Phase-controllable Nonlocal Spin Polarization in Proximitized Nanowires

X. P. Zhang,1, 2, ∗ V. N. Golovach,1, 2, 3 F. Giazotto,4 and F. S. Bergeret2, 1, † Donostia International Physics Center (DIPC), Manuel de Lardizabal, 4. 20018, San Sebastian, Spain Centro de Fisica de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, 20018 Donostia-San Sebastian, Basque Country, Spain IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain NEST Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy

Remarkably, such splitting field can be generated without applying an external field, but by the magnetic proximity effect from a magnetic insulator [27][28][29][30][31][32][33][34]. Indeed, a Zeemanlike splitting at zero field has been observed for a long time in superconducting Al layers in contact with the ferromagnetic insulator (FI), EuS [35][36][37][38][39][40]. A recent article reports the first hybrid epitaxial grow of InAs/EuS/Al nanowires [41], which is one of the pivotal systems in the studies on topological superconductivity. Even though the experiment is not conclusive with regard to Majorana physics, the wire shows signs of coexisting induced superconducting gap and Zeeman splitting.
Motivated by these recent experiments we demonstrate that an electronic spin polarization is induced in NWs adjacent to SCs and FIs (see sketch in Fig. 1a). The induced magnetic moment appears as a consequence of the interplay between the superconducting and magnetic proximity effects. The latter takes place at the NW/FI interface, where the localized magnetic moments of the FI interact with the conduction electrons in the NW via exchange. This interaction leads to the Pauli paramagnetic response of the conduction electrons, which manifests as finite magnetic moment localized at the interface over atomic distance. In addition, the superconducting proximity effect at the NW/SC interface allows for the leakage of the Cooper-pairs condensate into the NW. This condensate interacts with the localized Zeeman-like field and leads to a finite electron spin polarization which spreads in the NW over the pairs coherence length. The induced non-local magnetic moment is opposite in sign to the Pauli magnetization localized right at the FI/NW interface, and its strength is proportional to the condensate density in the NW. In the rest of this letter we calculate this non-local magnetic moment as a function of the system parameters, demonstrate its control by the phase difference in a loop geometry, and propose a way of measuring it via spin-dependent spectroscopy.
It is illustrative to begin by recalling the basic features of the response of a conventional SC to a Zeeman or exchange field h(r) [42][43][44]. In the normal state, the response is local arXiv:2002.08230v1 [cond-mat.mes-hall] 19 Feb 2020 and leads to a magnetic moment M P auli (r) = µ B ν F h(r), which follows from the Pauli paramagnetic response (dotblack curve in Fig. 1c). Here µ B is Bohr magneton, ν F is the normal density of states (DoS) at the Fermi level and the electron g-factor is set to be 2. When the temperature, T is below the critical superconducting temperature, T c , there exists an additional non-local contribution to magnetic moment, M N L (r) (dashed-red curve in Fig. 1c), from the superconducting condensate. In particular, in a homogeneous SC at zero temperature, this contribution exactly compensates the Pauli one, M N L = −M P auli , for fields h smaller than the superconducting gap, ∆. This explains the zero susceptibility of a SC [45]. In the presence of a spin-relaxation mechanism, for example SOC, the full magnetic moment cancellation does not occur (see blue curves in Fig. 1c), in accordance to Abrikosov and Gorkov theory of the Knight shift in SCs [42]. For h larger than ∆, the compensation is not complete and the total magnetic moment reads [46][47][48]. One can draw a connection between the non-local magnetic moment induced by the field and the modified spectrum of the SC, see Fig. 1(b). The exchange field h leads, to both a splitting of the quasi-particle density of states (DoS), and a reduction of the superconducting gap. As far as the latter is finite, the total magnetic moment is zero. For h > ∆, the gap closes and a finite magnetic moment appears as a consequence of an incomplete compensation |M N L | < M P auli . The previous discussion has been introduced for merely pedagogical purposes, as it is useful when discussing our main results. Nevertheless, it has to be noted that it is an oversimplification of the real situation. Strictly speaking, for a large enough field h, the superconducting gap has to be determined self-consistently, and striking phenomena, such as the inhomogenoues superconducting phase [43,44], might appear. However, the situation is simpler when superconductivity is induced in a nonsuperconducting material via the proximity effect. In this case the self-consistency is not needed and the exchange field can be arbitrary large. This is the case considered in this work.
