# Curvature-induced long ranged supercurrents in diffusive SFS Josephson
Junctions, with dynamic $0-\pi$ transition

Tancredi Salamone<EMAIL_ADDRESS>Center for Quantum Spintronics,
Department of Physics, NTNU Norwegian University of Science and Technology,
NO-7491 Trondheim, Norway  Mathias B.M. Svendsen  Center for Quantum
Spintronics, Department of Physics, NTNU Norwegian University of Science and
Technology, NO-7491 Trondheim, Norway  Morten Amundsen  Nordita, KTH Royal
Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
SE-106 91 Stockholm, Sweden  Sol Jacobsen  Center for Quantum Spintronics,
Department of Physics, NTNU Norwegian University of Science and Technology,
NO-7491 Trondheim, Norway

###### Abstract

We report that spin supercurrents can be induced in diffusive SFS Josephson
junctions without any magnetic misalignment or intrinsic spin orbit coupling.
Instead, the pathway to spin triplet generation is provided via geometric
curvature, and results in a long ranged Josephson effect. In addition, the
curvature is shown to induce a dynamically tunable $0-\pi$ transition in the
junction. We provide the analytic framework and discuss potential experimental
and innovation implications.

Introduction.— In the last two decades there have been substantial advances in
the experimental realization of curved nanostructures. Since the realization
of nanotubes by rolling up thin solid films [1], many new techniques of
bending, wrinkling and buckling nanostructures in up to three dimensions have
been developed [2, 3], as well as direct growth on curved templates [4],
electronbeam lithography [5, 6, 7] and many more (see e.g. [8] and references
therein). These techniques open a broad new range of spintronic device design,
and have already been shown to enable independent control of spin and charge
resistances [4].

Geometric curvature introduces two main effects: a quantum geometric
potential, producing many interesting phenomena at the nanoscale [9, 10, 11,
12], and a strain field leading to a curvature-induced Rashba spin-orbit
coupling (SOC), with strength proportional to the curvature [13]. Several
studies have investigated new properties triggered by curvature, e.g. in
semiconductors [14, 15, 16, 17, 18], magnets [8, 7, 4] and superconductors
[19, 20, 21]. Curved nanostructures with induced superconductivity can display
geometric control of spin-triplet correlations in the clean limit [22], and
proximizing a superconductor with a curved semiconductor can result in
topological edge states [15]. The curved topological superconductor/straight
semiconductor Josephson junction counterpart has been predicted to display a
$0-\pi$ transition and $\phi$-junction behaviour [20].

Hybrid structures of superconductors and ferromagnets are of great interest
for the field of superconducting spintronics [23, 24] since at the
superconductor/ferromagnet (SF) interface the proximity effect allows the
property of one material to “leak” into the other [25, 26, 27]. A coexistence
of superconductivity and magnetism may therefore enable data processing,
encoded in spin and charge degrees of freedom, to be performed without the
heat loss associated with traditional electronics. In diffusive
heterostructures, which cover a range of commonly available materials that may
have impurities or sub-optimal interface transparencies, conventional $s$-wave
superconducting correlations typically penetrate a ferromagnet for extremely
short distances, proportional to $\sqrt{D_{F}/h}$, with $D_{F}$ the diffusion
constant and $h$ the exchange field strength. Significant theoretical and
experimental effort has focused on the conversion of singlet correlations into
so-called long range triplet correlations (LRTC), which penetrate for longer
distances, on the order of $\sqrt{D_{F}/T}$, where $T$ is the temperature.
This conversion can take place in the presence of magnetic inhomogeneities
[28, 29, 30] or due to intrinsic spin-orbit coupling either in the
superconductor or in the ferromagnet [31, 32]. The role of geometric curvature
as a source of designable and dynamically alterable SOC in diffusive
structures has not been investigated in this context, and we address this
here. By considering a model SFS junction with constant curvature shown in
Fig. 1, we show that the curvature alone can induce long ranged supercurrents
due to the generation of triplet correlations. Moreover, we show that these
systems display a tunable $0-\pi$ transition.

Figure 1: Model system. SFS junction having a ferromagnet with constant
curvature as weak link. The three orthonormal unit vectors $\hat{T}(\theta)$,
$\hat{N}(\theta)$ and $\hat{B}(\theta)$ identifying the curvilinear
coordinates are also shown.

