# Evolution of ${4\pi}$-periodic Supercurrent in the Presence of In-plane
Magnetic Field

Bassel Heiba Elfeky1 Joseph J. Cuozzo2 Neda Lotfizadeh1 William F. Schiela1
Seyed M. Farzaneh1 William M. Strickland1 Dylan Langone1 Enrico Rossi2 Javad
Shabani1<EMAIL_ADDRESS>1Center for Quantum Information Physics, Department
of Physics, New York University, NY 10003, USA  2Department of Physics,
William $\&$ Mary, Williamsburg, VA, 23187, USA

###### Abstract

In the presence of a 4$\pi$-periodic contribution to the current phase
relation, for example in topological Josephson junctions, odd Shapiro steps
are expected to be missing. While missing odd Shapiro steps have been observed
in several material systems and interpreted in the context of topological
superconductivity, they have also been observed in topologically trivial
junctions. Here, we study the evolution of such trivial missing odd Shapiro
steps in Al-InAs junctions in the presence of an in-plane magnetic field
$B^{\theta}$. We find that the odd steps reappear at a crossover $B^{\theta}$
value, exhibiting an in-plane field angle anisotropy that depends on spin-
orbit coupling effects. We interpret this behavior by theoretically analyzing
the Andreev bound state spectrum and the transitions induced by the non-
adiabatic dynamics of the junction and attribute the observed anisotropy to
mode-to-mode coupling. Our results highlight the complex phenomenology of
missing Shapiro steps and the underlying current phase relations in planar
Josephson junctions designed to realize Majorana states.

Josephson junction, missing Shapiro steps, spin-orbit coupling, Landau-Zener
transitions, topological superconductivity

††preprint:

Josephson junctions (JJs) fabricated on semiconductor structures with
epitaxially grown superconductors have recently attracted attention due to
their propitious characteristics 1; 2; 3; 4; 5; 6; 7; 8; 9 and applications in
quantum computing 10; 11; 12; 13; 14; 15; 16; 17; 18. In the presence of a
Zeeman field 19; 20; 21 or a phase bias 22; 23; 24, and a strong spin-orbit
coupling (SOC) interaction, such high-quality JJs have shown signatures of
topological superconductivity 22; 23; 21; 24, which can host Majorana zero
modes useful for fault-tolerant quantum computation 25; 26. However, robust
implementation and signatures of topological superconductivity remain
ambiguous 27; 28; 29; 30; 31.

To harness the potential of topological superconductivity, it is essential to
be able to identify unambiguously the topological character of the states in a
JJ. Topological JJs exhibit a unique fractional Josephson effect which is
inaccessible with DC measurements due to relaxation processes to the ground
state. Consequently, detecting the fractional Josephson effect requires
measurements on timescales shorter than the relaxation time 32; 33; 34; 35;
36; 37; timescales that are accessible using microwave excitations 38; 39; 40;
41; 42.

Figure 1: Josephson junction geometry, Andreev spectrum and characterization.
a Schematic drawing of the material heterostructure with a junction of width
$w$ and length $l$ made of Al superconducting contacts and an InAs surface
quantum well. The 2D axis represents the direction of an applied in-plane
magnetic field, where $\theta$ is the in-plane field angle such that
$B^{0^{\circ}}$ is the in-plane field along the junction and $B^{90^{\circ}}$
is the in-plane field along the current. b Example of calculated energy
spectrum of the Andreev bound states in a wide junction with no applied
magnetic field. The results obtained are for a JJ with
$w=$500\text{\,}\mathrm{n}\mathrm{m}$$ and
$l=$100\text{\,}\mathrm{n}\mathrm{m}$$, superconducting gap
$\Delta=$300\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$, and
carrier density $n=4\times 10^{11}$\text{\,}\mathrm{c}\mathrm{m}$^{-2}$. The
long junction modes that contribute to $I_{4\pi}$ are separated by $\delta$
from the quasicontinuum at $E\sim\Delta$. c Energy spectrum in the presence of
a finite magnetic field with the spin-split long junction modes (red). The
dots on the modes indicate an occupied state. Arrows indicate possible mode to
mode (green) and mode to continuum (light-blue/yellow) transitions. The dark-
blue arrow indicates relaxation processes that fill low energy unoccupied
states. The purple box defines a forbidden transition due to both states being
occupied. d, e Differential resistance as a function of current bias, along
with a histogram of the voltage distribution, all as a function of RF power,
for frequency: d $f=$12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, and e
$f=$7\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ with no applied field,
$B^{\theta}=$0\text{\,}\mathrm{m}\mathrm{T}$$, for JJ1. The numbers correspond
to the index of the Shapiro steps. f Differential resistance as a function of
current bias and $B^{0^{\circ}}$ for JJ1.

When a microwave bias is applied to a JJ, the periodic modulation of the
current bias becomes phase locked with the dynamics of the junction and
results in constant voltage steps in the voltage-current characteristic known
as Shapiro steps. The Andreev bound states (ABSs) of a conventional JJ in the
short ballistic regime are 2$\pi$-periodic in phase $\phi$, resulting in
Shapiro steps at values of $n\frac{hf}{2e}$, where $f$ is the frequency of the
microwave drive, and $n$ is an integer. When the current phase relation (CPR)
is 4$\pi$-periodic, as expected for a topological JJs, the fractional
Josephson effect results in Shapiro steps only at $n\frac{hf}{e}$, resulting
in missing odd Shapiro steps. Missing Shapiro steps have been observed in
different material systems and are usually attributed to the presence of a
topological state 38; 39; 40; 43; 42; 44. In practice, even for a topological
JJ, a 4$\pi$-periodic component CPR coexists with a 2$\pi$-periodic component
in which case the absence of odd Shapiro steps depends on the details of the
junction, and the frequency and power of the microwave radiation 45; 46; 47.

