# Theory of scanning gate microscopy imaging
of the supercurrent distribution in a planar Josephson junction

K. Kaperek  AGH University of Science and Technology, Faculty of Physics and
Applied Computer Science, al. A. Mickiewicza 30, 30-059 Krakow, Poland  S.
Heun  NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San
Silvestro 12, 56127 Pisa, Italy  M. Carrega  CNR-SPIN, Via Dodecaneso 33,
16146, Genova, Italy  P. Wójcik  AGH University of Science and Technology,
Faculty of Physics and Applied Computer Science, al. A. Mickiewicza 30, 30-059
Krakow, Poland  M. P. Nowak<EMAIL_ADDRESS>AGH University of Science and
Technology, Academic Centre for Materials and Nanotechnology, al. A.
Mickiewicza 30, 30-059 Krakow, Poland

###### Abstract

We theoretically investigate the mapping of the supercurrent distribution in a
planar superconductor-normal-superconductor junction in the presence of a
perpendicular magnetic field via the scanning gate microscopy technique. We
find that the distribution of counter-propagating supercurrents aligned in
Josephson vortices can be mapped by the change of the critical current induced
by the tip of the scanning probe, if the flux in the junction is set close to
maxima of the Fraunhofer pattern. Instead, when the magnetic field drives the
junction to a supercurrent minimum in the Fraunhofer pattern, the
superconducting phase adapts, and the tip always increases the supercurrent.
The perpendicular magnetic field leads to the formation of Josephson vortices,
whose extension for highly transparent junctions depends on the current
circulation direction. We show that this leads to an asymmetric supercurrent
distribution in the junction and that this can be revealed by scanning gate
microscopy. We explain our findings on the basis of numerical calculations for
both short- and long-junction limits and provide a phenomenological model for
the observed phenomena.

##  I Introduction

The supercurrent in superconductor-normal-superconductor (SNS) Josephson
junctions is carried by quasiparticles forming Andreev bound states (ABS) in
the normal region. Upon application of a perpendicular magnetic field in SNS
junctions for which the self field of the supercurrent can be neglected Barone
and Paternò (1982); Gross  _et al._ (2016); Ostroukh  _et al._ (2016), the
vector potential of the field induces a spatial gradient of the
superconducting phase difference. This in turn introduces a spatial variation
of the supercurrent, leading to formation of negative and positive current
regions, and ultimately to the formation of supercurrent vortices. As such,
vortices correspond to a complete loops of the current and carry no net
transport of supercurrent. They are refereed to as Josephson vortices Tinkham
(2004). An increase in magnetic field leads to the formation of subsequent
vortices Rowell (1963), and accordingly the critical current of the junction
exhibits a Fraunhofer oscillation pattern.

The experimental confirmation of the formation of Josephson vortices was
obtained by scanning tunneling microscopy Roditchev  _et al._ (2015), and
their manipulation was performed by magnetic force microscopy Dremov  _et al._
(2019). Moreover, the change of the magnetic interference pattern from
Fraunhofer to SQUID-like when the density of states is tuned from bulk to edge
states in quasi-ballistic junction Sticlet  _et al._ (2020) was used for
indirect determination of the supercurrent distribution Hart  _et al._ (2014);
Pribiag  _et al._ (2015); Allen  _et al._ (2016); Ying  _et al._ (2020);
Guiducci  _et al._ (2019a).

Here we theoretically explore the possibility of a direct visualization of the
supercurrent distribution and the associated formation of Josephson vortices
by a combination of critical current measurements and the scanning gate
microscopy (SGM) technique. Our idea exploits the fact that in novel planar
SNS junctions Shabani  _et al._ (2016); Kjaergaard  _et al._ (2017); Fornieri
_et al._ (2019); Guiducci  _et al._ (2019a, b); de Vries  _et al._ (2018); Ren
_et al._ (2019); Telesio  _et al._ (2022); Ke  _et al._ (2019); Moehle  _et
al._ (2021); Zhi  _et al._ (2019); de Vries  _et al._ (2019); Salimian  _et
al._ (2021); Alidoust (2018); Zyuzin  _et al._ (2016) the normal
semiconducting regions between the superconductors remains exposed, which
opens the possibility for the application of the SGM technique. The latter
technique has been successfully used for over two decades for imaging of
normal current flow.

Indeed, in normal systems SGM is a widely used technique which relies on
measuring the conductance changes when a biased atomic force microscope tip
scans over the device Woodside and McEuen (2002), inducing a repulsive
potential in the two-dimensional electron gas (2DEG) of the sample and
affecting the trajectories of the propagating electrons. SGM allows for the
demonstration of, e.g., magnetic focusing Aidala  _et al._ (2007) or branched
electron flow Jalabert  _et al._ (2010); Gorini  _et al._ (2013); Nowak  _et
al._ (2014) in heterostructures Topinka  _et al._ (2001); Paradiso  _et al._
(2010); LeRoy  _et al._ (2005); Jura  _et al._ (2009) and monolayers Neubeck
_et al._ (2012); Connolly  _et al._ (2011); Mreńca-Kolasińska and Szafran
(2017); Garcia  _et al._ (2013); Prokop  _et al._ (2020).

Here we theoretically predict that the SGM technique enables spatial imaging
of the supercurrent in a Josephson junction realized in a ballistic 2DEG and
allows the identification of positive and negative supercurrent streams that
contribute to the formation of Josephson vortices. However, in contrast to
non-superconducting devices, the transport properties and the supercurrent
distribution in a SNS junction are controlled by the superconducting phase
difference. We show that SGM allows for the visualization of the supercurrent
distribution by measuring the critical current of the junction provided that
an external transverse magnetic field is set near the Fraunhofer maxima, i.e.,
when there is an even number of Josephson vortices and the phase is uniquely
defined despite changes of SGM tip position. Moreover, we show that the
supercurrent vortices in a ballistic junction are inherently nonsymmetric,
which in turn results in the asymmetry of the supercurrent flowing across the
junction, which can be revealed by the SGM technique. We test our findings in
an experimentally viable regime of short and long junctions, paving the way to
study the Josephson vortex structure and supercurrent distribution in a
variety of planar SNS devices, for which a standard Fraunhofer pattern is
currently observed. Furthermore, the proposed method of supercurrent
visualization can be applied to systems such as common 2DEGs where scanning
tunneling microscopy is not possible due to surface oxidization Chen (1993);
Voigtländer (2015).