Specifically we study the situation sketched in Fig. 1(a), in which a NW is in contact with SCs and FIs. To describe the superconducting proximity effect, we use the quasiclassical equations and assume the diffusive regime in the NW. The characteristic length over which the Cooper-pair correlations decay in the NW is denoted as ξ N , which is an energy dependent length. To describe the magnetic proximity effect in the FI/NW interface, we follow the approach of Ref. [49] and assume a region of thickness b in which the local magnetic moments of FI and the itinerant electrons of NW interact via a sd-exchange coupling. This interaction leads to an interfacial exchange field h ex acting on the conduction electrons. Because b ξ N the exchange field can be included in the quasiclassical equations as a term of the form h b (y) = h ex bδ(y), where we denote with y the coordinate axis perpendicular to the FI/NW interface [50]. At this stage we can already anticipate the appearance of a non-local magnetic moment in opposite direction to the one localized at the FI/NW interface. Indeed, the Cooper pairs in the NW consist of electrons with opposite spins (singlet state). Those pairs reaching the FI/NW interface interact with the local exchange field. Energetically it is favorable that the electron of the pair with spin parallel to the local exchange localize at the interface, while the second electron with opposite spin remains in the NW. As a result a non-local magnetic moment opposite to the interfacial one, is induced in the NW and extends over the characteristic Cooper size, ξ N . This physical picture resembles the inverse proximity effect in metallic superconductor-ferromagnetic junctions predicted in Refs. [51,52] and experimentally verified in Refs. [53][54][55].
To quantify this effect we calculate the non-local electronic equilibrium spin polarization, M N L , induced in the NW. This is given by the expression, where f (ω) = 1/(e ω/T +1) is the equilibrium Fermi distribution function, and N ↑/↓ (ω, X), are the local DoS for spin-up and -down electrons. The exchange field at the FI/NW leads to N ↑ = N ↓ and hence to a finite M N L . In addition to the non-local term there is the Pauli magnetic moment localized at the FI/NM interface M P auli = µ B ν F h ex bδ(y), such that the total magnetic moment equals M P auli + M N L . We consider first the SC/NW-FI/SC setup sketched in the inset of Fig. 2(c). The NW is in contact with a FI, and sandwiched between two SCs. The phase difference between the SCs, φ, can be tuned by a magnetic flux, when the junction is part of a superconducting loop. We denote with x the axis of the NW of length L N . The NW-FI interface is orthogonal to the y-axis and the NW width in this direction is W N . In this first example we assume that W N , L N ξ N and integrate the quasi-classical equations over the volume of the NW. The integration in y direction results in an effective exchange field h F = h ex b/W N , whereas the integration over x can be performed with help of the Kupriyanov-Lukichev boundary conditions [56] and accounts for the superconducting proximity effect. In this way we obtain a compact expression for the DoS [57]: where η =↑, ↓. This expression has the same structure as the BCS DoS of a spin-split superconductor with renormalized frequency, ω r = ω + 2i b G S and order parame- is an energy proportional to the tunneling rate across the NW/SC interface. R is the interface resistance per area, D is the diffusion coefficient, and σ N is the conductivity of the NW. From Eq. (2), one can calculate the gap induced in the NW by the superconducting proximity effect. In the limit of transparent contact, b ∆, this gap is of the same order as the SC gap and the spin splitting is negligibly small. In the case of a finite NW/SC barrier, when b ∆, Eq. (2) describes a NW with an induced minigap, ∆ N = ∆ 0 N cos(φ/2), with ∆ 0 N = 2 b , and a spin splitting in the DoS due to the effective exchange field, h F . In all cases the minigap induced in the NW is maximum when φ = 0 and vanishes at φ = π. By substituting Eq.
