# Superconducting triplet rim currents in a spin-textured ferromagnetic disk

Remko Fermin  Dyon van Dinter  Michel Hubert  Bart Woltjes  Huygens-Kamerlingh
Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The
Netherlands.  Mikhail Silaev  Department of Physics and Nanoscience Center,
University of Jyväskylä, P.O. Box 35 (YFL), Jyväskylä, Finland  Jan Aarts
Kaveh Lahabi  Huygens-Kamerlingh Onnes Laboratory, Leiden University, P.O. Box
9504, 2300 RA Leiden, The Netherlands<EMAIL_ADDRESS>
###### Abstract

Since the discovery of the long-range superconducting proximity effect, the
interaction between spin-triplet Cooper pairs and magnetic structures such as
domain walls and vortices has been the subject of intense theoretical
discussions, while the relevant experiments remain scarce. We have developed
nanostructured Josephson junctions with highly controllable spin texture,
based on a disk-shaped Nb/Co bilayer. Here, the vortex magnetization of Co and
the Cooper pairs of Nb conspire to induce long-range triplet (LRT)
superconductivity in the ferromagnet. Surprisingly, the LRT correlations
emerge in highly localized (sub-80 nm) channels at the rim of the ferromagnet,
despite its trivial band structure. We show that these robust rim currents
arise from the magnetization texture acting as an effective spin-orbit
coupling, which results in spin accumulation at the bilayer-vacuum boundary.
Lastly, we demonstrate that by altering the spin texture of a single
ferromagnet, both $0$ and $\pi$-channels can be realized in the same device.

Computational Physics Laboratory, Physics Unit, Faculty of Engineering and
Natural Sciences, Tampere University, P.O. Box 692, Tampere, Finland

##  1 Introduction

The appearance of localized supercurrents at the edges of a Josephson junction
is generally attributed to the topology of the electronic band structure and
edge states.1. Edge states and the accompanying edge currents are typically
found in ultraclean systems such as 2D electron gases,2 nanowires,3 and
graphene.4 Here, we report the emergence of highly localized (sub-80 nm) spin-
polarized supercurrents at the rim of disk-shaped Josephson junctions with a
diffusive ferromagnetic barrier (Co). As we demonstrate, however, the rim
currents are not related to the electronic band structure; but rather a direct
result of the interactions between spin-triplet Cooper pairs and the
nontrivial spin texture of the ferromagnet.

At the interface between a superconductor and a ferromagnet, short-range
triplet (SRT) Cooper pairs with zero spin projection emerge naturally via
spin-mixing of singlet pairs, and decay over a few nanometers
($\xi_{\text{F}}(\text{Co})\sim 3$ nm 5) inside the ferromagnet. Long-range
triplet (LRT) pairs can, on the other hand, propagate over substantially
larger distances.6, 7 Half-metallic systems can even show the LRT proximity
effect over hundreds of nanometers.11, 9, 10, 8 Due to their spin
polarization, the LRT Cooper pairs can provide the means to combine the
absence of Joule heating and decoherence with the functionality of spintronic
devices.12, 13 However, the controlled generation of LRT currents has proven
to be a demanding process, commonly realized in complex superconductor-
ferromagnet (S$-$F) hybrids, involving multiple F layers with noncollinear
magnetization.14, 17, 21, 22, 19, 24, 18, 15, 16, 20, 23. Furthermore, a
substantial body of research considered the possibility of generating and
controlling LRT correlations using spin-textured systems, such as domain
walls6, 25, 26, 27, 28 and vortices.29, 30 However, the experimental evidence
to verify such models remains scarce.31 In other recent developments, it was
suggested that theoretically spin mixing can also be achieved by spin-orbit
coupling (SOC)32, 33, 34, 35, 36. This led to researching long-range proximity
effects with Josephson junctions containing heavy metal interlayers.37, 38,
39, 40, 41, 42, 43. In addition, recent studies suggest that spin-orbit
coupling (SOC) can lead to spin accumulation at the edges of Josephson
devices35, 44, 45 and, in some cases, generation of LRT currents.46, 47, 48,
49 At present there is a complete lack of experiments that can examine the
influence of SOC on LRT transport. As a consequence, the interplay between
triplet pairing and magnetic texture as well as SOC remains elusive.