The outline of this paper is as follows. We first introduce the system in Sec.
II. In Sec. III, we present the results for the short-junction regime.
Phenomenological analysis of supercurrent probing is presented in Sec. III. A,
while in Sec. III. B we explain the asymmetry of the Josephson vortices.
Finally, in Sec. IV, we test the robustness of our results beyond the short-
junction limit and summarize our results in Sec. V.

##  II The investigated system

Figure 1: Illustration of the considered system. The grey color depicts the
superconducting contacts. Between them there is the normal region whose
surface is scanned by the SGM tip (purple). With the orange stream-plot we
show an exemplary supercurrent distribution with a Josephson vortex denoted
with the green arrow.

We study a planar SNS junction composed of two superconducting leads connected
through a semiconducting region in the ballistic regime, in the presence of a
magnetic field oriented perpendicular to the junction plane—see Fig. 1. The
junction dimensions are defined by the width ($W$) and the length ($L$) of the
semiconducting region. In experimental realizations, such planar junctions
have been defined in InAs Takayanagi  _et al._ (1995); Bauch  _et al._ (2005);
Shabani  _et al._ (2016); Kjaergaard  _et al._ (2017); Fornieri  _et al._
(2019); Guiducci  _et al._ (2019a, b); de Vries  _et al._ (2018), HgTe Ren
_et al._ (2019), bP Telesio  _et al._ (2022), graphene Heersche  _et al._
(2007); Calado  _et al._ (2015); Lee and Lee (2018); Ben Shalom  _et al._
(2016), InSb Ke  _et al._ (2019) or InAsSb Mayer  _et al._ (2020); Moehle  _et
al._ (2021) quantum wells, and InSb nanoflakes Zhi  _et al._ (2019); de Vries
_et al._ (2019); Salimian  _et al._ (2021) interfaced with aluminum or niobium
superconducting contacts. The supercurrent (visualized in orange in Fig. 1)
carried by the quasiparticles in the semiconductor is disturbed by the
repelling potential of a negatively charged SGM tip (purple in Fig. 1)
scanning above the surface of the semiconductor.

##  III Short-junction regime

We start with calculations in the short-junction approximation, i.e., in the
regime where the length of the normal scattering region is much shorter than
the superconducting coherence length, $L\ll\xi=\hbar v_{F}/\Delta$, with
$v_{F}$ the Fermi velocity in the semiconductor and $\Delta$ the
superconducting gap. This corresponds to the case where the dwell time of
quasiparticles inside the normal region $\tau_{\mathrm{dw}}$ is much smaller
than the time $\hbar/\Delta$ it spends inside the superconductor.sho

Between the superconducting contacts, the Andreev-reflected electrons and
holes create a set of bound states, whose energies can be determined Beenakker
(1991) from a matching condition
$S_{A}(E)S_{N}(E)\Psi_{\mathrm{in}}=\Psi_{\mathrm{in}}$ where
$\Psi_{\mathrm{in}}=(\Psi_{e},\Psi_{h})$ are the complex amplitudes of the
electron and hole waves incident on the junction defined in the basis of
normal region scattering modes. $S_{A}(E)$ describes an Andreev reflection
process at the NS interface

$S_{A}(E)=\zeta(E)\left(\begin{array}[]{cc}0&r_{A}^{*}\\\ r_{A}&0\\\
\end{array}\right),$ (1)

with $\zeta(E)=\sqrt{1-E^{2}/\Delta^{2}}+iE/\Delta$. Taking the outgoing modes
as time-reversed partners of the incoming states and assuming that Andreev
reflection does not mix the scattering modes, we can write

$r_{A}=\left(\begin{array}[]{cc}i\mathbf{1}&0\\\ 0&ie^{-i\phi}\mathbf{1}\\\
\end{array}\right),$ (2)

with $\phi$ the superconducting phase difference and $\mathbf{1}$ the identity
matrix spanning on the basis of the scattering modes.

The scattering properties for electrons and holes in the normal part are
captured in the block-diagonal matrix

$S_{N}(E)=\left(\begin{array}[]{cc}S(E)&0\\\ 0&S^{*}(-E)\\\
\end{array}\right),$ (3)

where $S(E)$ ($S^{*}(-E)$) corresponds to the electron (hole) scattering
block. Taking advantage of the short-junction approximation, i.e.
$S(E)=S(E=0)\equiv s$, we arrive at the eigenproblem van Heck  _et al._ (2014)

$\left(\begin{array}[]{cc}s^{\dagger}&0\\\ 0&s^{T}\\\
\end{array}\right)\left(\begin{array}[]{cc}0&r^{*}_{A}\\\ r_{A}&0\\\
\end{array}\right)\Psi_{\mathrm{in}}=\zeta(E)\,\Psi_{\mathrm{in}},$ (4)

whose solution yields the set of ABS eigenenergies and wave functions.

The supercurrent is obtained from the positive energy ABS,Beenakker (1991)

$I=-\frac{e}{\hbar}\sum_{E_{i}>0}\tanh\left(\frac{E_{i}}{2k_{\rm
B}T}\right)\frac{dE_{i}}{d\phi},$ (5)

and consequently, the critical current is $I_{c}=\max_{\phi}(I)$.

The scattering matrix is obtained for the normal region of the junction
described by the Hamiltonian

$H_{N}=\left(\frac{\hbar^{2}\mathbf{k}^{2}}{2m^{*}}-\mu+V_{\mathrm{tip}}(x,y)\right)\sigma_{0}+\alpha(\sigma_{x}k_{y}-\sigma_{y}k_{x}).$
(6)

The last term corresponds to Rashba spin-orbit coupling, $\mu$ is the chemical
potential, and $\sigma_{0}$ is an identity matrix acting on the spin degree of
freedom.

The impact of the tip positioned at ($x_{\mathrm{tip}},y_{\mathrm{tip}}$) on
the scattering of quasiparticles is included through the $V_{\mathrm{tip}}$
term in the Hamiltonian Eq. (6) that induces a Lorentzian potential island
Szafran (2011) in the 2DEG

$V_{\mathrm{tip}}(x,y)=\frac{V_{0}}{1+\frac{(x-x_{\mathrm{tip}})^{2}+(y-y_{\mathrm{tip}})^{2}}{d^{2}}},$
(7)

where we take $V_{0}=100$ meV as the effective potential magnitude
(corresponding to a negative voltage on the tip) and $d=50$ nm its width
Szafran (2011).

