithography of ballistic superconductor-semiconductor quantum devices, Nat. Commun. 12, 4914 (2021).
  * (34) F. Borsoi, G. P. Mazur, N. van Loo, M. P. Nowak, L. Bourdet, K. Li, S. Korneychuk, A. Fursina, J.-Y. Wang, V. Levajac, E. Memisevic, G. Badawy, S. Gazibegovic, K. van Hoogdalem, E. P. A. M. Bakkers, L. P. Kouwenhoven, S. Heedt, M. Quintero-Pérez, Single-shot fabrication of semiconducting-superconducting nanowire devices, Adv. Func. Mater. 2102388 (2021).
  * (35) R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov,P. Delsing, D. E. Prober, The Radio-Frequency Single-Electron Transistor (RF-SET): A Fast and Ultrasensitive Electrometer, Science 280, 1238 (1998).
  * (36) D. J. Reilly, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Fast single-charge sensing with a RF quantum point contact, Appl. Phys. Lett. 91, 162101 (2007).
  * (37) J. Stehlik, Y.-Y. Liu, C. M. Quintana, C. Eichler, T. R. Hartke, J. R. Petta, Fast Charge Sensing of a Cavity-Coupled Double Quantum Dot Using a Josephson Parametric Amplifier, Phys. Rev. Applied 4, 014018 (2015).
  * (38) J. Shen, G.W. Winkler, F. Borsoi, S. Heedt, V. Levajac, J.-Y. Wang, D. van Driel, D. Bouman, S. Gazibegovic, R. L. M. Op Het Veld, D. Car, J. A. Logan, M. Pendharkar, C. J. Palmstrøm, E. P. A. M. Bakkers, L. P. Kouwenhoven, B. van Heck, Full parity phase diagram of a proximitized nanowire island, Phys. Rev. B 104, 045422 (2021).
  * (39) A. E. G. Mikkelsen, P. Kotetes, P. Krogstrup, and K. Flensberg, Hybridization at Superconductor-Semiconductor Interfaces, Phys. Rev. X 8, 031040 (2018).
  * (40) S. D. Sarma, J. D. Sau, T. D. Stanescu, Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire, Phys. Rev. B 86, 220506(R) (2012).
  * (41) E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures, Nat. Nanotechnol. 9, 79-84 (2014).
  * (42) R. C. Dynes, V. Narayanamurti, and J. P. Garno, Direct Measurement of Quasiparticle-Lifetime Broadening in a Strong-Coupled Superconductor, Phys. Rev. Lett. 41, 1509 (1978).
  * (43) C.-X. Liu, J. D. Sau, T. D. Stanescu, and S. Das Sarma, Andreev bound states versus Majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaks, Phys. Rev. B 96, 075161 (2017).
  * (44) J. Shabani, M. Kjaergaard, H. J. Suominen, Y. Kim, F. Nichele, K. Pakrouski, T. Stankevic, R. M. Lutchyn, P. Krogstrup, R. Feidenhans’l, S. Kraemer, C. Nayak, M. Troyer, C. M. Marcus, and C. J. Palmstrøm, Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconducting networks, Phys. Rev. B 93, 155402 (2016).
  * (45) M. Pendharkar, B. Zhang, H. Wu, A. Zarassi, P. Zhang, C. P. Dempsey, J. S. Lee, S. D. Harrington, G. Badawy, S. Gazibegovic, R. L. M. Op het Veld, M. Rossi, J. Jung, A.-H. Chen, M. A. Verheijen, M. Hocevar, E. P. A. M. Bakkers, C. J. Palmstrøm, S. M. Frolov, Parity-preserving and magnetic field–resilient superconductivity in InSb nanowires with Sn shells, Science 372, 508-511 (2021).
  * (46) T. Kanne, M. Marnauza, D. Olsteins, D. J. Carrad, J. E. Sestoft, J. de Bruijckere, L. Zeng, E. Johnson, E. Olsson, K. Grove-Rasmussen, and J. Nygård, Epitaxial Pb on InAs nanowires for quantum devices, Nat. Nanotechnol. 16, 776-781 (2021).
  * (47) T. Ö. Rosdahl, A. Vuik, M. Kjaergaard, and A. R. Akhmerov, Andreev rectifier: A nonlocal conductance signature of topological phase transitions, Phys. Rev. B 97, 045421 (2018).
  * (48) G. C. Ménard, G. L. R. Anselmetti, E. A. Martinez, D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm, K. Flensberg, C. M. Marcus, L. Casparis and A. P. Higginbotham, Conductance-Matrix Symmetries of a Three-Terminal Hybrid Device, Phys. Rev. Lett. 124, 036802 (2020).
  * (49) D. Puglia, E. A. Martinez, G. C. Ménard, A. Pöschl, S. Gronin, G. C. Gardner, R. Kallaher, M. J. Manfra, C. M. Marcus, A. P. Higginbotham and L. Casparis, Closing of the Induced Gap in a Hybrid Superconductor-Semiconductor Nanowire, Phys. Rev. B 103, 235201 (2021).


