Non-constant geometric curvature for tailored spin–orbit coupling and chirality in superconductor-magnet heterostructures

We employ the quasiclassical formalism [64, 65] where observables are retrievable from a propagator $\check{g}$ in Keldysh ⊗ Nambu (particle-hole) ⊗ spin space,

$$\begin{align} \check{g} = \begin{pmatrix} \hat{g}^{R} & \hat{g}^{K} \\ 0 & \hat{g}^{A} \end{pmatrix}, \end{align} \tag{ 1 }$$

where the superscripts R, A and K label the retarded, advanced and Keldysh components, respectively. The elements are related by $\hat{g}^{A} = - \hat{\tau}_3 \hat{g}^{R\dagger} \hat{\tau}_3$ and $\hat{g}^{K} = \hat{g}^{R} \hat{h} - \hat{h} \hat{g}^{A}$, where we refer to $\hat{h}$ as the distribution matrix and $\hat{\tau}_3 = \text{diag}(1,1,-1,-1)$.

The propagator $\check{g}$ obeys the Usadel equation [65], which in curvilinear coordinates is given as [4]

$$\begin{align} D_{F} {G}^{\mu\nu} \tilde{D}_{\mu} \left(\check{g} \tilde{D}_{\nu} \check{g}\right) + i \left[ \epsilon \hat{\tau}_{3} \otimes I_{2} + \check{\Sigma}, \check{g} \right] = 0, \end{align} \tag{ 2 }$$

where D_F is the diffusion coefficient, ε is the energy, $\check{\Sigma}$ is the self-energy function, G^µν is the metric tensor, and I₂ is the $2\times 2$ identity matrix in Keldysh space. The coordinate-gauge covariant derivatives are given as

$$\begin{align} \tilde{D}_{\mu} v_{\nu} = \tilde{\partial}_{\mu} v_{\nu} - \Gamma^{\gamma}_{\mu\nu} v_{\gamma}, \end{align} \tag{ 3 }$$

with Christoffel symbols

$$\begin{align} \Gamma_{\mu\nu}^{\gamma} = \frac{1}{2} G^{\gamma\lambda} \left[\partial_{\nu} G_{\mu \lambda} + \partial_{\mu}G_{\lambda\nu} - \partial_{\lambda}G_{\mu\nu}\right], \end{align} \tag{ 4 }$$

and gauge-only covariant derivative $\tilde{\partial}_\mu v_\nu = \partial_\mu v_\nu-i[\hat{A}_\mu,v_\nu]$. The gauge field $\hat{A}_\mu = \textrm{diag}(A_\mu,-A_\mu^*)$ will depend on e.g. intrinsic SOC contributions.

The Christoffel symbols depend on the choice of coordinate system, and we consider a planar space curve in real, 3-dimensional space, $\boldsymbol{r}(s)$, at the center of a ferromagnetic nanowire. This space, and the nanowire, is parametrizable as $\boldsymbol{R}(s, n, b) = \boldsymbol{r}(s) + n \hat{{N}}(s) + b \hat{{B}}(s)$, where we refer to s, n and b as the arclength, normal and binormal coordinates respectively. The orthonormal basis vectors for the parametrization are $\hat{{T}}(s) = \partial_{s} \boldsymbol{r}(s)$, $\hat{{N}}(s) = \partial_{s} \hat{{T}}(s)/\kappa(s)$, and $\hat{{B}}(s) = \hat{{T}}(s) \times \hat{{N}}(s)$, which correspond to the tangential, normal and binormal direction as illustrated in figure 1. We refer to $\kappa(s)$ as the curvature function.