Recent work 48 has experimentally shown that topologically trivial JJs can
also exhibit missing odd Shapiro steps as predicted previously by other
theoretical works 49; 45; 50; 51; 44. This can happen when ABSs with a large
probability of undergoing a Landau-Zener transition (LZT) at $\phi\sim\pi$,
and a negligible probability of crossing into the continuum, are present.
Other mechanisms responsible for missing Shapiro steps have also been proposed
involving a bias-dependent junction resistance 52, or the presence of
multiband superconducting states 53. Therefore, the observation of
4$\pi$-periodic supercurrent I4π or missing Shapiro steps is a necessary
signature of topological superconductivity but is not conclusive. Given that
an in-plane magnetic field $B^{\theta}$ is one of the ingredients required to
drive a JJ to a topological transition, understanding how missing Shapiro
steps depend on $B^{\theta}$ is essential to distinguish a trivial JJ from its
topological counterpart.

In this work, we present measurements on highly-transparent epitaxial Al-InAs
JJs in the presence of an in-plane magnetic field $B^{\theta}$ and SOC
effects, conditions associated with inducing topological superconductivity.
For $B^{\theta}=$0\text{\,}\mathrm{m}\mathrm{T}$$, we observe missing odd
Shapiro steps with no applied field due to the presence of a topologically
trivial $I_{4\pi}$ as observed previously 48. As $B^{\theta}$ is increased,
these missing Shapiro steps eventually reappear and no topological signatures
are observed up to the junction critical field $B^{\theta}_{c}$. The
reappearance of the missing steps exhibits angle anisotropies that depend on
the angle-dependent $B^{\theta}_{c}$ and carrier density associated with SOC
interaction effects. Our results show the complex dependence of topologically
trivial $I_{4\pi}$ on the applied in-plane field magnitude and direction, and
SOC effects.

Figure 2: Missing Shapiro step reemergence at finite in-plane magnetic field.
Differential resistance as a function of current bias and RF power at a-d
$f=$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and e-h
$f=$6.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ for different $B^{0^{\circ}}$
values for JJ1.

Fig. 1a presents the junction heterostructure studied. An InAs near-surface
quantum well is grown between two layers of In0.81Ga0.19As which is then
capped with a thin layer of epitaxial Al grown in situ. Two JJs, JJ1 and JJ2,
are fabricated on two different wafers grown under slightly different growth
conditions (see Supporting Information). The junctions are defined using a
selective wet etch of the Al and are
$w=$4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ wide and
$l\sim$100\text{\,}\mathrm{n}\mathrm{m}$$ long. Given $l$ of the junctions,
the calculated mean free path to be $l_{\rm mfp}\approx
150-$250\text{\,}\mathrm{n}\mathrm{m}$$, and the superconducting coherence
length $\xi\approx 530-$630\text{\,}\mathrm{n}\mathrm{m}$$, the junctions are
expected to be in the short ($l<\xi$) ballistic ($l<l_{\rm mfp}$) regime.

To get insight into the dynamics of such highly transparent junctions, we
first perform tight binding simulations of an Al-InAs junction using realistic
parameters and calculate the energy spectrum of the ABSs shown in Fig. 1b
(simulation details are provided in Supporting Information). The calculations
of these wide junctions present a complex ABS spectrum with hundreds of modes.
For a junction with width larger than the coherence length ($w>\xi$), modes
with momentum primarily along the transverse direction behave effectively as
“long junction” modes 48. Consequently, these modes develop a detachment gap
$\delta$ from the continuum when the phase difference across the junction
$\phi$ is zero, as indicated in Fig. 1b. The number of long junction modes and
their $\delta$ is sensitive to several factors (density $n$, $w$,…). When the
junction is highly transparent, the gap at $\phi=\pi$ is sufficiently small to
allow Landau-Zener transitions (LZTs) when the system is diabatically driven
49; 45; 48. The combination of a large detachment gap and a small gap at
$\phi=\pi$ for these long junction modes gives rise to a $4\pi$-periodic
contribution to the CPR, causing a topologically trivial junction to have both
$2\pi$\- and $4\pi$-periodic supercurrent channels 46; 47. In the presence of
a magnetic field in the plane of the junction, the Zeeman effect splits the
ABSs and eventually leads to the closing of the detachment gap of the long
junction modes, as seen in Fig. 1c. The $4\pi$-periodic trajectory of long
junction modes is then suppressed due to transitions to the continuum.
Additionally, LZTs may occur between long junction modes and other modes with
negligible detachments gaps, leading to transitions to the continuum mediated
by conventional ABSs and suppressing $I_{4\pi}$.

To experimentally investigate such trivial $4\pi$-modes, we examine the
microwave response of JJ1 in a DC current-biased setup. The measurements are
carried out at $T=$30\text{\,}\mathrm{m}\mathrm{K}$$ where the junction
exhibits no hysteresis, as seen in Supporting Fig. S2. In Fig. 1d, we present
$\frac{dV}{dI}$ as a function of the DC current bias and RF power at
$f=$12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ in addition to a histogram of
the voltage distribution. For this value of $f$, we can identify all the
integer Shapiro steps along with subharmonic Shapiro steps. Subharmonic
Shapiro steps are expected at high frequencies due to the anharmoncity
associated with the forward skewness of the CPR in highly transparent
junctions 54; 55; 39; 56; 57; 58; 59. The presence of a $4\pi$-periodic
supercurrent channel, with critical current $I_{4\pi}$, is expected to result
in missing odd Shapiro steps 49; 45; 50; 48; 51; 44 when the energy of the
photon irradiating the JJ, $hf$, is less than $hf_{4\pi}\approx
2eI_{4\pi}R_{n}$ 46; 47. Fig. 1e shows a similar Shapiro map for
$f=$7\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ where we see that the first odd
Shapiro step is missing indicating the presence of a finite $I_{4\pi}$ even
though the JJ is in a topologically trivial regime. For JJ1, at
$B^{\theta}=$0\text{\,}\mathrm{m}\mathrm{T}$$, we find
$f_{4\pi}\sim$8.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ corresponding to
$I_{4\pi}=$52.1\text{\,}\mathrm{n}\mathrm{A}$$. Considering the Josephson
frequency, $f_{J}\equiv\frac{2e\text{I}_{c}\text{R}_{n}}{h}$, for JJ1, we get
$f_{4\pi}$/$f_{J}\cong I_{4\pi}$/$I_{c}$ corresponding to 6.5% of the
supercurrent being carried by a $4\pi$-periodic supercurrent channel.