11institutetext: BCN Medtech, Dept. of Information and Communication
Technologies,
Universitat Pompeu Fabra, Barcelona, Spain, 11email<EMAIL_ADDRESS>22institutetext: Australian Institute for Machine Learning, University of
Adelaide, Australia 33institutetext: ICREA, Barcelona, Spain

# Convolutional Nets Versus Vision Transformers for Diabetic Foot Ulcer
Classification

Adrian Galdran 1122 Gustavo Carneiro 22 Miguel A. González Ballester 1133

###### Abstract

This paper compares well-established Convolutional Neural Networks (CNNs) to
recently introduced Vision Transformers for the task of Diabetic Foot Ulcer
Classification, in the context of the DFUC 2021 Grand-Challenge, in which this
work attained the first position. Comprehensive experiments demonstrate that
modern CNNs are still capable of outperforming Transformers in a low-data
regime, likely owing to their ability for better exploiting spatial
correlations. In addition, we empirically demonstrate that the recent
Sharpness-Aware Minimization (SAM) optimization algorithm improves
considerably the generalization capability of both kinds of models. Our
results demonstrate that for this task, the combination of CNNs and the SAM
optimization process results in superior performance than any other of the
considered approaches.

###### Keywords:

Diabetic Foot Ulcer Classification, Vision Transformers,
Convolutional Neural Networks, Sharpness-Aware Optimization

##  1 Introduction and Problem Statement

Diabetes affects 425 million people with a projection to reach 629 million
affected people by 2045 [6]. A particularly concerning development of diabetes
are foot ulcers: the risk of a diabetic patient developing a foot ulcer along
their lifetime is estimated to be 19-34% [2].

Diabetic Foot Ulcers (DFU) presented with ischaemia and/or infections can lead
to limb gangrene, amputation, or even patient death. Iscahemia is
characterized by a deficient blood circulation, and it can be recognized by
poor vascular reperfusion or dark gangrenous patterns. On the other hand,
infection is associated to the presence of tissue inflammation or purulence
[12]. Although continuous monitoring and timely diagnosis is crucial to avoid
these serious complications, inappropriate care and lack of screening on
diabetic population, as well as the lack of medical specialists in developing
countries poses a barrier towards early discovery and appropriate management
of DFU. In this setting, computer-supported diagnosis by means of computer
vision techniques and conventional smartphone cameras represents a promising
and economically viable alternative, with some recent research focusing
already on developing this kind of technology [26, 25, 12]. Therefore,
previous research has addressed the need of automatic image understanding
towards DFU analysis. Ranging from conventional feature engineering-based
image classification [22] based on colour and texture to more advanced two-
stage approaches involving lesion detection or segmentation followed by
classification [4, 1], and more recently deep learning approaches [7, 17]. A
detailed recent review on machine learning applied to DFU can be found in
[20].