The possibility of $0-\pi$ state switching has been of much interest, in part
due to its potential role in solid state quantum computing [33, 34, 25, 35,
36, 37]. Investigations have confirmed the transition can be governed by
altering the length of the ferromagnetic weak link. However, this is not
practicable to do in-situ and must be done by preparing multiple samples of
different lengths. It has recently been predicted that the $0-\pi$ transition
can also be accessed out of equilibrium, by altering the strength of the SOC
via voltage gating [38]. In this Letter, we show that dynamically changing the
curvature of the magnet via in-situ strain manipulation, for example via
photostriction, piezoelectrics or thermoelectric effects [39, 40], allows for
a single-sample $0-\pi$ transition in the diffusive regime, without the need
to apply a varying voltage. Moreover, we show that curvature can yield long-
ranged Josephson currents without any magnetic inhomogeneities or intrinsic
SOC.

Theoretical framework.— A fundamental tool for the study of curved structures
is the thin-wall quantization procedure, where the quantum motion of a
particle in a 2D curved surface is treated as equivalent to the motion in a 3D
space with the addition of lateral quantum confinement [41, 42]. This
procedure allows one to derive the Hamiltonian for the motion of electrons
constrained to a curved planar one-dimensional structure [43, 44].

When dealing with a ferromagnet, further effects of the curvature must be
taken into account, namely in how it affects its exchange field. Recent
studies have developed a fully 3D approach for thin magnetic shells of
arbitrary shape and extended it to 2D shells and 1D wires [45, 46]. This
showed that curvature induces two effective magnetic interactions: an
effective magnetic anisotropy and an effective Dzyaloshinskii-Moriya
interaction (DMI). When dealing with a 1D curved wire below a certain critical
curvature, the magnetic anisotropy and the DMI, which results in an effective
Rashba SOC, both combine to give an effective field tangential to the wire.
The geometrically defined SOC is therefore both designable and tunable, and
gives greater freedom in the manipulation of superconducting proximity
effects.

We parametrize the curve by its arc length $s$, and define a set of three
orthonormal unit vectors $\hat{T}(s)$, $\hat{N}(s)$, $\hat{B}(s)$ representing
the tangential, normal and binormal curvilinear coordinates respectively, as
indicated in Fig. 1. These obey the following Frenet-Serret-type equation of
motion:

$\begin{pmatrix}\partial_{s}\hat{T}(s)\\\ \partial_{s}\hat{N}(s)\\\
\partial_{s}\hat{B}(s)\end{pmatrix}=\begin{pmatrix}0&\kappa(s)&0\\\
-\kappa(s)&0&0\\\ 0&0&0\end{pmatrix}\begin{pmatrix}\hat{T}(s)\\\ \hat{N}(s)\\\
\hat{B}(s)\end{pmatrix},$ (1)

where $\kappa(s)$ is the curvature of the wire, whose role and effect will be
discussed in detail below. Deriving the Hamiltonian for a wire, which may
include intrinsic SOC in general, we find [44]:

$\begin{split}H=&-\frac{\hbar^{2}}{2m}\partial_{s}^{2}-\frac{\hbar^{2}}{8m}\kappa(s)^{2}-i\hbar\alpha_{N}\sigma_{B}\partial_{s}\\\
&+i\hbar\alpha_{B}\left(\sigma_{N}\partial_{s}-\frac{\kappa(s)}{2}\sigma_{T}\right).\end{split}$
(2)

The SOC constants $\alpha_{N,B}$ represent the spin-orbit field with axis
along the normal and binormal direction respectively, and
$\sigma_{T,N,B}(s)=\bm{\sigma}\cdot\\{\hat{T},\hat{N},\hat{B}\\}(s)$ are the
set of three Pauli matrices in curvilinear coordinates. By using Eqs. (1) we
can incorporate the last three terms in Eq. (2) in a SU(2) spin-orbit field
term:

$\bm{A}=(\alpha_{N}\sigma_{B}-\alpha_{B}\sigma_{N},0,0),$ (3)

which has a vector structure in the geometric space and a $2\times 2$ matrix
structure in spin space. It is worth distinguishing between the two terms
entering the SU(2) field, namely $\alpha_{B}$ and $\alpha_{N}$. The former
represents the intrinsic, not induced by the curvature, SOC term which may or
may not exist according to the material taken into consideration. The latter
is curvature-induced, and is proportional to the curvature strength. In
natural units we have $\alpha_{N}=g\lambda\kappa(s)/(4m)$, where $g$ is the
g-factor and the parameter $\lambda>0$ is a characteristic energy scale for
the material. Inspection of the relevant diffusion equations for the system
shows that $\alpha_{N}$ and $\kappa(s)$ appear together in such a way that the
former always acts as a strengthening factor for the latter. Therefore,
considering a material with no intrinsic term we can ignore spin-orbit
coupling as a whole, and consider the $\kappa(s)$ term only.