We next consider the dependence of the critical current $I_{c}$ in JJ1 on a
magnetic field, without a microwave bias, as seen in the differential
resistance map in Fig. 1f where the in-plane magnetic field is applied along
the junction, $B^{0^{\circ}}$. The critical field, $B^{0^{\circ}}_{c}$, is
seen to be $\sim$620\text{\,}\mathrm{m}\mathrm{T}$$. Similar measurements
performed at different $\theta$ values are presented in Supporting Fig. S3.
The field dependence data show no topological signatures such as a minimum in
$I_{c}$ 21, indicating that the junctions are topologically trivial for all
the values of $B^{\theta}$ up to the critical field $B^{\theta}_{c}$.

Figure 3: Angle dependence of reemergence of missing Shapiro step. Shapiro
maps at $B^{0^{\circ}}=$200\text{\,}\mathrm{m}\mathrm{T}$$ for a
$\theta=30^{\circ}$ and b $\theta=90^{\circ}$. c Calculated $Q_{12}$ and d
$I_{c}R_{n}$ as a function of in-plane magnetic field $B^{\theta}$ for in-
plane field angles $\theta=0^{\circ}$ and $90^{\circ}$. e, f The crossover
field $B^{\theta}_{co}$, field value at which missing Shapiro step first fully
reemerges, presented in e units of Tesla and f normalized by the corresponding
critical field $B^{\theta}_{c}$, as a function of $\theta$.

In Fig. 2, we present Shapiro maps for various magnetic field strengths
applied along the junction for
$f=$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and
$f=$6.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$. At
$B^{0^{\circ}}=$0\text{\,}\mathrm{m}\mathrm{T}$$, the first Shapiro step is
seen to be missing for both frequencies since $f<f_{4\pi}$. At
$B^{0^{\circ}}=$200\text{\,}\mathrm{m}\mathrm{T}$$, the first step almost
completely emerges for $f=$6.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ while
still being missing for $f=$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$. At
$B^{0^{\circ}}\sim$300\text{\,}\mathrm{m}\mathrm{T}$$, the first step starts
emerging for $f=$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, eventually
completely appearing at $B^{0^{\circ}}=$400\text{\,}\mathrm{m}\mathrm{T}$$.
This behavior implies a decrease of $I_{4\pi}$ as a function of in-plane field
strength, consistent with the mechanisms described in Fig. 1c. We note that
the data presented in Fig. 2 imply that $f_{4\pi}$ does not scale
proportionally with $f_{J}$. In fact, the ratio $f_{4\pi}/f_{J}$ generally
increases as a function of in-plane field strength. This indicates that the
suppression of $I_{4\pi}$ is not simply proportional to the critical current
$I_{c}$, implying that the response of diabatically driven long junction modes
to an in-plane field is distinct from conventional “short junction” modes that
make up the rest of the spectrum in 2DEG JJs and the entire spectrum in narrow
junctions e.g., nanowire junctions.

Next, we consider the $I_{4\pi}$ dependence on the applied in-plane field
direction, $\theta$. A topologically non-trivial $I_{4\pi}$ is expected to be
sensitive 60 to $\theta$; on the other hand, the angle dependence of a trivial
$I_{4\pi}$ resulting from LZT is ambiguous and can depend on several
contributing effects from Zeeman, orbital and SOC interactions. Fig. 3a and b
show Shapiro maps with $f=$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ at
$B^{\theta}=$200\text{\,}\mathrm{m}\mathrm{T}$$ for $\theta=30^{\circ}$ and
$\theta=90^{\circ}$. Unlike the $\theta=0^{\circ}$ case presented in Fig. 2b,
the first step appears to partially reemerge for $\theta=30^{\circ}$ and
completely reemerges for $\theta=90^{\circ}$, which indicates an angle
anisotropy of $I_{4\pi}$. To determine more precisely the threshold value of
$B^{\theta}$ above which the first step reappears, we calculate $Q_{12}$ as a
function of $B^{\theta}$ where the ratio $Q_{12}=\frac{s_{1}}{s_{2}}$
represents the strength of the first step with respect to the second found by
binning the voltage distribution and calculating the max step size/bin count
of the first (second) step, $s_{1}$ ($s_{2}$). More details about the
extraction of $Q_{12}$ from the data are provided in the Supporting
Information. We then identify the crossover field $B_{co}^{\theta}$ for the
in-plane angle $\theta$ as the value of $B^{\theta}$ for which $Q_{12}\approx
1$. Fig. 3c shows the evolution of $Q_{12}$ with $B^{\theta}$ for
$\theta=0^{\circ}$ and $\theta=90^{\circ}$. In both cases, the first step is
suppressed up to the crossover value $B_{co}^{\theta}$ and is fully present
for values $B^{\theta}>B_{co}^{\theta}$. The scaling of $Q_{12}$ is seen to
exhibit clear anistropy with respect to $B^{\theta}$: $\theta=0^{\circ}$ shows
a $B^{0^{\circ}}_{co}\approx$400\text{\,}\mathrm{m}\mathrm{T}$$, whereas
$\theta=90^{\circ}$ shows a
$B^{\theta}_{co}\approx$200\text{\,}\mathrm{m}\mathrm{T}$$.

Figure 4: Evolution of missing Shapiro steps at an applied gate voltage for
JJ2. Shapiro maps at $f=$3.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ for a-d
$V_{g}=-5$V and e-h $V_{g}=+10$V for different $B^{0^{\circ}}$ and
$B^{90^{\circ}}$ values. i, j$Q_{12}$ as a function of $B^{\theta}$
(normalized by the respective critical field for each $\theta$ and $V_{g}$
value) for i $\theta=0^{\circ}$ and j $\theta=90^{\circ}$.