The basis vectors obey a Frenet–Serret like equation

$$\begin{align} \begin{pmatrix} \partial_{s} \hat{{T}}\left(s\right) \\ \partial_{s} \hat{{N}}\left(s\right) \\ \partial_{s} \hat{{B}}\left(s\right) \end{pmatrix} = \begin{pmatrix} 0 & \kappa\left(s\right) & 0 \\ -\kappa\left(s\right) & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \hat{{T}}\left(s\right) \\ \hat{{N}}\left(s\right) \\ \hat{{B}}\left(s\right) \end{pmatrix}, \end{align} \tag{ 5 }$$

and are accompanied by a non-trivial metric tensor [4, 66] $G_{\mu\nu} = \text{diag}(\eta^{2}, 1, 1)$, with $\eta(s, n) = 1 - \kappa(s) n$. When the system is in equilibrium, it is sufficient to consider only the retarded propagator, $\hat{g}^{R}$. Moreover, in the 1D transport limit, where $n, b \rightarrow 0$ with $\partial_{s} \kappa(s)$ finite, the non-zero Christoffel symbols do not contribute to the Usadel equation, and the basis vectors remain orthonormal, such that the Usadel equation for the retarded propagator takes the form [4]

$$\begin{align} D_{F}\tilde{\partial}_{s} \left( \hat{g}^{R} \tilde{\partial}_{s} \hat{g}^{R}\right) + i \left[ \epsilon \hat{\tau}_3 + \hat{\Sigma}, \hat{g}^{R} \right] = 0. \end{align} \tag{ 6 }$$

In this form, the effect of the curvature is encoded in the rotation of the basis vectors (and Pauli matrices) along the arclength.

Considering a superconductor/ferromagnet heterostructure, we take the self-energy $\hat{\Sigma} = \hat{\Delta}$ on the superconducting side, and $\hat{\Sigma} = h^{\mu}(s)\text{diag}\left[\sigma_{\mu}(s), \sigma_{\mu}^{*}(s)\right]$ on the ferromagnetic side. $h^{\mu}(s)$ is the exchange field and $\sigma_{\mu}(s)$ the Pauli matrices along the µ-direction, and $\hat{\Delta} = \text{antidiag}(\Delta, -\Delta, \Delta^{*}, -\Delta^{*})$, where Δ is the superconducting order parameter. In the following, we keep Δ fixed and equal to its value in an infinite superconductor, thereby ignoring the inverse proximity effect caused by the interface. All lengths are scaled relative to the diffusive superconducting coherence length, denoted by ξ, which is typically of the order of a few tens of nanometers.

For the boundary conditions we will employ the Kupryanov–Lukichev boundary conditions [67], which take the form

$$\begin{align} \check{g}_{j} \tilde{\partial}_{I} \check{g}_{j} = \frac{1}{2 L_{j} \zeta_{j}} \left[\check{g}_{L}, \check{g}_{R}\right], \end{align} \tag{ 7 }$$

where the index $j = \{L, R\}$ refers to the left and right side of the interface, L_j is the length of material, $\tilde{\partial}_{I}$ is the gauge covariant derivative at the interface, and $\zeta_{j} = R_B/R_j$ quantifies the interfacial resistance via the ratio of barrier resistance R_B to bulk resistance R_j .

We consider junctions consisting of straight and curved parts in series. Mathematically, this is done by considering curvature functions, $\kappa(s)$, consisting of a series of step-functions, as opposed to constants for constant curvature [4, 16]. We will demonstrate that approximate step-functions can be rigorously included in the formalism and use them to define the J-, C- and S-classes of curves.

We will consider planar curves that can be parameterized using curvature functions of the form

$$\begin{align} \kappa\left(s\right) = K \sum_{j} \alpha_{j} H\left(s - q_{j}\right), \end{align} \tag{ 8 }$$

where we refer to K as the curvature amplitude, and H is the Heaviside step function. Particular values from the sets $\{q_{j}\}$ and $\alpha_{j} = \pm 1$ determine where the curvature is present and/or reversed. The orientation of the unit vectors relative to their initial configuration, $\theta(s)$, can then be determined from the integral of the curvature function $\kappa(s)$:

$$\begin{align} \theta\left(s\right) & = \int_{\lambda = 0 }^{\lambda = s} \kappa\left(\lambda; K, q\right) d\lambda \nonumber \\ & = K \sum_{j} \alpha_{j} H\left(s - q_{j}\right) \cdot \left(s - q_{j}\right), \end{align} \tag{ 9 }$$