We discretize the Hamiltonian Eq. (6) which results in a set of onsite and
hopping elements that are used to describe the system on a square
computational mesh. We use a lattice constant $\delta x=\delta y=10$ nm. The
orbital effects of the magnetic field are included via Peierls substitution of
the hopping elements $t_{nm}\rightarrow
t_{nm}\exp\left[-ie/\hbar\int\mathbf{A}d\mathbf{l}\right]$ where the integral
is taken over the hopping direction. We adopt the vector potential in the
Landau gauge $\mathbf{A}=(0,xB,0)$. For the sake of simplicity, we neglect the
Zeeman interaction at this stage, whose impact will be included explicitly in
long-junction calculations in the following. The electron scattering matrix
$S(E=0)$ of the normal region is obtained using the Kwant package Groth  _et
al._ (2014). In the scattering matrix calculation, to take into account the
electrons incident on the normal region, we consider normal, semi-infinite
leads and assume zero magnetic field within them to account for the screening
effect.

For definiteness, we adopt the material parameters corresponding to InSb
Vurgaftman  _et al._ (2001), i.e., effective mass $m^{*}=0.014m$, with $m$ the
bare electron mass. We assume a wide junction with $W=1000$ nm, $L=200$ nm,
set the chemical potential to $\mu=40$ meV and take $\alpha=50$ meVnm. The
choice of such chemical potential guarantees population of the junction with a
large number of modes necessary to obtain the Fraunhofer pattern. We perform
calculations assuming a zero temperature limit ($T=0$). The critical current
maps are obtained via adaptive sampling of the parameter space using Adaptive
library Nijholt  _et al._ (2019). The code used for the calculations is
available in an on-line repository Kaperek and Nowak (2022).

Figure 2: Critical current versus flux and tip position (with
$y_{\mathrm{tip}}=0$) obtained in the short-junction limit.

The critical current versus the flux in the junction $\Phi=BLW$ and the tip
coordinate across the junction $x_{\mathrm{tip}}$ (with $y_{\mathrm{tip}}=0$)
is shown in Fig. 2. When the tip is outside the junction area
($|x_{\mathrm{tip}}|>W/2$) the critical current exhibits the usual Fraunhofer
pattern with the periodicity of a single flux quantum $\Phi_{0}=h/2e$. For
small magnetic fields, $\Phi<\Phi_{0}$, for which the critical current without
the tip is on the central (first) lobe of the Fraunhofer pattern, $I_{c}$
decreases as the tip approaches the center of the junction. For a stronger
magnetic field, we observe the formation of a pattern of lobes with quenched
or increased current emerging from the Fraunhofer pattern as the tip is moved
from the edge of the junction to its center. To better understand the physics
behind this observation, we now construct a phenomenological model of the SGM
$I_{c}$ response.

###  III.1 Phenomenological model of the SGM response

The critical current of a Josephson junction in a magnetic field can be
obtained by tracing the influence of the gauge-invariant phase difference on
the supercurrent distribution in the normal part. In a phenomenological model
we can thus write Guiducci  _et al._ (2019a)

$I_{c}=\max_{\phi}\int_{-W/2}^{W/2}J_{S}(x,x_{\mathrm{tip}})\,\mathcal{I}\left(\gamma(x)\right)dx,$
(8)

where the integral is carried out over the junction width. $J_{S}$ is the
local supercurrent density per unit length across the junction without the
magnetic field, constant along the $y$-direction and suppressed with a
Lorentzian function at the tip position $x_{\mathrm{tip}}$,

$J_{S}(x,x_{\mathrm{tip}})=J_{0}\left(1-\frac{1}{1+(x-x_{\mathrm{tip}})^{2}/d^{2}}\right).$
(9)

The supercurrent is modified by the current-phase relationship
$\mathcal{I}\left(\gamma(x)\right)$, which can be analytically expressed
Beenakker (1991) for a junction with transmission coefficient $\tau$ as,

$\mathcal{I}\left(\gamma(x)\right)=\frac{\tau\sin(\gamma(x))}{\sqrt{1-\tau\sin^{2}(\gamma(x)/2)}},$
(10)

where $\gamma(x)$ is the gauge-invariant local phase shift due to the vector
potential $\mathbf{A}$ and the superconducting phase difference pha ,

$\gamma(x)=\phi+\frac{2\pi}{\Phi_{0}}\int_{(x,0)}^{(x,L)}\mathbf{A}\cdot
d\mathbf{l}.$ (11)

Figure 3: (a) Supercurrent distribution across the junction calculated for
junctions with different transparency ($\tau$) obtained for constant
superconducting phase difference $\phi=-0.5\pi$. (b) Change in critical
current obtained in the short-junction calculations (red) and the
phenomenological model for various transparencies of the junction. The results
are obtained for $\Phi=1.5\Phi_{0}$.

The current-phase relation given by Eq. 10 is skewed for transmittive
junctions ($\tau\simeq 1$). This is in contrast to the case of junctions in
the tunneling regime ($\tau\ll 1$), where the current-phase relation
simplifies to the term $\sin(\gamma)$. Setting the junction transparency to a
constant value and inspecting the supercurrent density at given $x$ positions
in the junction in a non-zero magnetic field (the phase difference in an
external magnetic field depends on the $x$ position through Eq. (11) with
$\gamma=\phi+2\pi\Phi x/\Phi_{0}W$), we calculate the supercurrent
distribution across the junction, as depicted in Fig. 3(a). We observe that
for $\tau=0.6$ the supercurrent distribution is almost sinusoidal. On the
other hand, when the transmission probability is increased, the distribution
gets more skewed.

Figure 4: Supercurrent distribution $J=J_{0}\mathcal{I}(\gamma(x))$ across the
junction without the tip obtained in the phenomenological model for $\tau=1$
versus magnetic field. The current is obtained for the phase difference value
$\phi$ that maximizes the supercurrent in the junction.

The short-junction calculations shown above were obtained for a system in
which the only scatterer is constituted by the tip of the SGM. In the
phenomenological model, this translates into a high junction transparency. If
we consider a ballistic junction with $\tau=1$, Eq. (10) becomes
$\mathcal{I}\left(\gamma(x)\right)=\sin(\gamma(x))/4|\cos(\gamma(x)/2)|$. Let
us first neglect the tip influence and plot $J=J_{0}\mathcal{I}(\gamma(x))$
for $\phi$ set to obtain the maximum supercurrent at a given magnetic field
value. The resulting supercurrent distribution across the junction width
versus magnetic field is plotted in Fig. 4. At zero magnetic field ($\Phi=0$)
the supercurrent distribution is constant. When the magnetic field is
increased, the vector potential induces a spatially dependent phase shift
across the junction, and consequently the supercurrent becomes
position–dependent. When $\Phi=n\Phi_{0}$ (with $n$ an integer), the amount of
positive and negative current in the junction is equal. The resulting critical
current is zero, and thus a minimum of the Fraunhofer pattern is obtained. An
increase of the magnetic field is accompanied by an increasing number of
positive and negative supercurrent regions in the junction (as can be seen in
the map Fig. 4) and the number of zero crossings (that corresponds to the
Josephson vortices).