In Fig. 3e, we present a polar plot of $B^{\theta}_{co}$ (at
$f=$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$) as a function $\theta$. A
large variation in crossover field is observed; however, we note that the
critical field $B^{\theta}_{c}$ for $\theta=0^{\circ}$ and $90^{\circ}$ are
significantly different
($B^{0^{\circ}}_{c}=$620\text{\,}\mathrm{m}\mathrm{T}$$ and
$B^{90^{\circ}}_{c}=$320\text{\,}\mathrm{m}\mathrm{T}$$) as seen in Fig. 3d,
similar to other Al-InAs junctions 61. When normalized by their respective
critical fields to account for the angle-dependence of $B^{\theta}_{c}$, the
crossover fields become quantitatively similar and in fact match a fit of
$B^{\theta}_{co}/B^{\theta}_{c}$ = 67% as seen in Fig. 3f. This suggests that
the anisotropy observed in $B^{\theta}_{co}$ is likely due to the variation in
critical field and implies that JJ1 has weak SOC effects.

One of the unique advantages of using a semiconductor-based system is the
ability to have electrostatic tunability of the carrier density and SOC
interaction using a gate. To study the trivial $I_{4\pi}$ dependence on such
properties, we focus on JJ2 fabricated on the same heterostructure presented
in Fig. 1 but equipped with a top gate. JJ2 is expected to have a stronger SOC
interaction than JJ1 even at zero gate voltage
($V_{g}=$0\text{\,}\mathrm{V}$$) due to the presence of a gate dielectric
Al2O3 layer (see Supporting Information) that tends to increase the carrier
density and consequently SOC interaction. JJ2 is markedly hysteretic at
$30\text{\,}\mathrm{m}\mathrm{K}$ due to thermal effects 62, and so it is
studied at $800\text{\,}\mathrm{m}\mathrm{K}$ where it shows no hysteresis. At
$B^{\theta}=$0\text{\,}\mathrm{m}\mathrm{T}$$ and
$V_{g}=$0\text{\,}\mathrm{V}$$, JJ2 exhibits a missing first Shapiro step as
seen in Supporting Fig. S8 even though at
$T=$800\text{\,}\mathrm{m}\mathrm{K}$$ the overall transparency is expected to
be reduced. Further, Supporting Fig. S4 shows that JJ2 exhibits a similar
$B^{\theta}_{c}$ anisotropy to that of JJ1. However, we note that for JJ2,
$B^{\theta}_{c}$ also depends on $V_{g}$.

In Fig. 4, we present measurements performed on JJ2 at
$f=$3.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ for $V_{g}=-5$V and +10V at
different $B^{0^{\circ}}$ and $B^{90^{\circ}}$ values. For $\theta=0^{\circ}$,
$V_{g}=-5$V shows $B^{0^{\circ}}_{co}$ = $125\text{\,}\mathrm{m}\mathrm{T}$
while $V_{g}=+10$V shows
$B^{0^{\circ}}_{co}=$225\text{\,}\mathrm{m}\mathrm{T}$$. The difference
between $V_{g}=-5$V and +10V is reconciled when considering
$B^{0^{\circ}}/B^{0^{\circ}}_{c}(V_{g})$, as seen in Fig. 4i, where both
$V_{g}$ values exhibit a $B^{0^{\circ}}_{co}(V_{g})/B^{0^{\circ}}_{c}(V_{g})$
of $\sim$ 40%. For $\theta=90^{\circ}$, the data presented in Fig. 4j show a
$B^{90^{\circ}}_{co}/B^{90^{\circ}}_{c}$ ratio of $\sim 57\%$ and $\sim 65\%$
for $V_{g}=-5$V and +10V, respectively. While the $\theta=90^{\circ}$ case
exhibits similar $B^{90^{\circ}}_{co}/B^{90^{\circ}}_{c}$ values to that
reported for JJ1, the $\theta=0^{\circ}$ case shows a significant discrepancy
for both $V_{g}$ values. It is evident here that for JJ2, the angle anisotropy
is not simply accounted for by considering $B^{\theta}_{c}$ and that other
effects play a role in the suppression of $I_{4\pi}$, consistent with the
expectation of JJ2 having stronger SOC effects in comparison to JJ1. In the
following, we discuss the origin of such suppression of $I_{4\pi}$ and the
observed angle anisotropy by considering the ABS spectrum.

Following the picture presented in Fig. 1c, we first consider the suppression
of $I_{4\pi}$ in terms of transitions between the long junction modes to the
continuum, related mainly to the detachment gap $\delta$. Using tight-binding
simulations, we calculate the energy spectrum of the ABS spectrum in an InAs-
Al junction. Fig. 5a shows a linear decrease in $\delta$ as a function of the
Zeeman field $\Delta^{\theta}_{Z}$. The decrease in $\delta$ results in a
higher probability of undergoing LZTs to the continuum, suppressing the
$4\pi$-component of the CPR. In the absence of SOC effects
($\lambda_{SOC}=0$), corresponding to the black line in Fig. 5a, the
suppression of $\delta$ as a function of $B^{\theta}$ shows no
$\theta$-dependence.

In the presence of strong SOC effects, the Fermi surface of the quantum well
has an anisotropic response to an in-plane Zeeman field, creating an
anisotropic suppression of $\delta$ in the ABS spectrum. For
$\lambda_{SOC}=$7.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\cdot\mathrm{n}\mathrm{m}$$,
Fig. 5a illustrates that a larger $\Delta^{\theta}_{Z}$ in the
$\theta=0^{\circ}$ (green line) direction is needed than in the
$\theta=90^{\circ}$ (orange line) direction to suppress $\delta$ by the same
amount. However, Fig. 4i and j shows $B^{0^{\circ}}_{co}/B^{0^{\circ}}_{c}<$
$B^{90^{\circ}}_{co}/B^{90^{\circ}}_{c}$. This indicates that the presence of
strong SOC (as expected for JJ2) enhances the lack of correlation between the
suppression of $I_{4\pi}$ and of $\delta$.