(2) into Eq. (1), we obtain the non-local magnetic moment, M N L plotted in Fig. 2. As far as h F < ∆ N , M N L compensates the Pauli magnetic moment, M P auli = µ B ν F h F localized at the FI/NM interface (see the weakening of spin screening for b ∼ ∆ in [57]). At h F = ∆ N , M N L reaches a maximum value, µ B ν F ∆ N and decays as h [46][47][48]. This is the same behaviour as in a bulk superconductor discussed in Fig. 1(c), after identifying ∆ and h with the induced minigap ∆ N and effective exchange field h F , respectively. This analogy is clearly seen if we plot the curves of Fig. 2(a) as a function h F /∆ N . In this case all curves collapse into one (inset of Fig. 2a) coinciding with the behaviour shown in Fig. 1(c). In Fig. 2 Fig. 2b). In other words, as far as h F is smaller than the induced gap ∆ N = ∆ 0 N cos(φ/2), the M N L (φ) curve shows a plateau at the value opposite to M P auli . Interestingly, the value of M N L is proportional to the distance between the coherent peaks in the spin-splitting DOS, similar to those shown in Fig. 1(b). Indeed, in the present case when ∆ N ∆, according to Eq. (2), the peaks at positive energies occur [57]. The maximum modulation is achieved for h F = ∆ 0 N (green curve in Fig. 2b) in which the full screening of M N L only occurs at φ = 0. For larger values of h F , the NW is gapless and M N L (φ) is overall reduced (blue curve).
In the presence of SOC, electron spin channels are mixed. In this case the DoS of the NW is described by Eq. (2), after replacing ω r and ∆ r by ω η are the normal and anomalous parts of the retarded Green's function of the NW, respectively [57]. so is an energy describing the SOC. In the case of SOC induced by impurities, Elliott-Yafet SOC, so = 1/τ so , where τ so is the spin relaxation time due to the SOC [42,58]. In the case of intrinsic SOC of Rashba type so is proportional to the inverse of the Dyakonov-Perel relaxation time D(mα) 2 , where α is the Rashba SOC parameter [59]. The effect of a finite SOC is shown in Figs. 2(c-d). As expected from the analogy with the bulk SC, Fig. 1(c), the main effect of the SOC is the uncompensated screening of the Pauli magnetic moment, −M N L < M P auli , as shown by the green and blue curves in panel 2(c).
In addition the SOC leads to a shift of the maximum of the M N L (h F ) curves towards larger values of h F , such that, for h F > ∆ N , M N L is enhanced by the SOC. In fact, in the limit so h F , ∆ N , the effective exchange field acting on the con- h F < ∆ N . The reduction of the effective exchange field results into the right shift of δM N with respect to h F in analogy with the bulk case shown by the dot-dash-blue curve of Fig.  1(c). So far we have analyzed a short NW sandwiched between two SCs. In a more realistic setup, the length of the NW, L N can be larger than the ξ N . Moreover, in typical lateral structures the NW is partially covered by the SCs films of length L S . Such a lateral setup is sketched in Fig. 3(a). We assume that the NW is grown on top of a FI substrate, and that its cross-section dimensions are smaller than ξ N . In this case one can integrate the Usadel equation over the crosssection and reduce the problem to an effective 1D geometry with inhomogenoues fields (details are given in the supplementary material [57]). Hereafter, we assume a symmetric setup with L S = L N /3 and L F = L N (other situations are analyzed in the [57]). The distance between the SCs is then L = L N /3. Once induced, the minigap is constant in all the NW [60]. Its value depends on the distance between the superconducting leads and the characteristic barrier energy In the short limit, L N ξ N , M N L is almost constant in the NW and the results are similar to those shown in Figs. 2(a) and (b) (see [57]). More interesting is the case when L N is of the order of ξ N . Numerical results of the spatial dependence M N L (X) for L N = 4.7ξ 0 and different values of h F , are shown in Fig. 3(d). Remarkably, the shape of the M N L (X) curve depends on the strength of h F . These different behaviours can be explained in light of Eq. (1). The integrand in this expression can be well approximated by replacing the exact DoS, N (ω, X) by a BCS-like one, with a position-dependent pseudogap ∆ * N (X), defined as the energy where N (ω) intersects with the one in the normal state N 0 (ω) = 1, as shown in Fig. 2(b). Whereas the real minigap, ∆ N , is position independent, ∆ * N is not. In fact, the pseudogap is smaller in the middle of the wire becoming larger in the regions below the SCs (see also Fig. 5d in [57]). The shape of the M N L (X) is determined by the ration h F /∆ * N (X), in the same way as in the short junction limit h F /∆ 0 N determines M N , see Figs. 2 (a,c). Indeed, for a given h F with h F < ∆ * N (X) for all X, the values of |M N L | increases towards the middle of the wire (blue curve in Fig. 3(d)). In contrast, if ∆ * (−L/2) > h F > ∆ * (0) then a double-minima curve is obtained (green curve). Larger values of h F leads to |M N (X)| with a minimun at X = 0 (red curve). The