Having set up the Hamiltonian, we employ Green functions in the diffusive
limit at equilibrium. Here the dynamics are describable by the second-order
partial differential Usadel equation [47], which, with suitable boundary
conditions, describes the diffusion of superconducting correlations inside the
ferromagnet. Treating the case of diffusive equilibrium, it is sufficient to
consider just the retarded component $\hat{g}_{R}$ of the quasiclassical Green
function to describe the system [48]. Using Eq. (2) the Usadel equation in a
curved ferromagnet with constant curvature reads (from now on we set
$\hbar=1$):

$D_{F}\partial_{s}\left(\hat{g}_{R}\partial_{s}\hat{g}_{R}\right)+i\left[\varepsilon\hat{\tau}_{3}+\hat{M},\hat{g}_{R}\right]=0,$
(4)

with $\hat{\tau}_{3}=\operatorname{diag}(1,1,-1,-1)$, $\varepsilon$ the
quasiparticle energy and magnetization
$\hat{M}=\bm{h}\cdot\operatorname{diag}(\bm{\sigma},\bm{\sigma}^{*})$. The
components of both vectors $\bm{h}=(h_{T},h_{N},h_{B})$ and
$\bm{\sigma}=(\sigma_{T},\sigma_{N},\sigma_{B})$ are expressed in curvilinear
coordinates. To solve the Usadel equation we employ the Kuprianov-Lukichev
boundary conditions [49]:

$L_{j}\zeta_{j}\hat{g}_{Rj}\nabla_{I}\hat{g}_{Rj}=\left[\hat{g}_{R1},\hat{g}_{R2}\right].$
(5)

Here $\nabla_{I}$ is the derivative at the interface, $j$ refers to the
various components of the hybrid system, with $j=1,2$ denoting the materials
on the left and right side of the relevant interface, $L_{j}$ represents the
length of the material and $\zeta_{j}=R_{B}/R_{j}$ is the interface parameter
given by the ratio between the barrier resistance $R_{B}$ and its bulk
resistance $R_{j}$.

If desirable, the intrinsic SOC can be retained, in which case one also
introduces the gauge covariant derivative [32]:

$\partial_{s}(\,\cdot\,)\rightarrow\widetilde{\partial_{s}}(\,\cdot\,)\equiv\partial_{s}(\,\cdot\,)-i\left[\hat{A}_{T},\cdot\,\right],$
(6)

with $\hat{A}_{T}=\operatorname{diag}(A_{T},-A_{T}^{*})$ and $A_{T}$ is the
tangential component of the SO field of Eq. (3).

To treat the system we will use the Riccati parametrization [50, 51] for the
quasiclassical Green function:

$\hat{g}_{R}=\begin{pmatrix}N(1+\gamma\tilde{\gamma})&2N\gamma\\\
-2\tilde{N}\tilde{\gamma}&-\tilde{N}(1+\tilde{\gamma}\gamma)\end{pmatrix},$
(7)

where the normalization matrices are $N=(1-\gamma\tilde{\gamma})^{-1}$ and
$\tilde{N}=(1-\tilde{\gamma}\gamma)^{-1}$ and the tilde operation denotes
$\tilde{\gamma}(\varepsilon)=\gamma^{*}(-\varepsilon)$. The Usadel equation
(4) thus becomes:

$\begin{split}D_{F}\left\\{\partial^{2}_{s}\gamma+\right.&\left.2(\partial_{s}\gamma)\tilde{N}\tilde{\gamma}(\partial_{s}\gamma)\right\\}=\\\
&-2i\varepsilon\gamma-i\bm{h}\cdot(\bm{\sigma}(s)\gamma-\gamma\bm{\sigma}^{*}(s)).\end{split}$
(8)

Here the dependence on the curvature is implicitly contained in the Pauli
matrices $\sigma_{T,N,B}(s)$.