Figure 5: (a) Critical current versus position of the SGM tip across the
junction and magnetic flux. (b) Critical current cross-sections for two values
of the tip position marked with arrows in panel (a).

The critical current obtained in the phenomenological model against the
magnetic field and in the presence of the tip is presented in Fig. 5. In panel
(a) we see a similar oscillatory pattern as obtained in the short-junction
numerical calculation of Fig. 2 (note also the good agreement between the
black and red curves shown in Fig. 3(b)). In Fig. 5(b) we plot cross-sections
of $I_{c}$ versus the magnetic field for two positions of the tip. When the
tip is far from the junction center ($x_{\mathrm{tip}}=600$ nm), we observe
the usual Fraunhofer pattern as obtained from the integration of the
supercurrent distribution of Fig. 4 over the $x$ direction. On the other hand,
when the tip is located at the center of the junction, i.e.
$x_{\mathrm{tip}}=0$, the height of the lobes is modified depending on their
order. An even-odd pattern is obtained, in which odd maxima are suppressed and
the even ones are amplified. Furthermore, the periodicity of the positions of
the minima with flux quanta is lost.squ

Figure 6: Supercurrent distribution across the junction in the presence of the
SGM tip, as obtained from Eq. (8). $y_{\mathrm{tip}}=0$. (a)
$x_{\mathrm{tip}}=0$ nm, $\Phi=0.5\Phi_{0}$, $\phi=0.5\pi$. (b)
$x_{\mathrm{tip}}=0$ nm, $\Phi=\Phi_{0}$, $\phi=-0.676\pi$. (c)
$x_{\mathrm{tip}}=200$ nm, $\Phi=\Phi_{0}$, $\phi=0.89\pi$. (d)
$x_{\mathrm{tip}}=0$ nm, $\Phi=1.5\Phi_{0}$, $\phi=-0.5\pi$. (e)
$x_{\mathrm{tip}}=0$ nm, $\Phi=2.5\Phi_{0}$, $\phi=0.504\pi$.

To understand how the SGM mapping affects the critical current, let us focus
on supercurrent cross-sections obtained in the presence of the tip for four
values of the magnetic field, as presented in Fig. 6.

For a weak magnetic field $\Phi=0.5\Phi_{0}$ [Fig. 6(a)], we obtain the main
supercurrent lobe with an indentation at the position of the tip
($x_{\mathrm{tip}}=0$), as the tip induces a region of density depletion in
the junction. Note that the supercurrent value depends on the phase difference
$\phi$ in Eq. (8). Its maximal value is obtained for $\phi=0.5\pi$.

The supercurrent distribution for the flux $\Phi=\Phi_{0}$, such that $I_{c}$
exhibits the first minimum without the tip, is shown in Fig. 6(b). The amount
of positive and negative current is the same, independent of $\phi$, as
obtained by considering the magnetic field corresponding to an integer
multiple of the flux quantum in Eq. (8), i.e.
$\int_{-W/2}^{W/2}\mathcal{I}(\gamma(x))dx=0$ with $\gamma(x)=\phi+2\pi xn/W$
for integer $n$. When the SGM tip is positioned at the center of the junction,
the phase sets to $\phi=-0.676\pi$. Consequently, the tip cuts off a part of
the negative current and by that increases the total critical current.
Importantly, here the phase is free to adjust to maximize the supercurrent,
which results in the negative lobe following the tip position—see Fig. 6(c)
obtained for $x_{\mathrm{tip}}=200$ nm. As a result, for flux values set to a
Fraunhofer minimum, independent on the tip position, the presence of the SGM
always increases the critical current—see the red curve in Fig. 7(a).

Figure 7: (a) Change in critical current versus tip position across the
junction with respect to the critical current obtained in the absence of the
SGM tip ($I_{c0}$) for two values of the flux as obtained in the
phenomenological model. (b) Phase difference that results in maximal
supercurrent versus flux and tip position across the junction.

On the other hand, at each Fraunhofer maximum, where the number of positive
and negative lobes is unequal, there is a unique value of the phase difference
$\phi$ that assures the maximal critical current by setting the supercurrent
distribution with one extra positive lobe with respect to the number of
negative ones. The tip increases $I_{c}$ if it blocks the negative
supercurrent flow, while it decreases the critical current when it suppresses
the positive supercurrent lobe—see Figs. 6(d,e). The magnitude of the change
of the critical current is proportional to the amount of the suppressed
current. This phenomenon links the SGM critical current measurement to the
supercurrent distribution across the junction and allows to depict the
supercurrent flow [c.f. black curve in Fig. 7(a) with Fig. 6(d)] along with
its asymmetric distribution. The latter results in breaking of
$I_{c}(x_{\mathrm{tip}},\Phi)=I_{c}(-x_{\mathrm{tip}},\Phi)$ symmetry in the
map Fig. 5(a). The features in the map are rather more symmetric with
condition $I_{c}(x_{\mathrm{tip}},\Phi)=I_{c}(-x_{\mathrm{tip}},-\Phi)$
reflecting the skewness of the supercurrent distribution inherent to
transparent junctions.

We remark that for junctions with limited transparency, the skewness of the
supercurrent distribution is less pronounced, as can be seen in Fig. 3(a).
Accordingly, the asymmetry in the critical current versus tip position is less
visible, and the distribution becomes more symmetric when $\tau$ decreases, as
can be seen in Fig. 3(b). Results for opaque junction are given in the
appendix.

The aforementioned phenomena allow to connect the regions of strongly
amplified (suppressed) critical current in the map of Fig. 5(a) obtained
outside of the Fraunhofer minima with the regions of negative (positive)
supercurrent distribution in the junction. On the other hand, the process of
phase adjustment for flux values close to integer values of the flux quantum
is reflected in the strong superconducting phase modulation induced by the
tip, visible in Fig. 7(b).