We thus consider the suppression of $I_{4\pi}$ in terms of mode-to-mode
coupling. Due to the large number of ABS modes in our junctions, a result of
the large width $w$, we have a very dense ABS spectrum. Consequently, we have
several quasi-avoided crossings between ABSs and between ABSs and the
continuum. In the presence of a Zeeman field, the ABS spectrum becomes even
more complex, with more quasi-avoided crossings and new protected crossings. A
fully microscopic description of the JJ would require the determination of the
dynamics of a multi-level Landau-Zener problem. This is a problem that is
computationally prohibitive to solve.

Figure 5: Theoretical ABS spectrum analysis in the presence of a Zeeman field.
a Calculation of the detachment gap $\delta$ for a junction with
$w=$500\text{\,}\mathrm{n}\mathrm{m}$$ wide and
$l=$100\text{\,}\mathrm{n}\mathrm{m}$$ as a function of the Zeeman energy
$\Delta_{Z}^{\theta}$ for $\lambda_{SOC}=0$ and for $\theta=0^{\circ}$ and
$\theta=90^{\circ}$ with
$\lambda_{SOC}=$7.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\cdot\mathrm{n}\mathrm{m}$$.
b Schematic of the Andreev bound state spectrum illustrating which states are
used to calculate wavefunction overlaps. c Distribution of wavefunction
overlaps with
$\lambda_{SOC}=$7.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\cdot\mathrm{n}\mathrm{m}$$
between $\phi_{i}=0.6\pi$ and $\phi_{f}=1.41\pi$ for $\Delta_{Z}^{\theta}=0$
and 0.3$\Delta$ for $\theta=0^{\circ}$ and $90^{\circ}$. Inset: outlier
overlaps where $|\psi_{m}\rangle$ is a long junction mode.

However, to gain a qualitative understanding, we can estimate the relevant
multi-mode couplings by calculating the wave function overlap between a long
junction mode at $\phi=\phi_{i}$ and all positive energy Andreev mid-gap
states at $\phi=\phi_{f}$ far from the avoided crossing at $\phi=\pi$, as
shown schematically in Fig. 5b. This allows to estimate the probability that
an occupied ABS, when $\phi\approx\pi$, can either transition to an ABS with a
large detachment from the continuum and therefore contribute to $I_{4\pi}$, or
transition to an ABS with a small $\delta$ and therefore contribute solely to
$I_{2\pi}$. We provide a detailed discussion of the calculations in the
Supporting Information. In Fig. 5c, we present a histogram of the wave
function overlaps $|\langle\psi_{n}(\phi_{i})|\psi_{m}(\phi_{f})\rangle|^{2}$
between a long junction mode $|\psi_{n}\rangle$ and modes $|\psi_{m}\rangle$
for $\phi_{i}=0.6\pi$ and $\phi_{f}=1.41\pi$. At $\Delta_{Z}=0$, we observe a
distribution localized at zero except for a single outlier shown in the inset.
This outlier corresponds to an overlap with another long junction mode. At
finite $\Delta^{\theta}_{Z}$ and
$\lambda_{SOC}=$7.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\cdot\mathrm{n}\mathrm{m}$$,
more states develop a non-zero overlap with the long junction mode evident
from the histogram distribution. The histogram distribution also shows that
the system is more sensitive to $\Delta^{\theta}_{Z}$ in the
$\theta=0^{\circ}$ direction than the $\theta=90^{\circ}$ direction with the
$\theta=0^{\circ}$ case exhibiting a broader distribution. These results
suggest that the distribution of the overlaps between ABS states across
$\phi=\pi$, through their effect on Landau-Zener transitions, play an
important role in the anisotropy observed in Fig. 4i and j for JJ2, especially
where a strong SOC interaction is present.

By studying the microwave response of an epitaxial Al-InAs JJ, we observe
signatures of a 4$\pi$-periodic contribution to the CPR attributed to
topologically-trivial LZT between long junction modes. With the application of
an external magnetic field, the $I_{4\pi}$ is observed to be suppressed
differently to $I_{2\pi}$ and eventually disappears at a crossover field. In a
device with weak SOC (JJ1), we observe an isotropic suppression of $I_{4\pi}$
with an applied magnetic field when the device’s angle anisotropy in
$B^{\theta}_{c}$ is taken into account. In the gate tunable device (JJ2) with
a significantly larger SOC, an anisotropic suppression of $I_{4\pi}$ is
observed, which cannot be accounted for by the device’s $B^{\theta}_{c}$ angle
anisotropy. We attribute the anisotropy to SOC effects which introduce a non-
trivial angle $\theta$ dependence in the coupling of long junction modes to
other Andreev mid-gap states lacking a detachment gap, suggesting multi-level
LZTs. Our results indicate that such anisotropy in in-plane magnetic field and
dependence on SOC effects need to be considered when differentiating between
topologically trivial and non-trivial $I_{4\pi}$ and requires other correlated
signatures to make claims about topological superconductivity.

Supporting Information - Additional measurements that support our findings as
well as material growth, fabrication, measurement details and information
about the theoretical model are provided in the Supporting Information
section. This material is available free of charge via the internet at
http://pubs.acs.org.

Acknowledgments - We thank Matthieu C. Dartiailh for fruitful discussions. The
NYU team acknowledges support by DARPA TEE award no. DP18AP900007. We
acknowledge funding from DOE award no. DE-SC0022245. J.J.C also acknowledges
support from the Graduate Research Fellowship awarded by the Virginia Space
Grant Consortium (VSGC). J.J.C. and E.R. acknowledge William & Mary Research
Computing for providing computational resources and/or technical support that
have contributed to the results reported within this paper. URL:
https://www.wm.edu/it/rc. W.F.S. acknowledges funding from an ND- SEG
Fellowship. W.M.S. acknowledges funding from the ARO/LPS QuaCR Graduate
Fellowship.