actual shape of the curve can be inferred from the X dependence of ∆ * which is shown in Fig. 5c [57]. Finally, Fig. 3(c) shows the phase dependence of M N L calculated in the center of the wire for different values of h F . The result at low temperatures is qualitatively similar to the one obtained for the simpler setup analyzed in Fig. 2 Fig. 3c). Finally, we discuss possible ways of detecting M N L via its dependence on the phase-difference in a Josephson junction geometry. As discussed above the magnetic moment M N L depends crucially on the spectral properties of the proximitized NW, which in turn can be controlled by tuning the phase difference. This has been demonstrated experimentally in spectroscopy measurements, for example, by using a superconducting quantum interference proximity transistor (SQUIPT), sketched in Fig. 3a) [61][62][63][64], or by combining STM/AFM techniques [60]. In these experiments the phase difference, and hence the minigap, is controlled by the magnetic flux through a superconducting the loop [65,66]. In the present case the wire is in contact to a FI, and hence the DoS in the NW is spin-split due to the exchange field at the FI/NM interface. This should manifest as a splitting of the peaks at the edge of the gap. According to our predictions, if the SOC is negligible small, the observed splitting of the peaks remains almost constant as far as the phase-dependent pseudogap ∆ * N , is larger than the effective exchange field [see red curves in Figs. 2(b) and 3(c)]. The splitting in the DoS of the NW can be detected by measuring the differential conductance with a tunneling probe attached to the NW, as shown in Fig. 3a). When the phase difference is larger then arccos (h F /∆ 0 N ) then we predict a rapid suppression of the splitting as the phase difference is further increased. In the presence of SOC all sharp features vanish and hence this method would not provide an accurate measurement of M N L A more direct measurement of M N L and its phasedependence can be achieved by using a ferromagnetic probe tunnel-coupled to NW, as shown in Fig. 3(a) setup. We assume that the polarizations of the probe and the FI can be tuned between parallel (P) and antiparallel (AP) configurations. The measured differential conductance at low temperature is proportional to the DoS in the NW. In particular the difference between the conductances in the P and AP configurations is proportional to the spectral magnetic moment induced in the NW. Namely, where p is the polarization of the probe/NW tunnel junction and G 0 is normal-state tunneling conductance. The total induced magnetic moment can then be obtained from Eq. (1) by knowing the normal state properties of the tunneling contact. By using the SQUIPT setup of Fig.  3(a) one can tune the phase difference by an external magnetic field and measure the N N L (φ) curve. Comparison of experimental results with the curves in Fig. 3(c) may provide useful information about the exchange field at the FI/NW interface and SOC in the NW which are both important ingredients for the appearance of Majorana zero modes.
In conclusion, we predict the appearance of a non-local magnetic moment M N L in a NW when proximitized to SCs and a FI. This magnetic moment appears as a consequence of the interplay between the long-range superconducting correlations induced in the NW and the exchange field localized at the FI/NW interface. The sign of M N L is opposite to the local Pauli spin polarization right at the FI/NW interface and its value can be controlled by the phase difference between superconducting electrodes in a SC-NW-SC Josephson junction. We suggest ways of detecting M N L by electric measurements on realistic setups. σ k (τ k ) with k = 1, 2, 3 are the Pauli matrix for spin and Nambu spaces, respectively. D is the diffusion coefficient. ∆(r) is the gap of superconductor with phase, φ(r). h(r) is an exchange or Zeeman field. In this work, the order parameter, ∆(r) phase, φ(r) and Zeeman or exchange field, h(r) can be position-dependent. The right hand side of Eq. (S1) describes the effect of spin-orbit coupling (SOC), where so is the corresponding energy. For the sake of simplicity, both Planck and Boltzmann constants have been set to one, i.e., = 1 and k B = 1.
To described hybrid interfaces between different materials we used the Kupriyanov-Lukichev boundary conditions [56,67]: where g L,R are the Green's functions at the left and right side of the interface, σ L,R the corresponding conductivities, R the interface resistance per unit area, and n a vector normal to the interface. The first equality in Eq. (S2) corresponds to the current conservation at any interface. In particular if the interface is between a metal and vacuum the right hand side equlas to zero and the boundary condition reduces tǒ g(n∇)ǧ| int = 0.
In what follows we solve Eq. (S1) and determine the local density of states in different situations addressed in the main text. Because we are only interested in an equilibrium situation, it is enough to consider the retarded block of Eq. ( S1).