such that we may parameterize the basis vectors as

$$\begin{align} \hat{{T}}\left(s\right) & = - \text{sin} \theta \left(s\right) \hat{x} + \text{cos} \theta \left(s\right) \hat{y}, \end{align} \tag{ 10a }$$

$$\begin{align} \hat{{N}}\left(s\right) & = - \text{cos} \theta \left(s\right) \hat{x} - \text{sin} \theta \left(s\right) \hat{y} \end{align} \tag{ 10b }$$

$$\begin{align} \hat{{B}}\left(s\right) & = \hat{z}. \end{align} \tag{ 10c }$$

The conceptually simple case of constant curvature, with $\kappa(s) = K$, is shown in figure 2. When we have fixed the binormal basis vector, $\hat{{B}}(s)$, positive (negative) curvature functions $\kappa(s)$ correspond to counter-clockwise (clockwise) rotation.

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For this demonstration of non-constant curvature, we consider three conceptually simple cases: wires curved at one end (J-type), curved in the same direction at each end (C-type), or curved in opposite directions on each end (S-type), as shown in figure 1. The associated curvature functions are

$$\begin{align} \kappa^{J}\left(s; K, q\right) & = K H\left(s - q\right), \end{align} \tag{ 11a }$$

$$\begin{align} \kappa^{C}\left(s; K, q\right) & = \begin{cases} K \quad \hspace{2mm}\text{if } s < q \\ 0 \quad \hspace{3.5mm}\text{if } q < s < L - q \\ K \quad \hspace{2mm}\text{otherwise,}\\ \end{cases} \end{align} \tag{ 11b }$$

$$\begin{align} \kappa^{S}\left(s; K, q\right) & = \begin{cases} K \quad \hspace{2.7mm}\text{if } s < q \\ 0 \quad \hspace{4mm}\text{if } q < s < L - q \\ -K \quad \text{otherwise.}\\ \end{cases} \end{align} \tag{ 11c }$$

Using equation (9), this gives

$$\begin{align} \theta^{J} \left(s; K, q\right) & = \begin{cases} Ks \quad \hspace{15mm}\text{if } s < q \\ K q \quad \hspace{15mm}\text{otherwise,} \end{cases} \end{align} \tag{ 12a }$$

$$\begin{align} \theta^{C} \left(s; K, q\right) & = \begin{cases} Ks \quad \hspace{15mm}\text{if } s < q \\ K q \quad \hspace{15mm}\text{if } q < s < L - q\\ K \left(s \!+\! 2q \!-\! L\right) \quad \text{otherwise,} \end{cases} \hspace{-2mm} \end{align} \tag{ 12b }$$

$$\begin{align} \theta^{S} \left(s; K, q\right) & = \begin{cases} Ks \quad \hspace{15mm}\text{if } s < q\\ K q \quad \hspace{15mm}\text{if } q < s < L - q\\ K \left(L - s\right) \quad \hspace{5.5mm}\text{otherwise.} \end{cases} \end{align} \tag{ 12c }$$

Here L is the length of the parameterized section, and we refer to q as the shape parameter.

To ensure continuously varying observables, and to rigorously derive the Usadel equation in 1D, we require a continuous $\kappa(s)$ and finite $\partial_{s} \kappa(s)$. To achieve this we may approximate the Heaviside function by the hyperbolic tangent, and thus the curvature functions may, as an example, take the form

$$\begin{align} \kappa^{J}\left(s; K, a, q\right) = K\left(1 - \frac{e^{a\left(s-q\right)}}{1 + e^{a\left(s-q\right)}}\right). \end{align} \tag{ 13 }$$

Here we have defined a sharpness parameter a, indicating the abruptness of the change in curvature. In the limit $a \rightarrow \infty$ we retrieve the step-function, giving equation (11a ). We may further insert equation (13) into equation (9), to get

$$\begin{align} \theta^{J}\left(s; K, a, q\right) = K \left[ s + \frac{1}{a} \text{ln} \left(\frac{1 + e^{-aq}}{1 + e^{a\left(s-q\right)}}\right) \right].\end{align} \tag{ 14 }$$

There are closed form expressions for $\theta^{S,C}$, as well as for more generalized cases. There is, however, in general no closed form expression for the parameterized curve itself.