###  III.2 Probing Josephson vortices and supercurrent distribution asymmetry

Figure 8: (a,c,e) Supercurrent distribution for three values of the magnetic
field. (b,d,f) Critical current change (critical current in the junction minus
critical current obtained without the tip) versus SGM tip position. Column
(a-b) is obtained for $\Phi=0.5\Phi_{0}$ with $\phi=0.5\pi$ for (a); Column
(c-d): $\Phi=\Phi_{0}$ with $\phi=0$ for (c); Column (e-f): $\Phi=1.5\Phi_{0}$
with $\phi=-0.5\pi$ for (e).

The supercurrent distribution obtained in the phenomenological model nicely
reproduces the features of the supercurrent maps obtained in the short-
junction approximation. In Figs. 8(a,c,e) we show supercurrent maps for three
values of the magnetic field cur . Panel (a) is calculated for a flux near the
first Fraunhofer maximum with $\Phi=0.5\Phi_{0}$, for the phase difference
resulting in the maximal supercurrent. Panels (c) and (e) are obtained at the
first Fraunhofer minimum and the second maximum, respectively, where we
observe the formation of Josephson vortices.

In an experimentally viable situation, when the critical current is measured
in a current-bias configuration, the phase adjusts to provide the maximal
supercurrent—at the Fraunhofer minima we always observe a supercurrent
increase [Fig. 8(d)] by the tip. On the other hand, at $I_{c}$ maxima, the
structure of the Josephson vortices is visualized by the SGM [Fig. 8(f)] in
agreement with the results of Sec. III.A.

Importantly, we observe that in the array of Josephson vortices, the clockwise
and anticlockwise vortices have a different spatial span, as can be seen in
Fig. 8(e). In the limit $\tau=1$, the current abruptly changes sign in the
left part of the junction, while on the right it varies smoothly [see the
black curve in Fig. 3(a)]. This is mirrored in the asymmetry of the vortices.
The vortex on the left-hand side of the map Fig. 8(e) is compressed. On the
other hand, for positive $x$, the supercurrent increases smoothly as it goes
through zero, resulting in an expanded vortex on the right-hand side of Fig.
8(e). As for the short-junction calculation, the asymmetry in the supercurrent
vortices is reflected in the critical current measured, as visible in the map
of Fig. 2 as well as in the cross-section presented in Fig. 5(b) with the red
curve.

We notice that the asymmetry in counterpropagating vortices has been reported
by V. P. Ostroukh et al. Ostroukh  _et al._ (2016) for a square vortex
lattice. Here, however, for a linear array of vortices, we demonstrate that
the asymmetry bears an important consequence for the distribution of the
supercurrent flowing between the superconducting contacts—the current streams
are also not symmetric and shifted in the $x$ direction, i.e., the
supercurrent flowing in the direction parallel (antiparallel) to the $y$
direction is shifted towards positive (negative) $x$-direction.

##  IV Beyond the short-junction approximation

The length of the normal part can be of the order of hundreds of nanometers
Shabani  _et al._ (2016); Kjaergaard  _et al._ (2017); Guiducci  _et al._
(2019a); de Vries  _et al._ (2018); Telesio  _et al._ (2022); Ke  _et al._
(2019); Salimian  _et al._ (2021), due to fabrication techniques or the
requirements of the SGM method itself, i.e., the fact that the SGM tip has to
be able to fit between the superconductors, setting a lower bound to the
length of the normal part. This combined with a large induced gap when using,
e.g., Nb as the superconductor leads to structures that can exceed the short-
junction approximation, i.e., $L\simeq\xi$. It is important therefore to
assess whether the discussed effects occur also beyond that regime.

To this end, we numerically consider a finite system composed of
superconducting leads and a normal, ballistic region (which leads to a
creation of a junction with $\tau\simeq 1$). The system has a width $W$ and
consist of a scattering region of length $L$ sandwiched between two
superconducting regions, each of length $L_{\mathrm{SC}}$. The whole system is
described by the Hamiltonian

$H=\left(\begin{array}[]{cc}H_{N}+H_{Z}\sigma_{z}&\Delta(x,y)\sigma_{0}\\\
\Delta^{*}(x,y)\sigma_{0}&-H_{N}+H_{Z}\sigma_{z}\\\ \end{array}\right),$ (12)

where $H_{Z}$ stands for Zeeman term $H_{Z}=\frac{1}{2}g\mu_{B}B$. The
superconducting pairing potential is

$\Delta(x,y)=\begin{cases}\Delta&\mbox{if}\;y\leq-L/2\\\
0&\mbox{if}\;-L/2<y<L/2\\\ \Delta\exp[i\phi]&\mbox{if}\;y\geq L/2\\\
\end{cases}.$ (13)

We take $g=-50$, $W=1000$ nm and $\Delta=2$ meV. The estimated coherence
length is $\xi=330$ nm and thus shorter than the length of the normal region,
which we take as $L=600$ nm. At the same time, it is also shorter than the
assumed length of the superconducting leads $L_{\mathrm{SC}}=400$ nm, allowing
the evanescent quasiparticle wave functions to vanish in the superconducting
segments. We calculate the energy spectrum of the junction, performing an
exact diagonalization of Eq. (12) discretized on a square lattice. The tip is
introduced in a similar manner as in the short-junction calculations, and the
supercurrent is calculated from Eq. (5) taking into account the 300 lowest
positive energy eigenstates. Zeeman interaction and orbital effects of the
field are included only in the normal region.

Figure 9: (a) Supercurrent distribution for the magnetic field set to the
second Fraunhofer maximum and $\phi=-0.5\pi$. (b) Critical current map versus
the tip position for the first Fraunhofer maximum. (c) Critical current map
versus the tip position across the junction and the flux. The results are
obtained in the long-junction regime for $y_{\mathrm{tip}}=0$.

In Fig. 9(a) and (b) we show the supercurrent distribution and the
corresponding critical current map obtained with the SGM tip for the second
Fraunhofer maximum. The supercurrent exhibits similar features as found for
the short-junction [Fig. 8(e)] with a clear anisotropy in the supercurrent and
Josephson vortices. Most importantly, for elongated junctions, the vortices
form in a linear array at the half length of the junction. The SGM mapping as
shown in Fig. 9(b) not only determines the supercurrent flow between the
superconducting contacts (seen as the change in the critical current for
$y_{\mathrm{tip}}=0$) but also the distribution of the supercurrent along the
junction due to the quasiparticles flowing from the leads and circulating
around the vortices.