Within this context, the Diabetic Foot Ulcer Challenge (DFUC) 2021 was held in
conjunction with the MICCAI 2021 conference. The task at hand is multi-class
classification of foot images into four categories: no pathology, infection,
ischaemia, or presence of both pathologies, as illustrated in Fig. 1. This
paper describes our approach to solving the challenge, in which we obtained
the first position. In addition, we make to contributions: 1) we compare the
performance of two popular Convolutional Neural Networks (CNNs) [18, 13] with
that of two more recent Vision Transformers [9, 19] in this problem, revealing
that CNNs may still be the preferred choice for this kind of problems, and 2)
we experiment with the Sharpness-Aware (SAM, [10]) extension of the popular
Stochastic Gradient Descent optimization process, demonstrating that SAM
yields superior performance in all the analyzed cases. More details on the
DFUC 2021 organization, as well as live test leaderboard, can be found online
at https://dfu-2021.grand-challenge.org/ .

In the next section, we first describe the two CNN and the two Transformer
architectures we train for solving the task at hand. We then briefly recall
the SAM optimization algorithm, and provide a description of the training
process. Next, we present the data used for training and validating our
analysis, and the performance obtained by each of the considered approaches,
as well as the final test performance of our best model. We conclude the paper
with some discussion and further research directions.

(a)

(b)

(c)

(d)

Figure 1: Randomly sampled diabetic foot images from the DFUC 2021 dataset.
(a) No infection and no ischaemia, (b) presence of infection, (c) presence of
ischaemia, (d) presence of both conditions. Images provided by the
organization.

##  2 Methodology

We now describe the main methodological aspects of DFU classification in this
paper: CNN and Vision Transformer architectures, model optimization process
and training data.

###  2.1 Convolutional Neural Networks and Vision Transformers

In this work we experiment with two different kind of neural networks for
computer vision, namely conventional Convolutional Neural Networks (CNNs) and
more recent Vision Transformers. CNNs have become the de-facto approach to
most computer vision tasks in the last decade, since their initial dominance
of the 2012 ImageNet challenge [15]. Essential to the learning ability of CNNs
from image data are several distinctive features, namely their built-in
inductive bias (spatial translation invariance imposed through convolutions
with shared learnable filters across the image), or a representation hierarchy
arising from their layer-wise architecture design, by which larger patterns
are learned as sequential combinations of smaller patterns. On the other hand,
Transformers have emerged in the last few years as the most powerful
architecture for Natural Language Processing applications, dominating
virtually all public benchmarks111Transformers are based on the notion of
attention, the detailed description of which falls beyond the scope of this
paper [21]. Recently, the Transformer architecture has been extended to other
domains and data types, like graphs, speech, or vision [8, 27, 9]. After the
initial exploration of Transformers for computer vision in [9], where a
variety of image recognition tasks were successfully approached with this
architecture, the amount of research dedicated to proposing Vision
Transformers as an alternative to CNNs has increased exponentially. Generally
speaking, Vision Transformers consider images as sequences of small patches
akin to words or tokens, that can be supplied to standard Transformers.
However, and in contrast with CNNs, Transformers have no notion of distance
within an image, since patches are processed sequentially. Spatial
relationships need to be modeled by means of positional encodings/embeddings
and learned on extremely large datasets, in which case the increased
flexibility of Transformers could favor them over CNNs. In this paper, we test
Transformers’ ability to generalize for a relatively small dataset of lesion
images like those shown on Fig. 1. Specifically, the four architectures we
will analyze for our problem are enumerated below.

  * •

Big Image Transfer (BiT) [13]. We use the ResNeXt50 architecture from this
work, which contains 25,55M parameters, uses Group Normalization instead of
BatchNorm, and implements convolutional layers with Weight Standardization.

  * •

EfficientNet [18], which is a family of architectures designed by neural
architecture search, and carefully scaled so that they achieve state-of-the-
art performance with a fraction of parameters. In this paper we use the
EfficientNet B3 architecture, which contains 12,23M weights.

  * •

Vision Transformers (ViT) [9], the first architecture to replace convolutions
by attention, this is a pure transformer applied on sequences of image
patches. We use the ViT-base configuration, containing 22,20M weights.

  * •

Data-efficient Image Transformers (DeIT) [19], a refinement of ViT with
improved pre-training strategies. We use the DeiT-small variant, which
contains 22M learnable parameters, and again a patch size of $16\times 16$.