To numerically solve equation (6), it is helpful to employ a more tractable parameterization of the quasiclassical Green's function. We use the Riccati parametrization [15, 68]:

$$\begin{align} \hat{g}^{R} = \begin{pmatrix} N\left(1 + \gamma \tilde{\gamma}\right) & 2N\gamma \\ -2\tilde{N} \tilde{\gamma} & -\tilde{N}\left(1 + \tilde{\gamma} \gamma\right) \end{pmatrix}, \end{align} \tag{ 15 }$$

with $N = (1 - \gamma \tilde{\gamma})^{-1}$ and $\tilde{\gamma}(s, \varepsilon) = \gamma^{*}(s, -\varepsilon)$, i.e. tilde-conjugation implies inversion of the quasiparticle energy and element wise complex conjugation. The parametrization reduces the dimensionality of the problem from an equation of $(4\times4)$-matrices to coupled equations for $(2\times2)$-matrices, and bounds the norm of the matrix elements $\in[0,1]$.

In the ferromagnetic region, the Usadel equation is then parametrized as

$$\begin{align} & D_{F} \!\left\{\!\partial_{s} \gamma \!+\! 2\left(\partial_{s} \gamma\right) \tilde{N} \tilde{\gamma} \left(\partial_{s} \gamma\right)\!\right\} \nonumber\\ & \quad = -2i\varepsilon\gamma - i h^{\mu} \left[\sigma_{\mu}\left(s\right) \gamma - \gamma \sigma_{\mu}^{*}\left(s\right)\right], \end{align} \tag{ 16 }$$

with the index $\mu = \{\hat{T},\hat{N},\hat{B}\}$, and $\sigma_{\mu}(s)$ are the corresponding Pauli matrices. The unit vectors $\hat{{T}}(s), \hat{{N}}(s), \hat{{B}}(s)$ are determined from equation (10) with $\theta(s)$ being, e.g. equation (14) for a J-type curve or equation (12b ) for a C-type curve. Inserting the Riccati parametrization equation (15) into the boundary conditions equation (7) at the superconductor-ferromagnet interface, the boundary conditions for the matrix γ become [4, 15]:

$$\begin{align} \tilde{\partial}_{I}\gamma_S & = \frac{1}{L_S\zeta_S}\left(1\!-\!\gamma_S\tilde{\gamma}_F\right)N_F\left(\gamma_F\!-\!\gamma_S\right), \end{align} \tag{ 17a }$$

$$\begin{align} \tilde{\partial}_{I}\gamma_F & = \frac{1}{L_F\zeta_F}\left(1\!-\!\gamma_F\tilde{\gamma}_S\right)N_S\left(\gamma_F\!-\!\gamma_S\right).\end{align} \tag{ 17b }$$

The corresponding equations for $\tilde{\gamma}$ are obtained by tilde conjugation of equations (16) and (17).

We can then solve the complete, Riccati parameterized Usadel equation (16) in a 1D ferromagnetic region with either constant or non-constant curvature functions, coupled to bulk s-wave superconductors, employing the Kupriyanov–Lukichev boundary conditions (17) at the interfaces. The specific implementation is based on [69], modified to accommodate the non-constant curvature as specified here, and run using the bvp6c package for matlab.