$\begin{array}[]{c}\includegraphics[width=346.89731pt]{Figure_1_SD.png}\end{array}$

Figure 1: a. Schematic of the Josephson device. The Nb electrodes are
separated by a trench, forming a Co weak link. The pattern on the Co layer
corresponds to micromagnetic simulations of a micrometer-size disk. (for more
information on the micromagnetic simulations, see Supporting Information sec.
S2). The arrows correspond to the in-plane magnetization, while the out-of-
plane component is represented by color, which only appears at the vortex core
(blue region; less than 5 nm in diameter). b. False colored scanning electron
micrograph of a structured bilayer. The 20 nm gap indicates the Co weak link
at the bottom of the trench. The scale bar is equivalent to 400 nm.

$\begin{array}[]{c}\includegraphics[width=433.62pt]{Figure2_Final_Final_HD.pdf}\end{array}$

Figure 2: Overview of the basic transport properties of the devices. a.
Resistance as a function of temperature of a proximized disk device (pink) is
compared to a Co-Nb disk without a trench (blue) and a bar-shaped device with
uniform magnetization (orange). The bar-shaped device is not proximized due to
the lack of spin texture (for more information, see Supporting Information
sec. S3). b. $IV$-characteristics at 2 K of a proximized disk device, measured
while irradiating the sample with $f=$ 1.1 GHz of RF radiation for different
powers. The voltage is normalized in units of $hf/2e$. c. Evolution of the
Shapiro response as a function of RF power represented as
$\text{d}V/\text{d}I$ colormap.

To address these issues, we have developed a disk-shaped S$-$F$-$S Josephson
junction with a highly controllable ferromagnetic vortex spin texture, capable
of converting singlet Cooper pairs into LRT currents (see Figure 1). The
device consists of a Nb/Co bilayer, where a trench in the Nb layer introduces
a ($\sim 20$ nm) cobalt weak link, which eliminates any singlet or SRT
transport. We show that a magnetization gradient of the vortex can act as an
effective SOC, which leads to spin accumulation at the rims of the device.
This is verified by our transport experiments, which show that the LRT
transport is highly localized at the rims of the ferromagnet, resulting in a
distinct double-slit supercurrent interference pattern. By modifying the spin
texture in a controllable manner, we show that both $0$ and $\pi$ segments can
emerge in a single junction. Utilizing the linearized Usadel equation, we
examine the microscopic origin of the rim currents in the proximized
ferromagnet. Our findings suggest that, in addition to spin texture,
superconductor-vacuum boundary conditions play an important role in the
singlet to LRT conversion.

##  2 Results and Discussion

###  2.1 Establishing long-range triplet transport

Figure 2a shows resistance as a function of temperature for a typical disk
junction (see also Supporting Information sec. S1). A micrometer-wide weak
link has a typical resistance of 200 m$\Omega$ and becomes fully proximized at
low temperatures. We unambiguously establish the Josephson transport in our
device by observing their Shapiro response to microwave radiation. This is
carried out by measuring the current-voltage ($IV$)-characteristics while
irradiating the junction with microwaves from a nearby antenna. The
$IV$-curves show clear Shapiro steps (discrete voltage steps of $hf/(2e)$,
where $h$ is the Planck constant, $f$ is the frequency, and $e$ is the
electron charge), which is a result of the phase-locking between the applied
microwaves and the Josephson currents (see figure 2b). We also examined the
evolution of the width of the voltage plateaus as a function of microwave
power. The results are presented in Figure 2c as a color map of differential
resistance.