Finally, in Fig. 9(c) we show the critical current as a function of the flux
in the junction and the horizontal coordinate of the SGM tip (we set
$y_{\mathrm{tip}}=0$). We observe that the Fraunhofer minima are no longer
obtained for integer multiples of the flux quantum, as expected for elongated
junctions Barzykin and Zagoskin (1999). Most importantly, we obtain a very
clear critical current modulation as the tip scans across the sample that can
be used to determine the supercurrent distribution together with its asymmetry
despite the relaxation of the limits of short-junction approximation.

It should be noted that in very elongated junctions, where the length of the
normal region becomes comparable to the Josephson penetration depth, the non-
local electrodynamics starts to play a role affecting the supercurrent
distribution and also the formation of a Fraunhofer pattern Krasnov  _et al._
(1997); Boris  _et al._ (2013); Krasnov (2020). A study of these effects is
beyond the scope of this work, since SNS junctions realized on 2DEGs have
typical Josephson penetration lengths of the order of several microns Barone
and Paternò (1982).

##  V Summary and conclusions

We theoretically studied scanning gate microscopy imaging of supercurrent flow
in planar superconductor-semiconductor-superconductor Josephson junctions. We
considered a case where the perpendicular magnetic field induces a spatial
variation of the supercurrent and induces Josephson vortices. For systems in
both short- and long-junction regimes, we found that the repelling potential
of the SGM tip allows to visualize the supercurrent distribution and Josephson
vortices via critical current measurements. This is possible provided that the
flux in the junction is set outside of Fraunhofer minima for the case without
the tip. When the flux is set to a critical current minimum, the
superconducting phase difference adapts in such a way that the tip always
leads to an increase of the supercurrent. These results are valid both for
transmissive ($\tau\simeq 1$) and tunnel ($\tau\ll 1$) SNS junctions.

Furthermore, for highly transmissive junctions ($\tau\simeq 1$) we found that
the scanning gate microscopy critical current maps reveal an asymmetry in the
supercurrent distribution, which is caused by the asymmetry between the
clockwise and counterclockwise Josephson vortices in transparent junctions.
This effect should be already present in recently developed planar junctions
where the transmission coefficient takes high values of $\tau\simeq 0.9$
Kjaergaard  _et al._ (2017); Salimian  _et al._ (2021).

Our results were obtained assuming a short-junction regime and supported by a
phenomenological model, where the tip suppresses a phase-dependent current in
the junction. Extension beyond the short-junction limit was implemented to
confirm the robustness of our findings.

##  VI Acknowledgement

We acknowledge helpful discussions with Alina Mreńca-Kolasińska. This work was
supported by National Science Centre (NCN) agreement number
UMO-2020/38/E/ST3/00418. SH and MC were partially supported by the SUPERTOP
project, QUANTERA ERA-NET Cofound in Quantum Technologies (H2020 Grant No.
731473) and by the FET-OPEN project AndQC (H2020 Grant No. 828948).

*

##  Appendix A Phenomenological model in the tunneling limit

In the tunneling limit, when $\tau$ is much smaller than 1, the current phase
relation Eq. (10) can be approximated by $\mathcal{I}=\sin(\gamma(x))$. This
results in a symmetric supercurrent distribution in the junction as shown in
Fig. 3(a) with the green-dashed curve. Here we provide results for the
tunneling case that complement the phenomenological analysis done in the main
text.

Figure 10: Supercurrent distribution $J=J_{0}\mathcal{I}(\gamma(x))$ across
the junction without the tip obtained in the phenomenological model for
$\mathcal{I}=\sin(\gamma(x))$ versus magnetic field. The current is obtained
for the phase difference value $\phi$ that maximizes the supercurrent in the
junction.

The supercurrent distribution presented in Fig. 10 shows formation of positive
and negative supercurrent regions. When $\Phi=n\Phi_{0}$ (with $n$ an
integer), the amount of positive and negative current in the junction is
equal—a complete cycle of supercurrent is formed. The resulting critical
current is zero, and so a minimum of the Fraunhofer pattern is obtained. Every
time the complete cycle is crossed, the phase jumps by $\pi$ and the critical
current increases until it reaches the next Fraunhofer maximum—see Fig. 11.

Figure 11: (a) Critical current versus magnetic flux and position of the SGM
tip across the junction. (b) Critical current cross-sections for two values of
the tip position marked with arrows in panel (a). Figure 12: Supercurrent
distribution across the junction in the presence of the SGM tip at
$y_{\mathrm{tip}}=0$. (a) $x_{\mathrm{tip}}=0$ nm, $\Phi=0.5\Phi_{0}$,
$\phi=0.5\pi$. (b) $x_{\mathrm{tip}}=0$ nm, $\Phi=\Phi_{0}$, $\phi=-0.5\pi$.
(c) $x_{\mathrm{tip}}=200$ nm, $\Phi=\Phi_{0}$, $\phi=0.9\pi$. (d)
$x_{\mathrm{tip}}=0$ nm, $\Phi=1.5\Phi_{0}$, $\phi=-0.5\pi$. (e)
$x_{\mathrm{tip}}=0$ nm, $\Phi=2.5\Phi_{0}$, $\phi=0.51\pi$.

The symmetry in the current distribution shown in Fig. 10 translates into
symmetric features in the SGM critical current map shown in Fig. 11.

The process of critical current modification operates in the same manner as
for a transparent junction (cf. Fig. 6). At Fraunhofer minima it is always the
negative supercurrent region that is suppressed [Fig. 12(b,c)], which results
in an increase of $I_{c}$ by the tip. At Fraunhofer maxima [Fig. 12(d,e)] the
tip suppresses the supercurrent in its vicinity decreasing or increasing the
critical current, depending on the sign of the suppressed supercurrent.
Importantly, as now the supercurrent distribution is symmetric, so are the
critical current features, as show in the map Fig. 11(a).