A. Homogeneous Superconductors
We review first some basic features of the response of SC to a Zeeman field in the presence of SOC [42][43][44]. In spatially homogeneous situation the Usadel equation (S1) for the retarded component reduces to Here ω δ = ω+iδ, with delta being an infinitesimal small positive real number. η = ±1, correspond to the spin anti-parallel and parallel to the direction of exchange field,respectively. Thus,ǧ η S are matrices in the Nambu space. Hereafter, we consider only the retarded Green's function and omit δ for simplicity. The last term of the left hand side of Eq. (S4) describes the spin relaxation due to SOC. The general solution of Eq. (S4) where G S is the normal and F S the anomalous component. They can be written in a self-consistent form: Here spin flipping causes a spin-dependent renormalization of both, the frequency and the order parameter Once the Greens' function is determined the DoS can be obtained from its normal part, i.e., Eq. (S6) (S10) In the absence of SOC, the solution can be explicitly written with (S13) Therefore, the DoS (S10) reduces to which is nothing but the spectrum of a spin-split superconductor with coherent peaks in the DoS at: The (homogenoeus) non-local magnetic moment originated from the superconducting condensate is then given by where µ B is Bohr magneton, ν F is the normal DoS at the Fermi level, and the electron g-factor is set to be 2. f (ω) = 1/(e ω/T + 1) is equilibrium distribution function for frequency, ω and temperature, T . N ↑/↓ (ω) are the DoS for spinup and -down electrons. By substitution of Eq. (S14) in Eq. (S16) we obtain Hereafter, we consider the limit of T → 0. The Fermi-Dirac distribution function reduces a step function, i.e., f (ω) = θ(−ω). For h < ∆, we obtain M N L = −µ B ν F h = −M P auli , i.e. opposite to the Pauli spin response. Thus, the total magnetic moment becomes zero. In general we find a compact expression for magnetic moment: In the presence of SOC, the spin-dependent renormalization of frequency, as shown in Eqs. (S8), reveals that Zeeman field might be renormalized by normal Green function (S6). For the sake of simplicity, let us consider the case of small SOC, so ∆. The first order correction of normal Green function can be obtained by replacing the GFs, G η S and F η S on the right hand side of Eq. (S6), by the GFs, G η S and F η S in Eqs. (S12) and (S13) (S19) Therefore, in this limit, the effect of SOC is a further renormalization of the frequency, order parameter and Zeeman field The DoS of SC, in the first order of SOC, can be derived from Eq. (S19) Now the coherent peaks are shifted according to: In the present case, so ∆, we can approximately replace the ω η ± in the right hand side of Eq. (S25) by Eq. (S15). Then, we obtain The peaks at negeative energy are then given by and therefore the effective Zemman field becomes For h < ∆, we find For h > ∆, we reach The latter result explains the suppression of the effective Zeeman field in the presence of SOC which manifests as a shift of the δM S (h) curve in the Fig. 1(c) of the main text.

B. Hybrid Superconductor Structures
In this section, we consider hybrid structures with inhomogeneous fields. In particular we focus on the case when the exchange field is spatially localized, originated from the interaction between localized moments in the FI and the conduction electrons of the NW, and the superconducting correlations are induced in the NW via the proximity effect. The Usadel equation, Eq. (S1), determines an energy dependent length over which the pair correlations decay in the NW. We denote this length as ξ N .
To describe the magnetic proximity effect in the FI/NW, we follow the approach in Ref. [49] and assume a region of thickness b in which the local magnetic moments of FI and the itinerant electrons of NW coexist and interact via a sdexchange coupling. This interaction leads to an interfacial exchange field h ex acting on the latter which is localized at the interface. Because b ξ N the exchange field can be included in the quasiclassical equations as a localized field, h b (y) = h ex bδ(y), where y is the coordinate perpendicular to the FI/NW interface [50].

The SC/NW-FI/SC structure
We first focus on the setup, depicted in the inset of Fig. 2(c) of the main text. Here the FI is grown along one of the facets of the NW. In principle, we are dealing with a 3D problem. We simplify by assuming that the transverse dimensions of the NW are smaller than ξ N , such that we can assume the GFs being independent of y and z. We can then integrate the Usadel equation, (S1), first over z-direction, where the zero current BC at both Vacuum/NW interfaces applies, Eq. (S3), and second over the y-direction where at y = 0 there is a local exchange field from the FI. After these integrations the Usadel equation in the NW region reduces to a 1D equation: The magnetic proximity effect results in an effective exchange field h F = h ex b/W N , where W N is the width of NW in y direction.