A direct method to examine the presence of LRT correlations is to verify that,
once the mechanism for the emergence of LRT pairing is eliminated, the
proximity effect will disappear. For instance, in the case of S/F’/F/F’/S
multilayer junctions used in previous studies, where the generation of LRT
correlations requires a magnetic noncollinearity between the F and F’ layers,
the control experiment would show that the critical current ($I_{\text{c}}$)
is heavily suppressed if the F’ layers were either removed or magnetized
parallel (or antiparallel) to the F layer.17, 18, 19, 60, 20, 21 The same
argument applies here: if the proximity effect is due to LRT correlations
produced by the spin texture of the junction, the $I_{\text{c}}$ must vanish
once the magnetization is uniform. This actually happens when we remove the
spin texture by applying an in-plane field, typically around 100 mT, which
completely suppresses the $I_{\text{c}}$ (see Supporting Figure S3a). We also
verify this through two further experiments, described in Supporting
Information sec. S3. First, we examined the transport in bar-shaped control
samples, where shape anisotropy ensures that (even in the absence of in-plane
fields) the cobalt layer has a uniform magnetization along the long axis of
the bar (see Figure S1). These samples were fabricated via the same procedure
as the primary disk-shaped junctions and received the same FIB treatment to
structure their weak link. This is evident by the fact that the bar-shaped
junctions and the disk devices have a matching barrier resistance ($\approx$
200 m$\Omega$). Despite multiple attempts, however, the bar-shaped control
samples show no sign of long-range proximity. Additionally, we also prepared
disk-shaped control junctions where, by applying a lower dose of Ga ions when
structuring the weak link, we leave some residual Nb at the bottom of the
trench – forming a nonmagnetic channel for singlet transport. Such junctions
are completely insensitive to the magnetic state of the cobalt disk and are
robust against the in-plane fields used for altering the spin texture; they
maintain their $I_{\text{c}}$ at fields as high as 2 T (see Supporting Figure
S3b).

###  2.2 Triplet currents confined to the rims of the disk

We establish the presence of rim currents using superconducting quantum
interferometry (SQI), i.e., measuring the critical current as a function of a
magnetic field, applied perpendicular to the transport direction (out-of-
plane). Note that the out-of-plane fields used in our SQI experiments are too
small to disturb the stable vortex magnetization of the Co disk.

$\begin{array}[]{c}\includegraphics[width=433.62pt]{Figure_3_SD.png}\end{array}$

Figure 3:  a. and b. The superconducting interference patterns of two
junctions with different diameters. The pattern is obtained by measuring the
differential resistance as a function of d.c. current and magnetic field. The
disk diameter in a. and b. are 1.62 $\mu$m and 1.05 $\mu$m, respectively. The
period of the oscillations scales inversely with the junction area. In both
cases, the junctions show a clear two-channel interference pattern. c. and d.
depict the critical current density profiles obtained by the Fourier analysis
of the patterns in a. and b. respectively. The vertical lines indicate the
boundaries of the device.

In a conventional junction, the supercurrent is distributed uniformly across
the weak link. This results in the well-known Fraunhofer SQI pattern, where
the oscillation amplitude has a $1/B$ dependence, and the center lobe is twice
as wide as its neighbors. As shown in Figure 3, our devices show a completely
different behavior: two-channel interference patterns, characterized by equal-
width lobes and slow decay of oscillation amplitude.