## References

  * Krogstrup et al. 2015 P. Krogstrup, N. L. B. Ziino, W. Chang, S. M. Albrecht, M. H. Madsen, E. Johnson, J. Nygård, C. Marcus, and T. S. Jespersen, Nature Materials 14, 400 (2015), ISSN 1476-1122, 1476-4660, URL http://www.nature.com/articles/nmat4176.
  * Shabani et al. 2016 J. Shabani, M. Kjaergaard, H. J. Suominen, Y. Kim, F. Nichele, K. Pakrouski, T. Stankevic, R. M. Lutchyn, P. Krogstrup, R. Feidenhans’l, et al., Physical Review B 93, 155402 (2016), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.93.155402.
  * Kjaergaard et al. 2017 M. Kjaergaard, H. Suominen, M. Nowak, A. Akhmerov, J. Shabani, C. Palmstrøm, F. Nichele, and C. Marcus, Physical Review Applied 7, 034029 (2017), ISSN 2331-7019, URL http://link.aps.org/doi/10.1103/PhysRevApplied.7.034029.
  * Bøttcher et al. 2018 C. G. L. Bøttcher, F. Nichele, M. Kjaergaard, H. J. Suominen, J. Shabani, C. J. Palmstrøm, and C. M. Marcus, Nature Physics 14, 1138 (2018), ISSN 1745-2473, 1745-2481, URL http://www.nature.com/articles/s41567-018-0259-9.
  * Mayer et al. 2019 W. Mayer, J. Yuan, K. S. Wickramasinghe, T. Nguyen, M. C. Dartiailh, and J. Shabani, Applied Physics Letters 114, 103104 (2019), ISSN 0003-6951, 1077-3118, URL http://aip.scitation.org/doi/10.1063/1.5067363.
  * Lee et al. 2019 J. S. Lee, B. Shojaei, M. Pendharkar, A. P. McFadden, Y. Kim, H. J. Suominen, M. Kjaergaard, F. Nichele, H. Zhang, C. M. Marcus, et al., Nano Letters 19, 3083 (2019), ISSN 1530-6984, 1530-6992, URL https://pubs.acs.org/doi/10.1021/acs.nanolett.9b00494.
  * Mayer et al. 2020 W. Mayer, M. C. Dartiailh, J. Yuan, K. S. Wickramasinghe, E. Rossi, and J. Shabani, Nature Communications 11, 212 (2020), ISSN 2041-1723, URL http://www.nature.com/articles/s41467-019-14094-1.
  * Elfeky et al. 2021 B. H. Elfeky, N. Lotfizadeh, W. F. Schiela, W. M. Strickland, M. Dartiailh, K. Sardashti, M. Hatefipour, P. Yu, N. Pankratova, H. Lee, et al., Nano Letters 21, 8274 (2021), ISSN 1530-6984, 1530-6992, URL https://pubs.acs.org/doi/10.1021/acs.nanolett.1c02771.
  * Strickland et al. 2022a W. M. Strickland, M. Hatefipour, D. Langone, S. M. Farzaneh, and J. Shabani, _Controlling Fermi level pinning in near-surface InAs quantum wells_ (2022a), arXiv:2206.01057 [cond-mat], URL http://arxiv.org/abs/2206.01057.
  * Larsen et al. 2015 T. W. Larsen, K. D. Petersson, F. Kuemmeth, T. S. Jespersen, P. Krogstrup, J. Nygård, and C. M. Marcus, Phys. Rev. Lett. 115, 127001 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett.115.127001.
  * Luthi et al. 2018 F. Luthi, T. Stavenga, O. W. Enzing, A. Bruno, C. Dickel, N. K. Langford, M. A. Rol, T. S. Jespersen, J. Nygård, P. Krogstrup, et al., Phys. Rev. Lett. 120, 100502 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.100502.
  * Kringhøj et al. 2018 A. Kringhøj, L. Casparis, M. Hell, T. W. Larsen, F. Kuemmeth, M. Leijnse, K. Flensberg, P. Krogstrup, J. Nygård, K. D. Petersson, et al., Phys. Rev. B 97, 060508 (2018).
  * Casparis et al. 2018 L. Casparis, M. R. Connolly, M. Kjaergaard, N. J. Pearson, A. Kringhøj, T. W. Larsen, F. Kuemmeth, T. Wang, C. Thomas, S. Gronin, et al., Nature Nanotechnology 13, 915 (2018), ISSN 1748-3387, 1748-3395, URL http://www.nature.com/articles/s41565-018-0207-y.
  * Casparis et al. 2016 L. Casparis, T. W. Larsen, M. S. Olsen, F. Kuemmeth, P. Krogstrup, J. Nygård, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 116, 150505 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.116.150505.
  * O’Connell Yuan et al. 2021a J. O’Connell Yuan, K. S. Wickramasinghe, W. M. Strickland, M. C. Dartiailh, K. Sardashti, M. Hatefipour, and J. Shabani, Journal of Vacuum Science & Technology A 39, 033407 (2021a).
  * Danilenko et al. 2022 A. Danilenko, D. Sabonis, G. W. Winkler, O. Erlandsson, P. Krogstrup, and C. M. Marcus, _Few-mode to mesoscopic junctions in gatemon qubits_ (2022), URL https://arxiv.org/abs/2209.03688.
  * Hertel et al. 2022 A. Hertel, M. Eichinger, L. O. Andersen, D. M. T. van Zanten, S. Kallatt, P. Scarlino, A. Kringhøj, J. M. Chavez-Garcia, G. C. Gardner, S. Gronin, et al., _Gate-tunable transmon using selective-area-grown superconductor-semiconductor hybrid structures on silicon_ (2022), URL https://arxiv.org/abs/2202.10860.
  * Strickland et al. 2022b W. M. Strickland, B. H. Elfeky, J. O. Yuan, W. F. Schiela, P. Yu, D. Langone, M. G. Vavilov, V. E. Manucharyan, and J. Shabani, _Superconducting resonators with voltage-controlled frequency and nonlinearity_ (2022b), arXiv:2210.02491 [cond-mat, physics:quant-ph], URL http://arxiv.org/abs/2210.02491.