In this example, for the sake of clarity, we also assume that the length of the wire, L N , is smaller than ξ N such that we also can integrate the above Usadel equation over x. At the interfaces with the superconducting leads we use the BC in Eq. (S2) and assume that the superconductors are massive and are not modified by the inverse proximity effect. This results in a matrix algebraic equation: The superconducting proximity effect is described by the barrier energy andǧ S is the bulk BCS GF: and φ the corresponding phase-difference between the superconductors.
The solution of Eq. (S34) together with the normalization condition g 2 N = 1 for each spin block η = ±, can be written asĝ with These soluctions have the same form as the BCS GFs with a renormalized frequency and an induced gap The DoS of NW can be obtained from the normal part of the retarded Green function, i.e., Eq. (S40) In the absence of the SOC, the solutions, Eqs. (S39)-(S43), reduce toĝ and the corresponding DoS for each spin block, η = ± from Eq. (S46) Thus, we obtain the coherent peaks in the spin-splitting DOS Let us study the renormalization effect of minigap and spin splitting from the superconducting proximity effect, in the limit of b , h F ∆. The zero-order effect can be obtain by setting ω η ± = 0 in the right hand side of Eq. (S49). Thus, we obtain the coherent peaks with spin splitting of 2h F : where ∆ N (φ) is the minigap of NW in the absence of SOC which depends on the phase difference of two SCs, φ. Clearly ∆ N (φ) is zero at φ = π, while reaches is maximum value, ∆ 0 N = 2 b at φ = 0. Next, we consider the first order effect, which can be obtained by substituting Eq. (S50) in the right hand side of Eq. (S49). Hence we reach with We find both minigap and spin splitting decrease with increasing b . The later corresponds to the weakening of spin screening.   Fig. 4). On the other hand, we find the weakening of spin screening with increasing the barrier energy, b or minigap, ∆ 0 N . This can be understood from the spin resolved DoS in Fig. 4(b). Here, N ↑ (ω) and N ↓ (ω), are related to each other by a BCS-like DoS, N BCS (ω, ∆ N ) with a renormalized minigap, ∆ N (Eq. S53). Thus, ). The full spin screening corresponds to α r = 1 (Fig. 1b of main text). However, the failure of full screen is a result of the reduction of the spin-splitting due to the superconducting proximity effect. It becomes more obvious for larger ∆ N ( b ), (Eq. S54 and blue curves in Figs. 4).
In the presence of SOC, we find a spin dependent renormalization of the frequency, see Eq. (S42). This implies a renormalization of the effective exchange field. For the sake of simplicity, let us consider the case of small SOC, so , b ∆ N and hence ∆ 0 N 2 b . The first order correction to the normal Green function can be included by replacing the GFs, G η N and F η N on the right hand side of Eq. (S40), by the GFs, G η N and F η N in Eqs. (S46) and (S47). Then, we reach (S55) In the present limit, so ∆ N , the SOC then leads to following renormalization of frequency, minigap and effective excahnge field: with Thus, the DoS of NW in the first order of SOC, reads For spin block η, the coherent peaks in the spin-splitting DOS are given by In the limit of ∆ N ∆, we have G S (ω η ± ) 0 and F S (ω η ± ) 1. Hence, Eq. (S61) reduces into For small SOC, so ∆ N , we can approximately replace the ω η ± in the right hand side of Eq. (S63) by Eq. (S50). Thus, we reach For zero temperature, we are only interested in the spin splitting of negative frequency Thus the effective exchange field reads Clearly, we find effective exchange field decreases in the presence of SOC. This cause a right hand side shift of magnetic moments with exchange field, see Fig. 2(c) of main text.
The SC-FI-SC NW structure In this section we consider a more realistic setup, the lateral SC-FI-SC NW structure depicted in Fig. 3(a) of main text. Here, an arbitrary long normal wire (NW) is grown on the top of FI. Two superconductors (SCs) with phase difference, φ cover partially teh extremes of the NW. The starting point is agian the Usadel equation for the retarded quasiclassical Green's function in the NW: where we have neglected the effect of SOC. The magnetic proximity effect of FI can be described by a localized exchange field at FI/NW interface, h b (r) = bh ex θ F (x)δ(y), with where L F is the length of FI. On the other hand, the proximity effect of SCs is captured by the Kupriyanov-Lukichev boundary conditions (S2) at two NW/SC interfaces, which can be written in a compact form The positions of the left and right superconducting electrodes, in x direction, are respectively described by two step-like functions with L S being the length of both SCs. We do not consider the inverse proximity effect of FI on SCs and hence their GFs are the BCS oneš where we introduce phase difference, φ between SCs.