All the triplet junctions we measured (over ten devices) show such a two-
channel interference pattern. This is illustrated in Figure 3, where we show
the SQI patterns for two junctions with different diameters (1.62 $\mu$m and
1.05 $\mu$m). Note that the period of the oscillations scales inversely with
the area of the junction, which is determined by the radius of the disk. We
apply inverse Fourier transform to the SQI patterns to reconstruct the spatial
distribution of supercurrent density. 50 This is a well-established technique,
commonly applied to verify the existence of edge currents (see also Supporting
Information sec. S4).51, 52 Figure 3 shows the results of our Fourier analysis
for both devices. Regardless of the sample area, we consistently find the
supercurrent to be highly localized at the rim of the sample (70 nm or less in
width, limited by the resolution of the Fourier analysis). Furthermore, the
channels are highly symmetric, as indicated by the sharp cusps of the SQI
pattern. Note that the trench is deepest on the sides of the disk (due to a
higher milling rate, as can be seen by the small notches on sides of the disk
in Figure 1b), making the formation of accidental singlet edge channels even
less probable. More importantly, the two-channel behavior is completely absent
in all the singlet control samples (see Supporting Information Sec. S3). If
the barrier contains residual Nb or is made out of a normal metal (silver),
the junction yields a standard single-channel diffraction pattern (Supporting
Figures S2 and S4b respectively).

$\begin{array}[]{c}\includegraphics[width=411.93767pt]{Figure_4_SD.png}\end{array}$

Figure 4:  Supercurrent interference patterns (left column) measured at
different in-plane fields and the corresponding simulated spin textures (right
column). The gray line represents the position of the weak link, and the red
dot indicates the location of the vortex core. a. and b. At zero in-plane
field, the vortex is at the center of the disk, and a SQUID pattern is
observed, i.e., lobes of equal width and slow decay of peak height. c. and d.
Applying $\mu_{0}H_{\text{y}}$ = 10 mT breaks the axial symmetry of the vortex
magnetization. This results in the suppression of the middle peak in the
interference pattern, characteristic of a $0-\pi$ SQUID. e. and f. At
$\mu_{0}H_{\text{y}}$ = 40 mT the vortex core is displaced by over 100 nm. The
middle peak of the interference pattern is recovered and its width is doubled
with respect to the original pattern. The peak height increases as a function
of the out-of-plane field, regardless of sweep direction.

###  2.3 Altering the magnetic texture by an in-plane field

So far we have shown the unconventional distribution of supercurrents through
the spin-textured ferromagnetic weak link. We investigate this further by
modifying the spin texture using an in-plane magnetic field. As shown in our
micromagnetic simulations, in-plane fields can alter the spin texture by
effectively moving the vortex core along the axis perpendicular to the field
(see Figure 4). For small in-plane fields, the core displacement has an almost
linear response and is fully reversible. Using a vector magnet system, we are
able to apply a constant in-plane magnetic field while simultaneously
acquiring the SQI pattern as described above. Figure 4 shows the SQI patterns
measured for different in-plane fields, applied along the trench (y-axis),
together with the corresponding micromagnetic simulations.

At zero in-plane field (Figure 4a), we observe the aforementioned two-channel
(SQUID) interference pattern, with maximum $I_{\text{c}}$ at zero field. At
$\mu_{0}H_{\text{y}}$ = 10 mT we observe a strong suppression of
$I_{\text{c}}$ for zero out of plane field. Remarkably, however,
$I_{\text{c}}$ is recovered upon increasing the out-of-plane field in either
direction. The resulting SQI pattern bears resemblance of a $0-\pi$ SQUID: all
the lobes are similar in width and $I_{\text{c}}$ is suppressed around zero.
Increasing the $H_{\text{y}}$ to 40 mT, the vortex core has traveled over 100
nm away from the center of the junction (Figure 4e). Interestingly, we find
the $I_{\text{c}}$ to recover for zero out-of-plane field. This reentrant
behavior is accompanied by drastic changes to the SQI pattern. The central
lobe is now twice as wide, indicating a modified supercurrent distribution,
which no longer corresponds to the original two channels. Even more striking
is the amplitude of the $I_{\text{c}}$ oscillations: instead of decaying, the
lobes grow taller as we increase the magnitude of the out-of-plane field. This
is the universal characteristic of $0-\pi$ junctions, i.e., a junction
consisting of multiple $0$ and $\pi$ segments connected in parallel.53, 54, 55
Note that the observed evolution of the interference pattern with the in-plane
field cannot be attributed to stray fields or misalignment with the magnet
axes since the SQI patterns are independent of magnetic field sweep direction.
More importantly, the behavior is completely absent in the control samples
with no LRT (Supporting Information Sec. S3), which yield Fraunhofer patterns,
regardless of the amplitude or direction of the in-plane field.