In the weak proximity limit, we assume the anomalous Green's function to be small, such that

$$\begin{align} \hat{g}^{R} = \begin{pmatrix} 1 & f \\ -\tilde{f} & 1 \end{pmatrix}, \end{align} \tag{ 18 }$$

and parametrize the anomalous Green's function using the d-vector formalism:

$$\begin{align} f = \left(f_{0} + d_{\mu}\sigma_{\mu}\right) i \sigma_{y}. \end{align} \tag{ 19 }$$

Here f₀ and d_µ are the singlet and triplet amplitudes respectively. In this limit, we need only to consider the terms of the Riccati parametrized equation that are linear in γ, and the equation takes the following form in the ferromagnetic region:

$$\begin{align} \frac{iD_{F}}{2}\partial_{s}^{2}f_{0} \!& = \! \varepsilon f_{0} + h_{\mu}d_{\mu}, \end{align} \tag{ 20a }$$

$$\begin{align} \frac{iD_{F}}{2}\partial_{s}^{2}d_{T} \!&-\! iD_{F}\left(\partial_{s}\kappa \!+\! \kappa \partial_{s}\right)d_{N} = \left[ \varepsilon \!+\! \frac{iD_{F}\kappa^{2}}{2} \right]\! d_{T} \!+\! h_{T}f_{0}, \nonumber\\ \end{align} \tag{ 20b }$$

$$\begin{align} \frac{iD_{F}}{2}\partial_{s}^{2}d_{N} \!&+\! iD_{F}\left(\partial_{s}\kappa \!+\! \kappa \partial_{s}\right)d_{T} = \left[ \varepsilon \!+\! \frac{iD_{F}\kappa^{2}}{2} \right]\! d_{N} \!+\! h_{N}f_{0}, \nonumber\\ \end{align} \tag{ 20c }$$

$$\begin{align} \frac{iD_{F}}{2}\partial_{s}^{2}d_{B} \!& = \! \varepsilon h_{B} + h_{B}f_{0}.\end{align} \tag{ 20d }$$

With a non-zero κ, the triplet components undergo spin precession and spin relaxation. We identify the precession with the terms having a first order derivative, while the imaginary contributions to the energy represent the spin relaxation and loss of spin information from impurity scattering. While the curvature $\kappa(s)$ provides a mechanism for rotating between different triplet components, generating the spin-polarized triplets that are robust in magnetic fields (the so-called long-ranged component), it also induces spin relaxation. Having a non-constant curvature function, $\kappa(s)$, enables optimization of the long-range triplet generation and retention.

The triplet component with zero spin projection is short-ranged in a magnet, with d-vector parallel to the exchange field; the triplet component with spin polarization along the exchange field is long-ranged, with a d-vector perpendicular to the exchange field. For example, if the exchange field is directed along $\hat{T}$, the short-ranged triplets can be identified with d_T , while d_N and d_B are long ranged.

In the weak proximity equations, we can see a curvature dependent mixing between d_N and d_T . The mixing depends on $\partial_{s} \kappa$, which couples to the amplitudes themselves, d_µ , rather than their derivatives. That is, the mixing is enhanced in regions where the curvature changes abruptly. Moreover, it is possible to change the sign of $\partial_{s} \kappa$ without changing that of κ.

The simplest class of wires with non-constant curvature is the J-type ferromagnet (see figures 1(a) and 3). For any superconductor-ferromagnet (SF) bilayer, the ferromagnet's exchange field will convert a proportion of the superconducting singlet correlations into triplets with non-zero spin projection (short-ranged triplets), which is clear from equation (20a ). The curvature is then responsible for the rotation of the triplet vector, converting between tripets with and without spin projection. Since the short ranged triplet correlations decay rapidly in a ferromagnet, the best way of preserving long-ranged, spin-polarized triplets in the system is to have a region of sharp curvature near the superconducting interface for rapid conversion from short-ranged to long-ranged, and then a region of zero curvature, to avoid rotating the long-ranged components back into short-ranged triplets [16]. Although the conversion mechanism is in this case provided by the curvature, the process governing interconversion between components is the same as for straight multilayers with magnetic misalignment [70]. However, real-space curvature provides new mechanisms for design of variable exchange-field misalignment, and dynamic control of their relationship.