Because the transverse dimensions of the NW are smaller than the characteristic length ξ N , we can assume that the GFs do not depend on y and z. We can then integrate the Usadel equation, (S71), first over z-direction, where the zero current BC at both vacuum/NW interfaces applies, Eq. (S3), and second over the y-direction. In the second integration the local exchange field at the NW/FI at y = 0 results in an effective spin-splitting field h F , whereas at the SC/NW interface, ,y = W N , the boundary condition, Eq. (S73) introduces a term in the Usadel equation describing the induced superconducting condensate. The final 1D equation after these integrations reads: The strength of the superconducting proximity effect is parametrized by the energy: Eq. (S77) is complemented by the normalization condition, g 2 N (x) = 1. In order to solve numerically these two matrix equations it is convenient to use the Riccati parameterization to express the GFs in terms of two coherent functions γ andγ as follows:ǧ where F andF describe the Cooper pairs penetrating from both S regions. In Riccati parameterization, Usadel equation (S77), for each spin block η = ±, reads where α N = L 2 N /(W N σ N R ), and we have made the position coordinate dimensionless by introducing l = x/L N and energy is in the unit of th = D/L 2 N . In a more realistic setup, the length of the NW, L N can be larger than the characteristic length ξ N . Moreover, the NW can be partially covered by the SCs films of length, L S . We assume that the NW is grown on top of a FI substrate with length, L F . Hereafter, we assume a symmetric setup with L S = L N /3, and hence the distance between the SC leads is L = L N /3. The minigap induced in the NW, ∆ N depends on this distance and the NW/SC barrier resistance.
Let us begin with the case of weak superconducting proximity, b = 0.05∆ and short NW, L = ξ 0 /3, where ξ 0 = D/∆. Fig. 5(a) shows the h F dependence of M N L at the center of NW for different values of the phase difference, φ. As far as h F < ∆ N , M N L compensates the Pauli magnetic moment M P auli = µ B ν F h F localized at the FI/NM interface. At h F = ∆ N , M N L reaches a maximum value, µ B ν F ∆ N and decays as h F − h 2 F − ∆ 2 N for h F > ∆ N [46][47][48]. In Fig. 4(b), we show the phase difference,   Fig. 4b). In other words, as far as h F is smaller than the induced gap ∆ N = ∆ 0 N cos(φ/2), the M N L (φ) curve shows a plateau at the value opposite to M P auli . The maximum modulation is achieved for h F = ∆ 0 N (green curve in Fig. 4 b). For larger values of h F , M N L the NW is gapless and M N L (φ) is overall reduced (blue curve).
Let us now go beyond the limits of weak superconducting proximity and short NW. The results are plotted in Fig. 5 , where b = 0.5∆ and L = 4.7ξ 0 . In this case, the local DoS, N (ω, X) strongly depends on X (Fig. 5d here and Fig. 3b of main text). The induced minigap, ∆ N , though is spatially constant, as shown by the green line in Fig. 5(d). The local pseudogap, ∆ * N (X) defined by the energy in which the exact DoS, N (ω, X) coincide with the DOS in the normal state, N 0 (ω, X) = 1, is position-dependent, as shown by the red curves in Fig. 5(d). It is smallest at the center, ∆ * N (0) ∆ N , and becomes bigger closer to both ends. At zero temperature, the calculation of the local M N L (X), Eq. (1) of the main text, can be well approximated by replacing the exact DoS, N (ω, X) by a BCS-like one, N BCS (ω, ∆ * N (X)) with the position-dependent gap, ∆ * N (X). Panel 5(c) depicts M N L as a function of h F /∆ and X/L N . We find a interesting transition from a maximum to a minimum at X = 0 in the M N L (X) dependence. For small h F , the shape with a minimum is due to the weakening of spin screening with increasing pseudo gap, ∆ * N (X) from center to both ends.
In Fig. 6 we show the non-local magnetic moment in a setup when the FI is in contact only to certain portion of the NW, for example if L F /L N = 1/4. In Fig. 6 (a), we show the spatial dependence of M N L for different values of h F and in panel (b) the phase-dependence at x = 0.