We will consider our one-dimensional curved wire to be lying in the $xy$ plane
as represented in Fig. 1, so that the set of three unit vectors is:

$\displaystyle\hat{T}(s)$
$\displaystyle=-\sin\theta(s)\hat{x}+\cos\theta(s)\hat{y},$ (9a)
$\displaystyle\hat{N}(s)$
$\displaystyle=-\cos\theta(s)\hat{x}-\sin\theta(s)\hat{y},$ (9b)
$\displaystyle\hat{B}(s)$ $\displaystyle\equiv\hat{z},$ (9c)

with $\theta(s)=\kappa s$. It is useful to note that, when considering Eq.
(8), the curved ferromagnet can be regarded as equivalent to a straight wire
with a rotating exchange field, i.e. a tangential exchange field in a curved
wire is equivalent to a position dependent exchange field in a straight wire,
varying as $\vec{h}(s)=h_{0}(\sin\theta(s),-\cos\theta(s),0)$, with
$\theta(s)=\pi s/L_{F}$ and $L_{F}$ being the length of the ferromagnet.

Results.— Solving the Usadel equation, and therefore finding the
quasiclassical Green function of the system, allows us to calculate many
interesting quantities. In this work we will focus mainly on the charge
current given by:

$\frac{I_{Q}}{I_{Q0}}=\int_{-\infty}^{+\infty}\\!\\!d\varepsilon\mathrm{Tr}\left\\{\hat{\tau}_{3}\left(\hat{g}_{R}\partial_{s}\hat{g}_{R}-\hat{g}_{A}\partial_{s}\hat{g}_{A}\right)\right\\}\tanh(\beta\varepsilon/2).$
(10)

Here $\hat{g}_{A}=-\hat{\tau}_{3}\hat{g}_{R}^{\dagger}\hat{\tau}_{3}$ is the
advanced quasiclassical Green function and $\beta=(k_{B}T)^{-1}$ is the
inverse temperature, with $k_{B}$ being the Boltzmann constant. Moreover,
$I_{Q0}=N_{0}eD_{F}A\Delta_{0}/4L_{F}$, where $N_{0}$ is the density of states
at the Fermi energy, $A$ the interfacial contact area and $\Delta_{0}$ the
bulk gap of the two superconductors. Lengths and energies have been normalized
to $L_{F}$ (which in turn is scaled with the superconducting coherence length
$\xi_{S}$) and superconducting bulk gap $\Delta_{0}$ respectively, so that the
integral on the right side of Eq. (10) is dimensionless.

We investigate the system portrayed in Fig. 1 by solving numerically Eq. (8)
for various lengths $L_{F}$ of the ferromagnet and multiple curvatures
$\kappa$ for each length. We set the interface parameter with both
superconductors to be $\zeta=3$ and the temperature to $T=0.005T_{c}$. We
consider the exchange field inside the curved ferromagnet to be tangential to
its curvature profile at each point, $\bm{h}(s)\parallel\hat{T}(s)$, which we
expect to be the case in 1D curved structures below a certain critical
curvature [46].

Figure 2: Magnitude of the critical current as a function of the curvature for
different lengths $L_{F}$ of the ferromagnet, with $T=0.005T_{c}$,
$\vec{h}=\Delta_{0}\hat{T}$, $\zeta=3$. A $0-\pi$ transition occurs when
changing the curvature of the wire.

Two interesting effects of the curvature appear immediately from our results.
First, we show in Fig. 2 that it is possible to induce a $0-\pi$ transition in
the junction by changing the curvature of the ferromagnetic wire while keeping
its length fixed. Secondly, we will show in Fig. 3 that even for a long
junction, where the singlet contribution to the supercurrent is negligible, a
Josephson effect still appears for a non-zero $\kappa$ due to the presence of
long ranged triplets.

In Fig. 2 we plot the absolute value of the critical current as given by Eq.
(10) as a function of the curvature $\kappa$ of the ferromagnet across the
junction for different lengths $L_{F}$. From the figure we see that starting
in the $0$ state with a straight wire, increasing the curvature results in a
decreasing magnitude of the critical current, until it completely disappears
for a certain $\kappa$, indicating a $0-\pi$ transition. A further increase in
the curvature produces a revival of the critical current, which now flows in
the opposite direction with respect to the straight case. We also note that
increasing the length of the ferromagnet reduces not only the overall
magnitude of the critical current but also the curvature at which the $0-\pi$
transition takes place.