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The utility of curvature for controlling the generation of spin-polarized superconducting correlations becomes particularly clear when considering the equivalent field of two J-curves with opposite chirality, as in figure 3.

We see that the sign of the curvature at the superconducting interface governs the direction of the exchange field at the edge of the corresponding straight wire, which will be opposite for a curvature amplitude of $KL = \pi/2$. Transport through such wires will therefore experience opposite chiralities, and we can detect signatures of this chirality in the observables for superconducting correlations.

When we have a superconductor-ferromagnet-superconductor (SFS) junction instead of a bilayer, we must take into account the triplet conversion from both superconductors. Since the proximity effect at each interface will experience a chiral-dependent triplet conversion, combinations of different chiralities can therefore be expected to interfere constructively or destructively along the wire, which we explore below.

With our knowledge of the chirality-dependent polarization of a J-type bilayer, we can now examine and contrast the effects of combining different chiralities in SFS Josephson junctions, as illustrated by the C- and S-type wires in figure 1. In terms of the proximity effect, a C-type junction combines opposite chiralities with respect to the interface of each superconductor, and the S-type combines equal chiralities. That is, curving the regions close to the interface rotates the exchange field to give a component perpendicular to the direction of transport in the equivalent straight case (see figure 3). For the C-type junction, this component points in the opposite direction near each interface, and points in the same direction for S-type. Limiting cases now give the corresponding exchange field profiles of a junction with three misaligned ferromagnets [70–73], or a ferromagnet with domain walls at the interfaces [10]. As before, sharp curvatures at a superconducting interface promotes long-ranged triplet generation, and any further rotation along the wire will induce spin relaxation. Parametrization with the shape parameter, q, lets us vary the relative length of the straight and curved segments, and allows for separation of curvature-induced and length-induced effects.

We adopt the convention that with K > 0, we have $\kappa(s = 0) \gt 0$, and the wire curves counter-clockwise at the origin (as in figure 2). How the proximity-induced triplets will combine in a junction will therefore depend on the radius of curvature, and any change in sign of K along the arclength. Below, we will show that the full Usadel equation predicts a chirality-dependent magnetization and spin current density in C-type junctions with constant curvature in section 3.3.1, and go on to show the effect of constructive and destructive chirality combinations in the critical current of C- and S-type junctions in section 3.3.2.

3.3.1. Chiral signatures: magnetization and spin current density.

Once we have determined the Green's function, we may determine the equilibrium spin accumulation, or proximity-induced magnetization, M_µ in the wire in the µ direction from [74]

$$\begin{align} M_{\mu} = M_{0} \int_{-\infty}^{\infty} d\varepsilon \textrm{Tr} \left\{\text{diag}\left(\sigma_\mu, \sigma^{*}_\mu\right) \hat{g}^{K} \right\}, \end{align} \tag{ 21 }$$

where we use the ansatz $\hat{g}^{K} = (\hat{g}^{R} - \hat{g}^{A}) \text{tanh}{\frac{\beta\varepsilon}{2}}$ in equilibrium. The coefficient $M_{0} = g \mu_{B}N_{0}\Delta / 16$, with the Landé g-factor g ≈ 2 for electrons, µ_B is the Bohr magneton, $\beta = \frac{1}{k_{B}T}$, and N₀ is the normal-state density of states at the Fermi-level.

To demonstrate the chiral signatures, we begin by considering the case of constant curvature in C-type junctions, with different chiralities/sign of K, as depicted in figure 2. We show the induced magnetization along a wire for $K \in (-3\pi/2, 3\pi/2)$ in figure 4, and see that reversing the direction of the curvature also reverses the sign of the magnetization in the normal direction in figure 4(b), while the tangential magnetization (figure 4(a)) is unchanged. This leaves a chirality-dependent, observable signature of the spin polarization. We may understand this in terms of a straight SFS-junction with a rotating exchange field, which is equivalent to a curved wire with a tangential exchange field. The curved wire and its equivalent straight exchange field is illustrated in figure 3. In the straight case, reversing the chirality of the exchange field rotation would reverse the spin accumulation orthogonal to the transport direction. Note that the parameter determining the chirality, K, can vary continuously, and that the response, $M_{T,N}(s)$ varies continuously as a function of K, such that this still holds in the limit $K \rightarrow 0$.