###  2.4 Mapping spin texture to spin-orbit coupling

We now continue by describing the mechanism behind the formation of the rim
currents using a model that links the vortex spin texture to the SOC. Within
the cobalt weak link, the gradient of the local spin texture of the disk
junctions is, at the rims of the device, not substantially larger than that of
a bar-shaped device. This implies that even though the LRT currents emerge at
the rims, they are formed by a process that is sensitive to the global spin
texture of the disk.

It was demonstrated that the combination of SOC and exchange field 45, 35, 47
or Zeeman field 44, 56 can result in an equilibrium spin current (ESC) which
accumulates at the superconducting/vacuum boundaries. We show that a similar
process occurs in the presence of spin texture $\bm{m}(r)$, which produces the
pure gauge SU(2) field that acts as effective spin-orbit coupling denoted by
$i\hat{U}^{\dagger}\nabla\hat{U}$.57, 33, 34 Here, $\hat{U}(\bm{r})$ is the
spin-rotation matrix, determined by the transformation to the local spin
quantization axis. In the Supporting Information Section S5, where we present
the full technical details of the Usadel calculations, we show that the vortex
spin texture $\bm{m}=(-\sin\theta_{v},\cos\theta_{v},0)$ can be transformed
into a uniform one, with an effective SOC term
$i\hat{U}^{\dagger}\nabla\hat{U}=-\hat{\sigma}_{z}\nabla\theta_{v}/2$ (here
$\hat{\sigma}_{z}$ is the spin Pauli matrix and
$\theta_{v}=\arctan[(y-y_{v})/(x-x_{v})]$, where $x_{v},y_{v}$ are the
coordinates of the vortex center). Hence, our system is analogous to one with
a uniform magnetization and an intrinsic, spatially inhomogeneous, SOC with
the amplitude $|\nabla\theta_{v}|=1/r_{v}$ (with
$r_{v}=\sqrt{(x-x_{v})^{2}+(y-y_{v})^{2}}$), and therefore hosts the
aforementioned ESC.

The ESC is carried by the SRT Cooper pairs, which spontaneously appear both at
the bottom of the S electrodes and the top of the F layer. The ESC can be
thought of as a spin-imbalance in this SRT condensate. It can be parametrized
in terms of the spin vector $\bm{f}$, which characterizes the spin component
of the triplet condensate. The SRT pairing corresponds to
$\bm{f}=\bm{m}f_{\text{SRT}}$, while the LRT one is described by
$\bm{f}_{\text{LRT}}\perp\bm{m}$. The direction of the ESC is determined by
the in-plane gradients of the magnetic texture and flows parallel to the S$-$F
interface. In terms of $\bm{f}$ it then becomes
$J^{\gamma}_{j}\propto|f_{\text{SRT}}|^{2}(\bm{m}\times\nabla_{j}\bm{m})_{\gamma}$
(here $\gamma=x,y,z$ is the index in spin space and $j=x,y,z$ is the index in
coordinate space). A ferromagnetic vortex texture yields
$J^{z}_{j}\propto|f_{\text{SRT}}|^{2}\nabla_{i}\theta_{v}=|f_{\text{SRT}}|^{2}m_{j}/r_{v}$
which is in accordance with the general gauge-invariant expression for the
spin current.58