###  2.2 Sharpness-Aware Optimization and Training Details

In this paper we are also interested in comparing standard Stochastic Gradient
Descent (SGD, [3]) to a recently introduced approach for neural network
optimization that tries to improve their generalization ability, namely
Sharpness-Aware Optimization ( SAM, [10, 14]) . SAM simultaneously minimizes
the value of a loss function and the loss sharpness, thereby seeking to find
parameters lying in neighborhoods with a uniformly low loss. This way, the
goal of SAM is to improve model generalization, and it has been shown to
deliver State-of-the-Art performance for different applications, while also
enforcing robustness against label noise. Further technical details about this
optimization algorithm can be found in [10].

Other than the optimization algorithm choice, all the models trained in this
work follow the same process, similarly to [11]: network weights are optimized
so as to minimize the cross-entropy loss with each of the optimizers adopting
an initial learning rate of $l=0.01$ and a batch-size of $32$. Note that these
two values were selected by grid search on an initial train-validation split.
The learning rate is decayed following a cosine law from its initial value to
$l=1e$-$8$ during $3$ epochs, which defines a training cycle. This cycle is
repeated $25$ times, restarting the learning back at the beginning. During
training images are augmented with standard techniques (random rotations,
vertical/horizontal flipping, contrast/saturation/brightness changes). The
F1-score is monitored on an independent validation set and the best performing
model is kept for testing purposes. For testing, we generate four different
versions of each image by horizontal/vertical flipping, predict on each of
them, and average the results.

Note that in all cases, optimization starts from weights pre-trained on the
ImageNet classification task222All architectures and pretrained weights are
taken from [23] at a $224\times 224$ resolution, which happens to coincide
with the size of the image patches provided in the DFUC 2021 challenge, as
explained in the next section.

###  2.3 Training Data

In DFUC 2021, participants were provided with a dataset composed of 15,683 DFU
RGB image patches of a fixed 224$\times 224$ resolution, with 5,955 training
and 5,734 testing unlabelled DFU patches for final ranking purposes. An
additional set of 3,994 unlabeled DFU patches was released for training,
although it was unused in this work. These patches were extracted from close-
ups of diabetic feet at a distance of approximately 30-40 cm perpendicularly
to the plane of the ulcer. Natural data augmentation was included (repeated
takes of the same lesion), and controlled for with automatic image similarity
techniques while splitting the data into training and test subsets. More
details on this dataset can be found in [24].

##  3 Experimental Results and Analysis

Below we analyze the performance of the four architectures described in
section 2.1 in the context of the DFUC 2021. The organizers provided
participants with a labeled training set and offered an unlabeled validation
set as well as a public leaderboard to which preliminary submissions could be
submitted. After the validation period was over, the participants had to
submit their predictions on a second hidden test set, and the final ranking
was based on the latter. In all cases, the models described below were trained
on a 4-fold split of the labeled training data, with predictions generated by
average ensembling. Models were implemented in PyTorch [16] and trained on a
workstation equipped with a NVIDIA GeForce RTX 3080 with 10 Gb memory size.

###  3.1 Impact of SAM optimization

In this section we compare the performance of the aforementioned four
architectures when trained as specified above, employing both standard SGD and
Sharpness-Aware loss minimization with SGD as base optimizer. Performance
measures include F1-score (the metric that drives the DFU competition), as
well as AUC, recall and precision on its macro versions. For this experiment,
these quantities are measured on the intermediate validation set released by
the organizers. Table 1 shows such results for all cases.

Model/Performance | F1-score | AUC | Recall | Precision
---|---|---|---|---
BiT-ResNeXt50 SGD | 51.34 | 84.79 | 54.99 | 51.05
BiT-ResNeXt50 SAM | 57.71 | 87.68 | 61.88 | 57.74
[2pt/5pt] Perf. Diff. | +6.37 | +2.89 | +6.89 | +6.69
EfficientNet B3 SGD | 47.55 | 82.92 | 49.49 | 48.47
EfficientNet B3 SAM | 57.65 | 84.81 | 60.74 | 57.13
[2pt/5pt] Perf. Diff. | +10.05 | +1.89 | +11.25 | +8.66
ViT SGD | 48.84 | 85.35 | 53.13 | 51.07
ViT SAM | 51.94 | 87.85 | 56.52 | 54.38
[2pt/5pt] Perf. Diff. | +3.10 | +2.50 | +3.39 | +3.31
DeiT SGD | 53.87 | 84.92 | 58.01 | 53.01
DeiT SAM | 53.98 | 85.97 | 58.12 | 54.03
[2pt/5pt] Perf. Diff. | +0.11 | +1.05 | +0.11 | +1.02
Table 1: Performance analysis of different combinations of architectures with
SGD and SAM optimization procedures. Note that all metrics are macro-averaged
across categories.