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The effect of the exchange field chirality also gives a signature in the spin current density $\boldsymbol{J}_{s}^{\mu}$, which is given in the µ-direction by the Keldysh component

$$\begin{align} \boldsymbol{J}_{s}^{\mu} = \frac{\hbar N_{0} D}{4} \int_{-\infty}^{\infty} d\varepsilon\textrm{Tr}\left\{ \text{diag}\left(\sigma_\mu, \sigma^{*}_\mu\right) \hat{\tau}_{3} \left(\check{g}\partial_{s}\check{g}\right)^{K} \right\}. \end{align} \tag{ 22 }$$

In equilibrium, we can see the contributions of the triplet components in weak proximity in the binormal component:

$$\begin{align} \boldsymbol{J}_{s}^{B} & = -\frac{\hbar N_{0} D}{4} \!\!\int_{-\infty}^{\infty}\!\!\! d\varepsilon \text{Im} \left[ d_{T} \partial_{s} \tilde{d}_{N} \!-\! \tilde{d}_{N} \partial_{s} {d}_{T} \!+\! \text{t.c.}\! \right]\!\text{tanh}{\frac{\beta\varepsilon}{2}}\nonumber\\ & \quad -\kappa\left(s\right)\frac{\hbar N_{0} D}{2} \!\!\int_{-\infty}^{\infty}\!\!\! d\varepsilon \text{Im} \left[ d_{T}\tilde{d}_{T} \!+\! {d}_{N}\tilde{d}_{N}\! \right]\!\text{tanh}{\frac{\beta\varepsilon}{2}}, \end{align} \tag{ 23 }$$

where t.c. indicates the tilde conjugate of the preceding terms. The normal and tangential contributions to the spin current density are zero, since the terms couple to d_B and the derivative of σ_B , which give zero contribution.

By inspecting the weak proximity expression (23), we can infer that the spin current density should also change sign under reversal of K: the second term is directly odd in κ, and we saw from the magnetization that reversing K reverses d_N and $\tilde{d}_N$, which will reverse the sign of the first term. This is shown in figure 4(c), for a phase difference of $\phi = \pi/2$, where the triplet contribution is maximal. We also see a second sign change for higher curvatures, here around $KL/\pi\approx 0.6$, due to a $0-\pi$ transition [16].

3.3.2. Mixed chiralities in the critical charge current and ground state transitions.

We can consider the combination of mixed chiralities with respect to the local superconducting interface in SFS junctions by comparing the behaviour of C- and S-type junctions with varying degrees of curvature, and will find that the triplet response to the changing spin quantization axis will interfere either constructively or destructively.

The curvature-induced $0-\pi$ transition for in-plane SFS junctions with constant curvature is due to the singlet-triplet conversion [16]. Here, we will compare the behaviour of junctions with mixed chiralities at the interfaces. The charge current in an SFS junction is given by the Keldysh component

$$\begin{align} \frac{I_{Q}}{I_{Q_{0}}} = \int_{-\infty}^{\infty} d\varepsilon \textrm{Tr} \left( \hat{\tau}_{3}\check{g} \partial_{s} \check{g}\right)^{K}, \end{align} \tag{ 24 }$$

which, in equilibrium, simplifies to

$$\begin{align} \frac{I_{Q}}{I_{Q_{0}}} = \int_{-\infty}^{\infty}\!\!\! d\varepsilon \textrm{Tr}\left\{ \hat{\tau}_{3}\left( \hat{g}^{R} \partial_{s} \hat{g}^{R} \!-\! \hat{g}^{A} \partial_{s} \hat{g}^{A} \right) \right\}\text{tanh}{\frac{\beta\varepsilon}{2}}. \end{align} \tag{ 25 }$$

Here $I_{Q_{0}} = N_{0}eD_{F}A\Delta_{0}/4L$, where e is the electron charge, A the cross sectional area of the wire, and Δ₀ is the gap of the two identical bulk-like superconductors. The magnitude of the critical current is given in figure 5 as a function of the curvature amplitude K, for different ferromagnet lengths L and shapes (C, S). It is clear that here the C-type curves have a curvature-induced $0-\pi$ transition, whereas the S-type curves only allow for a modulation of the current.