In order to better understand how this $0-\pi$ transition appears, and to show
that the role of the triplets is crucial in tuning it, we split the charge
current into singlet and triplet contributions, $I_{0}$ and $I_{t}$
respectively. It can be shown that the charge current given by Eq. (10) only
depends on the anomalous Green function $f$ which is the off-diagonal block
matrix in the retarded Green function. We define
$f=(f_{0}+\bm{d}\cdot\bm{\sigma})i\sigma_{y}$, with $f_{0}$ representing the
singlet contribution and $\bm{d}=(d_{T},d_{N},d_{z})$ the d-vector
representing the triplet contribution, and obtain that the charge current can
be written as $I_{Q}/I_{Q0}=I_{0}+I_{t}$, where
$I_{t}=I_{T}+I_{N}+I_{z}+I_{\kappa}$ and:

$\displaystyle I_{0}$
$\displaystyle=-8\\!\\!\int_{0}^{\infty}\\!\\!d\varepsilon{\mathrm{Re}}\left\\{\tilde{f}_{0}\partial_{s}f_{0}-f_{0}\partial_{s}\tilde{f}_{0}\right\\}\tanh(\beta\varepsilon/2),$
(11a) $\displaystyle I_{j}$
$\displaystyle=8\\!\\!\int_{0}^{\infty}\\!\\!d\varepsilon{\mathrm{Re}}\left\\{\tilde{d}_{j}\partial_{s}d_{j}-d_{j}\partial_{s}\tilde{d}_{j}\right\\}\tanh(\beta\varepsilon/2),$
(11b) $\displaystyle I_{\kappa}$
$\displaystyle=16\kappa\\!\\!\int_{0}^{\infty}\\!\\!d\varepsilon{\mathrm{Re}}\left\\{\tilde{d}_{N}d_{T}-\tilde{d}_{T}d_{N}\right\\}\tanh(\beta\varepsilon/2),$
(11c)

with $j=(T,N,z)$. The terms $I_{0}$ and $I_{j}$ represent the contribution
coming from the singlet and triplets with spin aligned in the $j$ direction
respectively. The last term $I_{\kappa}$ instead defines an inverse Edelstein
term due to the curvature. This kind of contribution appears whenever the
d-vector undergoes a rotation and is therefore non zero only in the presence
of finite curvature and/or spin-orbit coupling [52].

Figure 3: Charge current as a function of the phase difference $\phi$, showing
separately the singlet (solid lines) and triplet (dashed lines) contributions
with $T=0.005T_{c}$, $\vec{h}=\Delta_{0}\hat{T}$, $\zeta=3$, for a straight
($\kappa=0$) and semi-circular ($\kappa L_{F}=\pi$) ferromagnetic wire. (a)
$L_{F}=2\xi_{S}$ Increasing the curvature causes the triplet contribution to
change sign. (b) $L_{F}=6\xi_{S}$ Increasing the curvature causes the singlet
contribution to be neglegible with respect to the triplet one, signaling that
the charge current is transported almost exclusively by the triplet
correlations.

In Fig. 3(a) we plot these different contributions to the charge current for
two different values of $\kappa$ and $L_{F}=2\xi_{S}$. It can be seen that for
$\kappa=0$ triplets and singlet charge currents have opposite sign, with the
triplets contribution, which comes only from the short-range component
$I_{T}$, being generally bigger than the singlet one. Interestingly however,
when increasing the curvature the triplet current changes sign, i.e. starts
flowing in the opposite direction, because of the appearance of the long-range
component $I_{N}$, while the singlet contribution does not. Hence, the $0-\pi$
transition is tuned by the curvature through its effect on the triplets.
Furthermore, we note from Fig. 3(a) that in the $\pi$-phase for
$\kappa=\pi/L_{F}$ the singlet and triplet currents have the same sign and
thus flow in the same direction. Consequently, the two contributions add up,
resulting in a larger critical current in the $\pi$-phase at
$\kappa=\pi/L_{F}$ compared to the $0$-phase at $\kappa=0$.

We point out that curvature also introduces a spin current to the system,
which is absent in a straight nanowire. This _exchange_ spin current, as it is
known in the literature, is caused by the misalignment of the magnetization in
the system and is non-zero even at phase differences of $\phi=0$ and
$\phi=\pi$, where there are no charge currents [53, 54]. The magnitude of the
spin current is affected by the curvature, thereby providing means by which it
can be externally manipulated.