###  3.2 CNNs vs Transformers

We also study the performance of each of our four architectures in the final
hidden test set. For this, Table 2 shows the metrics obtained by each model,
whereas Table 3 describes the result of our final submission for the challenge
classification, compared to the results obtained by the top-ranked
contestants.

Model/Performance | F1-score | AUC | Recall | Precision
---|---|---|---|---
BiT-ResNeXt50 | 61.53 | 88.49 | 65.59 | 60.53
EfficientNet B3 | 59.71 | 87.01 | 61.79 | 59.34
ViT | 58.48 | 87.64 | 62.27 | 58.91
DeiT | 57.29 | 87.98 | 61.35 | 57.01
Table 2: Performance analysis in the final test set for the four different architectures considered in our comparison. Note that all metrics are macro-averaged across categories. Participant/Performance | F1-score | AUC | Recall | Precision
---|---|---|---|---
agaldran (this work) | 62.16 | 88.55 | 65.22 | 61.40
Louise.Bloch | 60.77 | 86.16 | 62.46 | 62.07
S. Ahmed | 59.59 | 86.44 | 59.79 | 59.84
Abdul | 56.91 | 84.88 | 61.04 | 58.14
orhunguley | 56.10 | 87.02 | 57.59 | 59.17
Table 3: Performance analysis in the final test set for the five top
competitors sorted by ranking in terms of F1-score. Note that our submission
in this table corresponds to an ensemble of the predictions from the BiT-
ResNeXt50 and EfficientNet B3 models in Table 2. Note that all metrics are
macro-averaged across categories.

###  3.3 Discussion of the Results

The first conclusion we can draw from the above results is that the SAM
optimization generally benefits the end performance of both CNNs and
Transformers, with CNNs always appearing to be superior to Transformers in all
cases. It should be noted that SAM requires two forward passes instead of one
before weight update, which turns it into a relatively expensive alternative
to SGD. However, results in Table 1 appear to indicate that the extra
computational load is widely compensated, particularly when working with small
datasets for which the cost of one training run is not substantial.

Next, we can see in Table 2 that, as expected, the relative ranking of our
four models in the validation set is preserved when evaluating them in the
final test set. However, there is a noticeable increase in the performance of
Vision Transformers, which attain results close to those of the EfficientNet
B3 architecture. It is also worth stressing that, just as in the validation
experiments on Table 1, the BiT-ResneXt50 model still achieves the highest
performance, surpassing by a good margin the second best model, which is
EfficientNet B3.

Finally, we see from the final ranking of the DFUC 2021 challenge333Ongoing
(live) leaderboard can be accessed at https://dfu-2021.grand-
challenge.org/evaluation/live-testing-leaderboard/leaderboard/. in Table 3
that our approach is validated when compared to other contestants’ results.
Our final highest score was achieved by a linear combination of the
predictions extracted from BiT-ResneXt50 and EfficientNet B3, but it is worth
noting that BiT-ResneXt50 alone would still be the ranked the best method, and
EfficientNet B3 would have been ranked second.

##  4 Conclusion

This paper describes the winning solution to the DFUC 2021 grand-challenge on
Diabetic Foot Ulcer Classification, held in conjunction with MICCAI 2021,
which is based on an ensemble of CNNs (BiT-Resnext50 and EfficientNet B3)
trained on different data folds. In addition, for weight optimization we
employ Sharpness-Aware Minimization, which provides a noticeable improvement
in performance for all considered models. Note that, in the literature, SAM
has been benchmarked mostly with CNNs, which could introduce a bias of this
optimization algorithm towards that option in this paper. However, it has been
recently shown that SAM has equally good impact in performance for Vision
transformers [5]. Finally, we also include a comparison of the performance of
CNNs and Vision Transformers for this task, which reveals that despite the
recent popularity of Transformers for Computer Vision tasks, CNNs may still be
the preferred choice in few-data scenarios.