###  2.5 Mechanism for generating LRT rim currents

Having established the equivalency between spin texture and SOC, we now
provide a possible mechanism that relates the ESC to the emergence of LRT rim
currents (see Supporting Figure S7 for a schematic representation). In the
absence of spin texture, there is no ESC (Figure S7a). If the spin texture
gradient is nonzero, the ESC adiabatically follows the local spin gauge field
($\bm{J}^{z}\parallel\nabla\theta_{v}=(\bm{m}\times\nabla\bm{m})_{z}$ see
Figure S7b). When the spin current encounters the bilayer-vacuum boundary (for
instance, due to deviations from the ideal circular geometry) the adiabatic
approximation breaks down, resulting in an accumulation of spin at the rims of
the device. Naturally, the spin accumulation decays over the spin diffusion
length, which for cobalt is approximately 60 nm.59 Near the interface, the
adiabatic ESC can develop a nonzero normal component
$\bm{m}\times(n_{j}\nabla_{j})\bm{m}\neq 0$, where $\bm{n}$ is the interface
normal. Since the total spin current is zero across the boundary,
$n_{j}J_{j}^{z}=0$, the adiabatic approximation breaks down and the ESC is
compensated by a spin current carried by an LRT condensate, which emerges near
the vacuum boundary (Figure S7c). Indeed, considering the local spin basis
with $\bm{m}\parallel\bm{x}$ the vector product of
$\bm{f}_{\text{SRT}}\parallel\bm{m}$ and
$\nabla_{j}\bm{f}_{\text{LRT}}\perp\bm{m}$ provides the contribution to the
$z$-component of the spin current (mediated by the condensate
$J_{j}^{z}\sim(\bm{m}\times\nabla_{j}\bm{m})_{z}$) due to the first term of
Eq.14 in the Supporting Information sec. S5. This contribution compensates the
ESC near the rim. A similar SRT to LRT conversion process has been proposed to
occur at the sample boundaries of SNS junctions with intrinsic SOC and a spin
active interface46 or in one-dimensional systems with a geometric curvature.47

By solving the linearized Usadel equation for a 2D disk-shaped S$-$F bilayer
(without the trench), we simulate the distribution of the LRT amplitude
$\Psi_{\text{LRT}}$, where
$\bm{f}_{\text{LRT}}=\Psi_{\text{LRT}}(m_{y},-m_{x},0)$. The results for three
different spin textures are presented in Figure 5. For a uniform
magnetization, the LRT correlations are completely absent, regardless of
sample geometry (Figure 5a). For a perfectly symmetric vortex pattern, any
deviation from the ideal circular geometry at the sample-vacuum boundary
(i.e., rim roughness or disorder) results in the emergence of LRT
correlations. In our simulation (Figure 5a,b), we use notches on the sides of
the disk, also present in our devices, to demonstrate the effect of nonideal
boundaries. However, in practice, any deviation from the perfect circular
geometry or disorder at boundaries results in a similar outcome.
Interestingly, even in the ideal circular geometry with flawless boundaries
(i.e., atomically clean and smooth edges), the LRT currents would still appear
if

$\begin{array}[]{c}\includegraphics[width=433.62pt]{Figure_6_SD.png}\end{array}$

Figure 5:  Simulated distribution of $\Psi_{\text{LRT}}$ in a S/F bilayer for
three different spin configurations, where the amplitude is normalized to
$(\gamma\xi_{0})f_{\text{SRT}}$ ($\gamma$ is the S$-$F interface
transparency). The direction of the in-plane spin texture is shown by arrows.
The simulated geometry includes notches that arise from FIB processing of the
actual devices. a. A homogeneous magnetization, where LRT correlations are
completely absent. b. In the presence of a magnetic vortex, LRT Cooper pairs
are generated at the notches, forming two LRT current channels. c. If the
vortex core is shifted from the disk center, an asymmetry arises between the
signs of the LRT channels, resulting in $0-\pi$ SQUID-like configuration, also
in the absence of the notches.

the magnetic vortex is not perfectly centered (see Figure 5c). Note that for
perfect circular symmetry, with the vortex at the center, $J^{z}_{j}$ will
always remain parallel to $m_{j}$ and no LRT is generated.