  * Hell et al. 2017 M. Hell, M. Leijnse, and K. Flensberg, Physical Review Letters 118, 107701 (2017), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.118.107701.
  * Pientka et al. 2017 F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Physical Review X 7, 021032 (2017), ISSN 2160-3308, URL http://link.aps.org/doi/10.1103/PhysRevX.7.021032.
  * Dartiailh et al. 2021a M. C. Dartiailh, W. Mayer, J. Yuan, K. S. Wickramasinghe, A. Matos-Abiague, I. Žutić, and J. Shabani, Physical Review Letters 126, 036802 (2021a), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.126.036802.
  * Ren et al. 2019 H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp, et al., Nature 569, 93 (2019), ISSN 0028-0836, 1476-4687, URL http://www.nature.com/articles/s41586-019-1148-9.
  * Fornieri et al. 2019 A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portolés, A. C. C. Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. Kallaher, et al., Nature 569, 89 (2019), ISSN 0028-0836, 1476-4687, URL http://www.nature.com/articles/s41586-019-1068-8.
  * Banerjee et al. 2022 A. Banerjee, O. Lesser, M. A. Rahman, H. R. Wang, M. R. Li, A. Kringhøj, A. M. Whiticar, A. C. C. Drachmann, C. Thomas, T. Wang, et al., arxiv (2022), publisher: arXiv Version Number: 1, URL https://arxiv.org/abs/2201.03453.
  * Nayak et al. 2008 C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Reviews of Modern Physics 80, 1083 (2008), ISSN 0034-6861, 1539-0756, URL https://link.aps.org/doi/10.1103/RevModPhys.80.1083.
  * Aasen et al. 2016 D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, et al., Physical Review X 6, 031016 (2016), ISSN 2160-3308, URL https://link.aps.org/doi/10.1103/PhysRevX.6.031016.
  * Mohapatra et al. 2019 S. Mohapatra, S. Mathimalar, S. Chaudhary, and K. V. Raman, Journal of Physics Communications 3, 045005 (2019), ISSN 2399-6528, URL https://iopscience.iop.org/article/10.1088/2399-6528/ab14a7.
  * Pan et al. 2020 H. Pan, W. S. Cole, J. D. Sau, and S. Das Sarma, Physical Review B 101, 024506 (2020), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.101.024506.
  * Cayao and Burset 2021 J. Cayao and P. Burset, Physical Review B 104, 134507 (2021), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.104.134507.
  * Yu et al. 2021 P. Yu, J. Chen, M. Gomanko, G. Badawy, E. P. A. M. Bakkers, K. Zuo, V. Mourik, and S. M. Frolov, Nature Physics 17, 482 (2021), ISSN 1745-2473, 1745-2481, URL http://www.nature.com/articles/s41567-020-01107-w.
  * Pan and Das Sarma 2022 H. Pan and S. Das Sarma, Physical Review B 105, 115432 (2022), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.105.115432.
  * Kwon et al. 2003 H.-J. Kwon, K. Sengupta, and V. M. Yakovenko, The European Physical Journal B - Condensed Matter 37, 349 (2003), ISSN 1434-6028, 1434-6036, URL http://link.springer.com/10.1140/epjb/e2004-00066-4.
  * Shaw et al. 2008 M. D. Shaw, R. M. Lutchyn, P. Delsing, and P. M. Echternach, Physical Review B 78, 024503 (2008), ISSN 1098-0121, 1550-235X, URL https://link.aps.org/doi/10.1103/PhysRevB.78.024503.
  * Martinis et al. 2009 J. M. Martinis, M. Ansmann, and J. Aumentado, Physical Review Letters 103, 097002 (2009), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.103.097002.
  * Pikulin and Nazarov 2012 D. I. Pikulin and Y. V. Nazarov, Physical Review B 86, 140504 (2012), ISSN 1098-0121, 1550-235X, URL https://link.aps.org/doi/10.1103/PhysRevB.86.140504.
  * Badiane et al. 2013 D. M. Badiane, L. I. Glazman, M. Houzet, and J. S. Meyer, Comptes Rendus Physique 14, 840 (2013), ISSN 16310705, URL https://linkinghub.elsevier.com/retrieve/pii/S1631070513001692.
  * van Woerkom et al. 2015 D. J. van Woerkom, A. Geresdi, and L. P. Kouwenhoven, Nature Physics 11, 547 (2015), ISSN 1745-2473, 1745-2481, URL http://www.nature.com/articles/nphys3342.
  * Rokhinson et al. 2012 L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nature Physics 8, 795 (2012), ISSN 1745-2473, 1745-2481, URL http://www.nature.com/articles/nphys2429.
  * Wiedenmann et al. 2016 J. Wiedenmann, E. Bocquillon, R. S. Deacon, S. Hartinger, O. Herrmann, T. M. Klapwijk, L. Maier, C. Ames, C. Brüne, C. Gould, et al., Nature Communications 7, 10303 (2016), ISSN 2041-1723, URL http://www.nature.com/articles/ncomms10303.
  * Bocquillon et al. 2017 E. Bocquillon, R. S. Deacon, J. Wiedenmann, P. Leubner, T. M. Klapwijk, C. Brüne, K. Ishibashi, H. Buhmann, and L. W. Molenkamp, Nature Nanotechnology 12, 137 (2017), ISSN 1748-3387, 1748-3395, URL http://www.nature.com/articles/nnano.2016.159.
  * Deacon et al. 2017 R. Deacon, J. Wiedenmann, E. Bocquillon, F. Domínguez, T. Klapwijk, P. Leubner, C. Brüne, E. Hankiewicz, S. Tarucha, K. Ishibashi, et al., Physical Review X 7, 021011 (2017), ISSN 2160-3308, URL http://link.aps.org/doi/10.1103/PhysRevX.7.021011.