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In the case of a straight ferromagnet, the $0-\pi$ transition is governed by the length of the ferromagnet, due to the modulation in the acquired phase difference between the correlated spins [8, 75]. This length-based phase acquisition is modified by the phases acquired due to curvature-induced rotation, but the length will clearly still be a factor in controlling the ground state. For example, for the parameters chosen in figure 5, longer junctions (e.g. $L = 6\xi$) would already be in the π ground state for K = 0, with initial reversal of the sign of the critical current. In that case, the S-type would display a $\pi-0$ transition, whereas the C-type would remain in the π-state. We can compare the singlet and triplet contributions to the current by decomposing the anomalous Green's function using equation (19) and writing $I_{Q}/I_{Q_0} = I_{0} + I_{\mu} + I_{\kappa}$, with

$$\begin{align} \frac{I_{0}}{I_{Q_{0}}} & = -8\!\! \int_{0}^{\infty}\!\!\! d\varepsilon \text{Re}\left\{ \tilde{f}_{0}\partial_{s} f_{0} \!-\! f_{0}\partial_{s} \tilde{f}_{0} \right\}\!\text{tanh}{\!\frac{\beta\varepsilon}{2}}, \end{align} \tag{ 26a }$$

$$\begin{align} \frac{I_{\mu}}{I_{Q_{0}}} & = +8\!\! \int_{0}^{\infty}\!\!\! d\varepsilon \text{Re}\left\{ \tilde{d}_{\mu}\partial_{s} d_{\mu} \!-\! d_{\mu}\partial_{s} \tilde{d}_{\mu} \right\}\!\text{tanh}{\!\frac{\beta\varepsilon}{2}}, \end{align} \tag{ 26b }$$

$$\begin{align} \frac{I_{\kappa}}{I_{Q_{0}}} & = 16 \kappa\left(s\right) \!\!\int_{0}^{\infty}\!\!\!d\varepsilon \text{Re}\left\{ \tilde{d}_{N}d_{T} \!-\! \tilde{d_{T}}{d}_{N} \right\}\!\text{tanh}{\!\frac{\beta\varepsilon}{2}}.\end{align} \tag{ 26c }$$

Here I₀ represents the singlet contribution, I_µ is the triplet contribution in the µ direction, and I_κ is the inverse Edelstein contribution, present when the d-vector rotates [76]. That is, there will be no inverse Edelstein contribution in a straight segment of the wire here.

As discussed above, the curvature induces generation of the long range triplet correlations by rotating the d-vector from the short ranged correlations, giving the long-range current component I_N . The first thing to notice about figure 6 is that I_N is almost antisymmetric with respect to the geometry being either of the S or the C type. We can understand this by employing the construction discussed in figure 3. By deforming the curved wire to a straight equivalent, while keeping the exchange field orientation fixed, one finds that for C-type geometries, the effective orthogonal exchange field (i.e. the component along the normal direction evaluated in the middle of the junction, $\boldsymbol{h}\cdot \hat{N}(s = L/2)$) points in opposite directions at the ends, whereas for S-type geometries, it points in the same direction. In the former case, the d_N triplets generated at the ends, are expected to have a phase difference of π, which comes in addition to the Josephson effect, due to the sign change of h_N . This produces a current contribution I_N which flows in the opposite direction to the latter case, where there is no such sign-change. As we increase the curvature amplitude K, the C-type geometry therefore features a $0-\pi$ transition in the charge current I_Q , which is shown in figure 6.

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