## References

  * Barone and Paternò (1982) A. Barone and G. Paternò,  _Physics and Applications of the Josephson Effect_ (John Wiley & Sons, Inc., 1982).
  * Gross  _et al._ (2016) R. Gross, A. Marx, and F. Deppe,  _Applied Superconductivity: Josephson Effect and Superconducting Electronics_, De Gruyter Textbook Series (Walter De Gruyter Incorporated, 2016).
  * Ostroukh  _et al._ (2016) V. P. Ostroukh, B. Baxevanis, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B 94, 094514 (2016).
  * Tinkham (2004) M. Tinkham,  _Introduction to Superconductivity_, 2nd ed. (Dover Publications, 2004).
  * Rowell (1963) J. M. Rowell, Phys. Rev. Lett. 11, 200 (1963).
  * Roditchev  _et al._ (2015) D. Roditchev, C. Brun, L. Serrier-Garcia, J. C. Cuevas, V. H. L. Bessa, M. V. Milošević, F. Debontridder, V. Stolyarov, and T. Cren, Nat. Phys 11, 332 (2015).
  * Dremov  _et al._ (2019) V. V. Dremov, S. Y. Grebenchuk, A. G. Shishkin, D. S. Baranov, R. A. Hovhannisyan, O. V. Skryabina, N. Lebedev, I. A. Golovchanskiy, V. I. Chichkov, C. Brun, T. Cren, V. M. Krasnov, A. A. Golubov, D. Roditchev, and V. S. Stolyarov, Nat. Commun. 10, 4009 (2019).
  * Sticlet  _et al._ (2020) D. Sticlet, P. Wójcik, and M. P. Nowak, Phys. Rev. B 102, 165407 (2020).
  * Hart  _et al._ (2014) S. Hart, H. Ren, T. Wagner, P. Leubner, M. Mühlbauer, C. Brüne, H. Buhmann, L. W. Molenkamp, and A. Yacoby, Nat. Phys 10, 638 (2014).
  * Pribiag  _et al._ (2015) V. S. Pribiag, A. J. A. Beukman, F. Qu, M. C. Cassidy, C. Charpentier, W. Wegscheider, and L. P. Kouwenhoven, Nat. Nanotechnol. 10, 593 (2015).
  * Allen  _et al._ (2016) M. T. Allen, O. Shtanko, I. C. Fulga, A. R. Akhmerov, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, L. S. Levitov, and A. Yacoby, Nat. Phys 12, 128 (2016).
  * Ying  _et al._ (2020) J. Ying, J. He, G. Yang, M. Liu, Z. Lyu, X. Zhang, H. Liu, K. Zhao, R. Jiang, Z. Ji, J. Fan, C. Yang, X. Jing, G. Liu, X. Cao, X. Wang, L. Lu, and F. Qu, Nano Lett. 20, 2569 (2020).
  * Guiducci  _et al._ (2019a) S. Guiducci, M. Carrega, F. Taddei, G. Biasiol, H. Courtois, F. Beltram, and S. Heun, Phys. Rev. B 99, 235419 (2019a).
  * 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, S. Kraemer, C. Nayak, M. Troyer, C. M. Marcus, and C. J. Palmstrøm, Phys. Rev. B 93, 155402 (2016).
  * Kjaergaard  _et al._ (2017) M. Kjaergaard, H. J. Suominen, M. P. Nowak, A. R. Akhmerov, J. Shabani, C. J. Palmstrøm, F. Nichele, and C. M. Marcus, Phys. Rev. Applied 7, 034029 (2017).
  * 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, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern, C. M. Marcus, and F. Nichele, Nature 569, 89 (2019).
  * Guiducci  _et al._ (2019b) S. Guiducci, M. Carrega, G. Biasiol, L. Sorba, F. Beltram, and S. Heun, Phys. Status Solidi RRL 13, 1800222 (2019b).
  * de Vries  _et al._ (2018) F. K. de Vries, T. Timmerman, V. P. Ostroukh, J. van Veen, A. J. A. Beukman, F. Qu, M. Wimmer, B.-M. Nguyen, A. A. Kiselev, W. Yi, M. Sokolich, M. J. Manfra, C. M. Marcus, and L. P. Kouwenhoven, Phys. Rev. Lett. 120, 047702 (2018).
  * 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, B. I. Halperin, and A. Yacoby, Nature 569, 93 (2019).
  * Telesio  _et al._ (2022) F. Telesio, M. Carrega, G. Cappelli, A. Iorio, A. Crippa, E. Strambini, F. Giazotto, M. Serrano-Ruiz, M. Peruzzini, and S. Heun, ACS Nano 16, 3538 (2022).
  * Ke  _et al._ (2019) C. T. Ke, C. M. Moehle, F. K. de Vries, C. Thomas, S. Metti, C. R. Guinn, R. Kallaher, M. Lodari, G. Scappucci, T. Wang, R. E. Diaz, G. C. Gardner, M. J. Manfra, and S. Goswami, Nat. Commun. 10, 3764 (2019).
  * Moehle  _et al._ (2021) C. M. Moehle, C. T. Ke, Q. Wang, C. Thomas, D. Xiao, S. Karwal, M. Lodari, V. van de Kerkhof, R. Termaat, G. C. Gardner, G. Scappucci, M. J. Manfra, and S. Goswami, Nano Lett. 21, 9990 (2021).
  * Zhi  _et al._ (2019) J. Zhi, N. Kang, F. Su, D. Fan, S. Li, D. Pan, S. P. Zhao, J. Zhao, and H. Q. Xu, Phys. Rev. B 99, 245302 (2019).
  * de Vries  _et al._ (2019) F. K. de Vries, M. L. Sol, S. Gazibegovic, R. L. M. o. h. Veld, S. C. Balk, D. Car, E. P. A. M. Bakkers, L. P. Kouwenhoven, and J. Shen, Phys. Rev. Research 1, 032031 (2019).
  * Salimian  _et al._ (2021) S. Salimian, M. Carrega, I. Verma, V. Zannier, M. P. Nowak, F. Beltram, L. Sorba, and S. Heun, Appl. Phys. Lett. 119, 214004 (2021).
  * Alidoust (2018) M. Alidoust, Phys. Rev. B 98, 245418 (2018).
  * Zyuzin  _et al._ (2016) A. Zyuzin, M. Alidoust, and D. Loss, Phys. Rev. B 93, 214502 (2016).
  * Woodside and McEuen (2002) M. T. Woodside and P. L. McEuen, Science 296, 1098 (2002).
  * Aidala  _et al._ (2007) K. E. Aidala, R. E. Parrott, T. Kramer, E. J. Heller, R. M. Westervelt, M. P. Hanson, and A. C. Gossard, Nat. Phys 3, 464 (2007).
  * Jalabert  _et al._ (2010) R. A. Jalabert, W. Szewc, S. Tomsovic, and D. Weinmann, Phys. Rev. Lett. 105, 166802 (2010).
  * Gorini  _et al._ (2013) C. Gorini, R. A. Jalabert, W. Szewc, S. Tomsovic, and D. Weinmann, Phys. Rev. B 88, 035406 (2013).
  * Nowak  _et al._ (2014) M. P. Nowak, K. Kolasiński, and B. Szafran, Phys. Rev. B 90, 035301 (2014).