## Acknowledgments

Adrian Galdran was funded by a Marie Skłodowska-Curie Global Fellowship (No
892297). Gustavo Carneiro was partially supported by Australian Research
Council grants (DP180103232 and FT190100525)

## References

  * [1] Amin, J., Sharif, M., Anjum, M.A., Khan, H.U., Malik, M.S.A., Kadry, S.: An Integrated Design for Classification and Localization of Diabetic Foot Ulcer Based on CNN and YOLOv2-DFU Models. IEEE Access 8, 228586–228597 (2020). https://doi.org/10.1109/ACCESS.2020.3045732, conference Name: IEEE Access
  * [2] Armstrong, D.G., Boulton, A.J.M., Bus, S.A.: Diabetic Foot Ulcers and Their Recurrence. The New England Journal of Medicine 376(24), 2367–2375 (Jun 2017). https://doi.org/10.1056/NEJMra1615439
  * [3] Bottou, L., Bousquet, O.: The Tradeoffs of Large Scale Learning. In: Platt, J., Koller, D., Singer, Y., Roweis, S. (eds.) Advances in Neural Information Processing Systems. vol. 20. Curran Associates, Inc. (2008), https://proceedings.neurips.cc/paper/2007/file/0d3180d672e08b4c5312dcdafdf6ef36-Paper.pdf
  * [4] Brüngel, R., Friedrich, C.M.: DETR and YOLOv5: Exploring Performance and Self-Training for Diabetic Foot Ulcer Detection. In: 2021 IEEE 34th International Symposium on Computer-Based Medical Systems (CBMS). pp. 148–153 (Jun 2021). https://doi.org/10.1109/CBMS52027.2021.00063, iSSN: 2372-9198
  * [5] Chen, X., Hsieh, C.J., Gong, B.: When Vision Transformers Outperform ResNets without Pre-training or Strong Data Augmentations. arXiv:2106.01548 [cs] (Oct 2021), http://arxiv.org/abs/2106.01548, arXiv: 2106.01548
  * [6] Cho, N.H., Shaw, J.E., Karuranga, S., Huang, Y., Fernandes, J.D.d.R., Ohlrogge, A.W., Malanda, B.: IDF Diabetes Atlas: Global estimates of diabetes prevalence for 2017 and projections for 2045. Diabetes Research and Clinical Practice 138, 271–281 (Apr 2018). https://doi.org/10.1016/j.diabres.2018.02.023, publisher: Elsevier
  * [7] Das, S.K., Roy, P., Mishra, A.K.: Recognition of ischaemia and infection in diabetic foot ulcer: A deep convolutional neural network based approach. International Journal of Imaging Systems and Technology n/a(n/a). https://doi.org/10.1002/ima.22598, _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/ima.22598
  * [8] Dong, L., Xu, S., Xu, B.: Speech-Transformer: A No-Recurrence Sequence-to-Sequence Model for Speech Recognition. In: 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). pp. 5884–5888 (Apr 2018). https://doi.org/10.1109/ICASSP.2018.8462506, iSSN: 2379-190X
  * [9] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., Houlsby, N.: An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale. In: International Conference on Learning Representations (2021)
  * [10] Foret, P., Kleiner, A., Mobahi, H., Neyshabur, B.: Sharpness-aware Minimization for Efficiently Improving Generalization. In: International Conference on Learning Representations (2021), https://openreview.net/forum?id=6Tm1mposlrM
  * [11] Galdran, A., Anjos, A., Dolz, J., Chakor, H., Lombaert, H., Ayed, I.B.: The Little W-Net That Could: State-of-the-Art Retinal Vessel Segmentation with Minimalistic Models (Sep 2020), http://arxiv.org/abs/2009.01907
  * [12] Goyal, M., Reeves, N.D., Rajbhandari, S., Ahmad, N., Wang, C., Yap, M.H.: Recognition of ischaemia and infection in diabetic foot ulcers: Dataset and techniques. Computers in Biology and Medicine 117, 103616 (Feb 2020)
  * [13] Kolesnikov, A., Beyer, L., Zhai, X., Puigcerver, J., Yung, J., Gelly, S., Houlsby, N.: Big Transfer (BiT): General Visual Representation Learning. In: ECCV (2020)
  * [14] Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)