Our simulations also provide insights into the phase of LRT correlations. When
the vortex core is aligned with the trench (at $x=0$), the LRT currents will
result in two $\pi$-channels, as indicated by the sign change of
$\Psi_{\text{LRT}}$. In Figure 5c the vortex is displaced from the center
(e.g., due to an in-plane field) the LRT channels develop opposite signs at
the trench. This asymmetry is consistent with the observed $0-\pi$ SQUID
interference pattern, measured under a constant in-plane field (Figure 4).

###  2.6 Discussion

While the model presented here can describe the emergence of LRT correlations
at the bilayer-vacuum boundaries and the appearance of spontaneous
supercurrents in our junctions (i.e., $0-\pi$ segments), we should point out
that this formalism is restricted to 2D slices of the bilayer. Accounting for
the superconductor-vacuum interface formed by the trench is more challenging,
as it requires a full three-dimensional model and the knowledge of the exact
trench dimensions (e.g., its extent in the Co layer). We discuss this further
in Supporting Information sec. S5.

It should be noted that there is a fundamental difference between the devices
presented here and those reported in a previous work, where the disk-shaped
junctions consisted of a magnetic multilayer
(S$-$F${}^{\prime}-$F$-$F${}^{\prime\prime}-$S).15 In contrast to the
Nb/Ni/Cu/Co/Cu/Ni/Nb junctions, where long-range proximity was the result of
the magnetic noncollinearity between the Co and Ni layers, here the LRT
correlations are generated directly by the spin texture of a single
ferromagnet. This is evident by the fact that the multilayer devices were
highly sensitive to magnetic conditioning of the Ni (1.5 nm) layer (e.g., the
$I_{\text{c}}$ was irreversibly enhanced when the sample was conditioned)
whereas in the case of disks with a single ferromagnet, the transport
characteristics are unaltered by magnetic conditioning, since the vortex
magnetization is the global ground state of the Co-disk; regardless of the
magnetic history, the disk will revert to the vortex magnetization at zero
field. This is also confirmed by our micromagnetic simulations. Furthermore,
devices with and without the nickel layer exhibit radically different behavior
as a function of in-plane fields. However, there are similarities: both
devices show a double slit interference pattern, although the current channels
are considerably more confined in the case of a single ferromagnetic weak
link.

##  3 Conclusions

In summary, we have revealed an unexpected interplay between triplet
superconductivity and magnetic texture, which manifests itself as LRT
supercurrents localized at the rim of the ferromagnet. We elucidate the origin
of the rim currents by mapping the magnetic texture to an effective SOC, which
leads to the emergence of equilibrium spin currents – carried by the triplet
Cooper pairs present at the S$-$F interface. We also propose a mechanism for
converting the spin currents into equal-spin LRT correlations based on the
breakdown of the adiabatic approximation at the sample-vacuum boundaries.
Lastly, we show that the nature of LRT transport undergoes drastic changes
when the spin texture is modified. As illustrated here, by application of
relatively small magnetic fields, the same Josephson junction can be tuned to
function as both standard ($0-0$) and $0-\pi$ SQUIDs. The capacity to control
supercurrents with the spin texture of a single ferromagnetic layer opens
exciting prospects for regulating transport in superconducting devices.
Ferromagnetic vortices, in particular, can be manipulated by microwave
frequencies in a controllable manner, making them a promising candidate for
ultrafast dissipationless electronics.

This work was supported by the project ”Spin texture Josephson junctions”
(Project Number 680-91-128) and by the Frontiers of Nanoscience (NanoFront)
program, which are both (partly) financed by the Dutch Research Council (NWO).
It was also supported by EU Cost Action CA16218 (NANOCOHYBRI) and benefited
from access to The Netherlands Centre for Electron Nanoscopy (NeCEN) at Leiden
University.

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