  * Topinka  _et al._ (2001) M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski, and A. C. Gossard, Nature 410, 183 (2001).
  * Paradiso  _et al._ (2010) N. Paradiso, S. Heun, S. Roddaro, L. Pfeiffer, K. West, L. Sorba, G. Biasiol, and F. Beltram, Physica E Low Dimens. Syst. Nanostruct. 42, 1038 (2010).
  * LeRoy  _et al._ (2005) B. J. LeRoy, A. C. Bleszynski, K. E. Aidala, R. M. Westervelt, A. Kalben, E. J. Heller, S. E. J. Shaw, K. D. Maranowski, and A. C. Gossard, Phys. Rev. Lett. 94, 126801 (2005).
  * Jura  _et al._ (2009) M. P. Jura, M. A. Topinka, M. Grobis, L. N. Pfeiffer, K. W. West, and D. Goldhaber-Gordon, Phys. Rev. B 80, 041303 (2009).
  * Neubeck  _et al._ (2012) S. Neubeck, L. Ponomarenko, A. Mayorov, S. Morozov, R. Yang, and K. Novoselov, Physica E Low Dimens. Syst. Nanostruct. 44, 1002 (2012).
  * Connolly  _et al._ (2011) M. R. Connolly, K. L. Chiu, A. Lombardo, A. Fasoli, A. C. Ferrari, D. Anderson, G. A. C. Jones, and C. G. Smith, Phys. Rev. B 83, 115441 (2011).
  * Mreńca-Kolasińska and Szafran (2017) A. Mreńca-Kolasińska and B. Szafran, Phys. Rev. B 96, 165310 (2017).
  * Garcia  _et al._ (2013) A. G. F. Garcia, M. König, D. Goldhaber-Gordon, and K. Todd, Phys. Rev. B 87, 085446 (2013).
  * Prokop  _et al._ (2020) M. Prokop, D. Gut, and M. P. Nowak, J. Condens. Matter Phys. 32, 205302 (2020).
  * Chen (1993) C. J. Chen,  _Introduction to Scanning Tunneling Microscopy_ (Oxford University Press, 1993).
  * Voigtländer (2015) B. Voigtländer,  _Scanning Probe Microscopy_ (Springer-Verlag Berlin Heidelberg, 2015).
  * Takayanagi  _et al._ (1995) H. Takayanagi, T. Akazaki, and J. Nitta, Phys. Rev. Lett. 75, 3533 (1995).
  * Bauch  _et al._ (2005) T. Bauch, E. Hürfeld, V. M. Krasnov, P. Delsing, H. Takayanagi, and T. Akazaki, Phys. Rev. B 71, 174502 (2005).
  * Heersche  _et al._ (2007) H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. Vandersypen, and A. F. Morpurgo, Nature 446, 56 (2007).
  * Calado  _et al._ (2015) V. E. Calado, S. Goswami, G. Nanda, M. Diez, A. R. Akhmerov, K. Watanabe, T. Taniguchi, T. M. Klapwijk, and L. M. K. Vandersypen, Nat. Nanotechnol. 10, 761 (2015).
  * Lee and Lee (2018) G.-H. Lee and H.-J. Lee, Rep. Prog. Phys. 81, 056502 (2018).
  * Ben Shalom  _et al._ (2016) M. Ben Shalom, M. J. Zhu, V. I. Fal’ko, A. Mishchenko, A. V. Kretinin, K. S. Novoselov, C. R. Woods, K. Watanabe, T. Taniguchi, A. K. Geim, and J. R. Prance, Nat. Phys. 12, 318 (2016).
  * Mayer  _et al._ (2020) W. Mayer, W. F. Schiela, J. Yuan, M. Hatefipour, W. L. Sarney, S. P. Svensson, A. C. Leff, T. Campos, K. S. Wickramasinghe, M. C. Dartiailh, I. Zutic, and J. Shabani, ACS Appl. Electron. Mater. 2, 2351 (2020).
  * (51) In the literature there is another meaning of ”short-junction” which refers to the junction shorter than the Josephson penetration depth, where the self field effects can be neglected. In this sense all the junctions considered in this work belong to this regime.
  * Beenakker (1991) C. W. J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991).
  * van Heck  _et al._ (2014) B. van Heck, S. Mi, and A. R. Akhmerov, Phys. Rev. B 90, 155450 (2014).
  * Szafran (2011) B. Szafran, Phys. Rev. B 84, 075336 (2011).
  * Groth  _et al._ (2014) C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, New J. Phys. 16, 063065 (2014).
  * Vurgaftman  _et al._ (2001) I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, Journal of Applied Physics 89, 5815 (2001).
  * Nijholt  _et al._ (2019) B. Nijholt, J. Weston, J. Hoofwijk, and A. Akhmerov,  (2019), 10.5281/zenodo.1182437.
  * Kaperek and Nowak (2022) K. Kaperek and M. P. Nowak, Zenodo (2022), 10.5281/zenodo.6641899.
  * (59) Note that the gauge invariant phase results from a gauge invariant phase gradient $\nabla\gamma=\nabla\phi-q/\hbar\mathbf{A}$ which leads to a plus sign in the last term of Eq. (11) when using nomenclature $\Phi_{0}=h/2e>0$.
  * (60) By properly tuning tip height and applied bias, so that a large area of the junction becomes depleted, the SGM could be used to continuously tune the junction from a Fraunhofer pattern to a SQUID-like pattern.
  * (61) The wave functions in the normal region obtained from the solution of the scattering problem defined by Eq. (6) are phase independent. On the other hand, the ABS wave-function amplitudes (obtained as eigenvectors of Eq. (4)) are orthonormalized—as a result the information about the magnitude of the current carried by each ABS is not contained in its wave function. For a proper calculation of the supercurrent distribution one needs to normalize the supercurrent carried by each ABS accordingly to its value obtained directly from the ABS energy spectra.
  * Barzykin and Zagoskin (1999) V. Barzykin and A. M. Zagoskin, Superlattices and Microstruct. 25, 797 (1999).
  * Krasnov  _et al._ (1997) V. M. Krasnov, V. A. Oboznov, and N. F. Pedersen, Phys. Rev. B 55, 14486 (1997).
  * Boris  _et al._ (2013) A. A. Boris, A. Rydh, T. Golod, H. Motzkau, A. M. Klushin, and V. M. Krasnov, Phys. Rev. Lett. 111, 117002 (2013).
  * Krasnov (2020) V. M. Krasnov, Phys. Rev. B 101, 144507 (2020).