To highlight that the triplets generate a long range Josephson effect, we
consider a long junction, $L_{F}=6\xi_{S}$, and plot in Fig. 3(b) separately
singlet and triplet contributions for a straight ($\kappa=0$) and semi-
circular ($\kappa L_{F}=\pi$) wire. We see that, while for $\kappa=0$ the
triplet term is essentially zero and the singlet term is finite, for $\kappa
L_{F}=\pi$ the singlet contribution is negligible compared to the triplet one,
which additionally presents a long-range component $I_{N}$ dominating over the
short-range one $I_{T}$. Going from a straight to a semi-circular ferromagnet
produces a significant singlet to triplet conversion, of which component in
the normal direction is long-ranged, i.e. $d_{N}=|\bm{d}\times\bm{\hat{h}}|$,
since in the case considered the $d_{z}$ component is zero. In the simple
example of a long wire with constant curvature chosen here, the magnitude of
this LRT component is quite small, but we explain how this can be increased
and manipulated below.

To better understand the role played by the curvature, it is useful to
consider the weak proximity effect, meaning that $\left|\gamma_{ij}\right|\ll
1$ and $N\simeq 1$. The $\gamma$ matrix can be then expressed in terms of $f$:
$\gamma=f/2$. We then obtain the following linearized version of the Usadel
equation:

$\displaystyle\frac{D_{F}}{2}\partial_{s}^{2}f_{0}$
$\displaystyle=-i\varepsilon f_{0}-i\bm{h}\cdot\bm{d},$ (12a)
$\displaystyle\frac{D_{F}}{2}\\!\left(\partial_{s}^{2}d_{T}\\!-\\!2\kappa\partial_{s}d_{N}\\!\right)\\!\\!$
$\displaystyle=\\!\left(\\!-i\varepsilon\\!+\\!\frac{D_{F}\kappa^{2}}{2}\\!\right)\\!d_{T}\\!-\\!if_{0}h_{T},$
(12b)
$\displaystyle\frac{D_{F}}{2}\\!\left(\partial_{s}^{2}d_{N}\\!+\\!2\kappa\partial_{s}d_{T}\\!\right)\\!\\!$
$\displaystyle=\\!\left(\\!-i\varepsilon\\!+\\!\frac{D_{F}\kappa^{2}}{2}\\!\right)\\!d_{N}\\!-\\!if_{0}h_{N},$
(12c) $\displaystyle\frac{D_{F}}{2}\partial_{s}^{2}d_{z}$
$\displaystyle=-i\varepsilon d_{z}-if_{0}h_{z}.$ (12d)

By inspecting the linearized Usadel equation for the triplet components, given
in Eqs. 12a, 12b and 12c, we see that the curvature produces a Dyakonov-Perel
term, describing the spin-relaxation due to precession around the exchange
field. A curvature of $\kappa L_{F}=\pi$ gives a strong spin-relaxation term
which causes a fast decay even for the LRT component. From a qualitative
perspective we can see that, since the exchange field varies with the
position, a LRT component flowing through the wire will acquire an increasing
component parallel to $\bm{h}$, i.e. a quickly decaying short-range component.
The SRT component likewise acquires a LRT, but the conversion region is
restricted to the typical decay of the SRT $\sim 1/\sqrt{h}$. In order to
maximise the LRT generation from the SRT, one should therefore have a region
of high curvature over the spatial decay of the SRT near the superconducting
interface, and then minimal or zero curvature beyond. Alternatively, one may
start with an intrinsically triplet superconductor, or have a compensating
spin-orbit field in the ferromagnet that can negate the effect of the
curvature.

Concluding remarks.— We have shown that curvature is a designable and tunable
parameter that can generate and control long-ranged supercurrents in diffusive
SFS Josephson junctions without any magnetic inhomogeneities or intrinsic SOC.
The system displays a curvature-controlled $0-\pi$ transition, which can be
manipulated dynamically in-situ with a single sample. This can facilitate
experimental investigation of the transition, and improve our understanding of
the coexistence of superconductivity and magnetism in different phases. In the
longer term this opens a diverse new toolkit for design and control of
diffusive superconducting spintronic systems, and may be a useful
implementation in solid state quantum computing. Since this field is still in
its infancy, with several exciting directions still to be explored, we
anticipate that curvature in such systems will be integral to the new
generation of spintronic designs.

###### Acknowledgements.

We thank P. Gentile for useful discussions and H. G. Hugdal for his helplful
insights. The computations have been performed on the SAGA supercomputer
provided by UNINETT Sigma2 - the National Infrastructure for High Performance
Computing and Data Storage in Norway. We acknowledge funding via the
“Outstanding Academic Fellows” programme at NTNU, the Research Council of
Norway Grant number 302315, as well as through its Centres of Excellence
funding scheme, project number 262633, “QuSpin”.

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