  * [15] Krizhevsky, A., Sutskever, I., Hinton, G.E.: ImageNet Classification with Deep Convolutional Neural Networks. In: Pereira, F., Burges, C.J.C., Bottou, L., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems. vol. 25. Curran Associates, Inc. (2012), https://proceedings.neurips.cc/paper/2012/file/c399862d3b9d6b76c8436e924a68c45b-Paper.pdf
  * [16] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S.: PyTorch: An Imperative Style, High-Performance Deep Learning Library. In: Wallach, H., Larochelle, H., Beygelzimer, A., Alché-Buc, F.d., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems 32, pp. 8024–8035. Curran Associates, Inc. (2019), http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf
  * [17] Rostami, B., Anisuzzaman, D.M., Wang, C., Gopalakrishnan, S., Niezgoda, J., Yu, Z.: Multiclass wound image classification using an ensemble deep CNN-based classifier. Computers in Biology and Medicine 134, 104536 (Jul 2021). https://doi.org/10.1016/j.compbiomed.2021.104536
  * [18] Tan, M., Le, Q.: EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks. In: International Conference on Machine Learning. pp. 6105–6114. PMLR (May 2019)
  * [19] Touvron, H., Cord, M., Douze, M., Massa, F., Sablayrolles, A., Jegou, H.: Training data-efficient image transformers & distillation through attention. In: Proceedings of the 38th International Conference on Machine Learning (Jul 2021)
  * [20] Tulloch, J., Zamani, R., Akrami, M.: Machine Learning in the Prevention, Diagnosis and Management of Diabetic Foot Ulcers: A Systematic Review. IEEE Access 8, 198977–199000 (2020). https://doi.org/10.1109/ACCESS.2020.3035327, conference Name: IEEE Access
  * [21] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., Polosukhin, I.: Attention is All you Need. In: Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R. (eds.) Advances in Neural Information Processing Systems. vol. 30. Curran Associates, Inc. (2017), https://proceedings.neurips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf
  * [22] Veredas, F., Mesa, H., Morente, L.: Binary Tissue Classification on Wound Images With Neural Networks and Bayesian Classifiers. IEEE Transactions on Medical Imaging 29(2), 410–427 (Feb 2010). https://doi.org/10.1109/TMI.2009.2033595, conference Name: IEEE Transactions on Medical Imaging
  * [23] Wightman, R.: PyTorch Image Models (2019). https://doi.org/10.5281/zenodo.4414861, https://github.com/rwightman/pytorch-image-models, publication Title: GitHub repository
  * [24] Yap, M.H., Cassidy, B., Pappachan, J.M., O’Shea, C., Gillespie, D., Reeves, N.: Analysis Towards Classification of Infection and Ischaemia of Diabetic Foot Ulcers (Jun 2021), http://arxiv.org/abs/2104.03068
  * [25] Yap, M.H., Chatwin, K.E., Ng, C.C., Abbott, C.A., Bowling, F.L., Rajbhandari, S., Boulton, A.J.M., Reeves, N.D.: A New Mobile Application for Standardizing Diabetic Foot Images. Journal of Diabetes Science and Technology 12(1), 169–173 (Jan 2018). https://doi.org/10.1177/1932296817713761
  * [26] Yap, M.H., Ng, C.C., Chatwin, K., Abbott, C.A., Bowling, F.L., Boulton, A.J.M., Reeves, N.D.: Computer Vision Algorithms in the Detection of Diabetic Foot Ulceration. Journal of Diabetes Science and Technology 10(2), 612–613 (Oct 2015)
  * [27] Yun, S., Jeong, M., Kim, R., Kang, J., Kim, H.J.: Graph Transformer Networks. In: Wallach, H., Larochelle, H., Beygelzimer, A., Alché-Buc, F.d., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems. vol. 32. Curran Associates, Inc. (2019), https://proceedings.neurips.cc/paper/2019/file/9d63484abb477c97640154d40595a3bb-Paper.pdf