  * Laroche et al. 2019 D. Laroche, D. Bouman, D. J. van Woerkom, A. Proutski, C. Murthy, D. I. Pikulin, C. Nayak, R. J. J. van Gulik, J. Nygård, P. Krogstrup, et al., Nature Communications 10, 245 (2019), ISSN 2041-1723, URL http://www.nature.com/articles/s41467-018-08161-2.
  * Li et al. 2018 C. Li, J. C. de Boer, B. de Ronde, S. V. Ramankutty, E. van Heumen, Y. Huang, A. de Visser, A. A. Golubov, M. S. Golden, and A. Brinkman, Nature Materials 17, 875 (2018), ISSN 1476-1122, 1476-4660, URL http://www.nature.com/articles/s41563-018-0158-6.
  * Fischer et al. 2022 R. Fischer, J. Picó-Cortés, W. Himmler, G. Platero, M. Grifoni, D. A. Kozlov, N. N. Mikhailov, S. A. Dvoretsky, C. Strunk, and D. Weiss, Physical Review Research 4, 013087 (2022), ISSN 2643-1564, URL https://link.aps.org/doi/10.1103/PhysRevResearch.4.013087.
  * Domínguez et al. 2012 F. Domínguez, F. Hassler, and G. Platero, Physical Review B 86, 140503 (2012), ISSN 1098-0121, 1550-235X, URL https://link.aps.org/doi/10.1103/PhysRevB.86.140503.
  * Domínguez et al. 2017 F. Domínguez, O. Kashuba, E. Bocquillon, J. Wiedenmann, R. S. Deacon, T. M. Klapwijk, G. Platero, L. W. Molenkamp, B. Trauzettel, and E. M. Hankiewicz, Physical Review B 95, 195430 (2017), ISSN 2469-9950, 2469-9969, URL http://link.aps.org/doi/10.1103/PhysRevB.95.195430.
  * Picó-Cortés et al. 2017 J. Picó-Cortés, F. Domínguez, and G. Platero, Physical Review B 96, 125438 (2017), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.96.125438.
  * Dartiailh et al. 2021b M. C. Dartiailh, J. J. Cuozzo, B. H. Elfeky, W. Mayer, J. Yuan, K. S. Wickramasinghe, E. Rossi, and J. Shabani, Nature Communications 12, 78 (2021b), ISSN 2041-1723, URL http://www.nature.com/articles/s41467-020-20382-y.
  * Billangeon et al. 2007 P.-M. Billangeon, F. Pierre, H. Bouchiat, and R. Deblock, Physical Review Letters 98, 216802 (2007), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.98.216802.
  * Sau and Setiawan 2017 J. D. Sau and F. Setiawan, Physical Review B 95, 060501 (2017), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.95.060501.
  * Galaktionov and Zaikin 2021 A. V. Galaktionov and A. D. Zaikin, Physical Review B 104, 054521 (2021), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.104.054521.
  * Mudi and Frolov 2021 S. R. Mudi and S. M. Frolov, arXiv:2106.00495 [cond-mat] (2021), arXiv: 2106.00495, URL http://arxiv.org/abs/2106.00495.
  * Cuozzo et al. 2022 J. J. Cuozzo, W. Yu, P. Davids, T. M. Nenoff, D. B. Soh, W. Pan, and E. Rossi, _Leggett modes in dirac semimetals_ (2022), eprint 2205.15995.
  * Lee et al. 2015 G.-H. Lee, S. Kim, S.-H. Jhi, and H.-J. Lee, Nature Communications 6, 6181 (2015), ISSN 2041-1723, URL http://www.nature.com/articles/ncomms7181.
  * Askerzade 2015 I. N. Askerzade, Low Temperature Physics 41, 241 (2015), ISSN 1063-777X, 1090-6517, URL http://aip.scitation.org/doi/10.1063/1.4916071.
  * Snyder et al. 2018 R. Snyder, C. Trimble, C. Rong, P. Folkes, P. Taylor, and J. Williams, Physical Review Letters 121, 097701 (2018), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.121.097701.
  * Kringhøj et al. 2018 A. Kringhøj, L. Casparis, M. Hell, T. W. Larsen, F. Kuemmeth, M. Leijnse, K. Flensberg, P. Krogstrup, J. Nygård, K. D. Petersson, et al., Physical Review B 97, 060508 (2018), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.97.060508.
  * Panghotra et al. 2020 R. Panghotra, B. Raes, C. C. de Souza Silva, I. Cools, W. Keijers, J. E. Scheerder, V. V. Moshchalkov, and J. Van de Vondel, Communications Physics 3, 53 (2020), ISSN 2399-3650, URL http://www.nature.com/articles/s42005-020-0315-5.
  * O’Connell Yuan et al. 2021b J. O’Connell Yuan, K. S. Wickramasinghe, W. M. Strickland, M. C. Dartiailh, K. Sardashti, M. Hatefipour, and J. Shabani, Journal of Vacuum Science & Technology A 39, 033407 (2021b), ISSN 0734-2101, 1520-8559, URL https://avs.scitation.org/doi/10.1116/6.0000918.
  * Scharf et al. 2019 B. Scharf, F. Pientka, H. Ren, A. Yacoby, and E. M. Hankiewicz, Physical Review B 99, 214503 (2019), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.99.214503.
  * Suominen et al. 2017 H. J. Suominen, J. Danon, M. Kjaergaard, K. Flensberg, J. Shabani, C. J. Palmstrøm, F. Nichele, and C. M. Marcus, Physical Review B 95, 035307 (2017), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.95.035307.
  * Courtois et al. 2008 H. Courtois, M. Meschke, J. T. Peltonen, and J. P. Pekola, Physical Review Letters 101, 067002 (2008), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.101.067002.