Introduction

Coherent quantum tunneling of Cooper pairs through a thin barrier is one of the first examples of macroscopic quantum coherent phenomena. As predicted by Josephson more than 50 years ago1, it has important applications in quantum circuits used in metrology, quantum sensing, and quantum information processing2.

Most of the previous studies focused on conventional Josephson junctions (JJs) consisting of two s-wave superconductors (S) that are connected by an insulating (I) or a normal (N) region3,4. The flow of supercurrent through a JJ depends on the superconducting phase difference ϕ between two superconductors and, in general, is characterized by the current-phase relationship J(ϕ) (CPR). In conventional JJs CPR should be periodic with 2π, I(ϕ) = I(ϕ + 2π) which follows from the Bardeen–Cooper–Schrieffer theory5. This result is a manifestation of a 2e charge of Cooper pairs and is used in metrology to measure electron charge. Time-reversal symmetry requires that I(ϕ) = − I( − ϕ) which imposes a constraint that the supercurrent should be zero for ϕ = πn where n is an integer. In general, CPR can be expanded in Fourier harmonics, \(I(\phi )={\sum }_{n}{I}_{n}\sin (n\phi )\).

In many cases, however, CPR is well approximated by the first harmonic \(I(\phi )\approx {I}_{{{{\rm{c}}}}}\sin (\phi )\) with Ic being the maximum supercurrent that can flow through the junction, i.e., the critical current. At a microscopic level, the supercurrent through a short SNS junction is determined by bound states forming in the constriction due to Andreev’s reflection at the NS interfaces. In the Andreev reflection process, an incident electron-like quasiparticle with spin ↑ gets reflected at the NS interface as a hole-like quasiparticle with spin ↓ and a Cooper pair is emitted into the condensate. When time-reversal symmetry is not broken, electrons and holes propagate with the same velocity in the normal region. In this case, no phase shift accumulates between this pair of quasiparticles along their trajectories in the N region, and the sign of Ic is fixed. When Ic > 0 we refer to this case as 0-junction.

In a magnetic Josephson junction (MJJ), exchange splitting breaks time-reversal symmetry and leads to an interesting interplay of superconductivity and magnetism4,6,7,8,9. In superconductor–ferromagnet–superconductor (SFS) junctions [Fig. 1a] the correlated quasi-particles and quasi-holes forming Andreev bound states propagate through the junction under the exchange field of the ferromagnet (F). In many ferromagnets, such as Fe or Ni, the exchange splitting is large (of the order of eV) and significantly perturbs the band structure of metal and, consequently, significantly modifies Fermi velocities of minority (spin ↓) and majority (spin ↑) carriers. Strong time-reversal symmetry breaking leads to the appearance of characteristic superconducting correlations with oscillatory dependence determined by the difference in wavenumbers, \({k}_{{{{\rm{F}}}}}^{\uparrow }-{k}_{{{{\rm{F}}}}}^{\downarrow }\)10,11. This effect opens a possibility for the supercurrent reversal as a function of the thickness of the ferromagnetic region, the so-called Josephson π-junction12,13. The correlation between the phase shift of the supercurrent and the magnetization provides a possibility for realizing magnetic spin valves, see Fig. 1b, which may have promising novel applications for cryogenic superconducting digital technologies14,15,16. Understanding the microscopic physics of MJJs is of great scientific interest as well as technological importance. 0–π transitions in SFS junctions have been extensively studied experimentally since the early 2000s and have been observed in different material systems13,17,18,19,20,21,22,23,24,25. While qualitatively these observations are consistent with the previous phenomenological theories26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42 the roles of the microscopic band structure arising from the atomic lattice on the supercurrent suppression with junction thickness remain unclear. Our primary goal is to address these essential points. The supercurrent suppression that is exponential in the junction length is often associated with the presence of disorder in the ferromagnetic region6,43. However, significant supercurrent suppression can also appear in relatively clean metals (e.g., Ni) whose mean free path is larger than the junction thickness22,23,44.

Fig. 1: MJJ stacks.
figure 1

a Schematic view of the SFS junction. b SFNFS junction. Arrows indicate possible magnetization of ferromagnets. The supercurrent through the spin-valve JJ depends on the relative magnetization of the ferromagnets, which governs the properties of the Josephson magnetic random-access memory (JMRAM)16. c Ball-and-stick representation of the Nb(110)/Ni(111)/Nb(110) junction with five layers of Ni. Nb atoms are light grey, Ni atoms are blue. The top and bottom atomic planes of Nb(110) are repeated periodically in the z-direction to create the semi-infinite Nb leads of the junction through which the current flows. Periodic boundary conditions are used in the xy-plane. The corresponding reciprocal space defines two-dimensional k∥ = (kx, ky) vectors, i.e., the transverse modes, used in the calculations.

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Here, we study the suppression of the critical current in the MJJs shown in Fig. 1a and identify two microscopic mechanisms for its suppression, both a consequence of the band structure asymmetry of the majority and minority carriers in the F region. First, there is an asymmetry in the structure of the Fermi surface, see Fig. 2a. For certain bands and momenta, the Fermi surface present in one spin channel may be absent in the other. This is typical in ferromagnetic materials like Fe, Co, and Ni because the bandwidth for d-electrons is relatively small and is often comparable to the exchange splitting. As a result, the wavenumber of one of the constituent quasi-particles forming Andreev bound states in MJJ becomes imaginary and the supercurrent becomes suppressed. We label this scenario as mechanism (i). In a second mechanism (ii), we show there is a dephasing of a harmonic signal originating from different Fourier components of the supercurrent due to the Fermi velocity dispersion, see Fig. 3a. Both these mechanisms lead to an exponential suppression of the supercurrent, which was previously believed to occur due to the presence of disorder in the magnetic region. We show that band structure effects are important and maybe even dominant in many cases.

Fig. 2: Bound states in SFS junctions.
figure 2

a Simplified band structure of a ferromagnet, with the majority and minority bands, split by Vex. \({k}_{{{{\rm{F}}}}}^{\uparrow }\) and \({k}_{{{{\rm{F}}}}}^{\downarrow }\) are the Fermi momenta for the majority and minority carriers, respectively. Fermi level \({E}_{{{{\rm{F}}}}}^{{{{\rm{(i)}}}}}\) corresponds to large Vex, where the minority band is pushed above \({E}_{{{{\rm{F}}}}}^{{{{\rm{(i)}}}}}\). \({E}_{{{{\rm{F}}}}}^{{{{\rm{(ii)}}}}}\) corresponds to small Vex. Thus, the propagation of minority quasiparticles is characterized by an imaginary momentum \({k}_{{{{\rm{F}}}}}^{\downarrow }\) and is suppressed. b Supercurrent in SFS junction is carried by Andreev bound states localized in the junction. Solid red line represents a quasi-classical trajectory corresponding to an Andreev bound state. The spectrum of Andreev states depends on the relative superconducting phase difference across the junction as well as the phase, \(\delta \varphi =| {k}_{{{{\rm{F}}}}}^{\uparrow }-{k}_{{{{\rm{F}}}}}^{\downarrow }| w\), accumulated due to the difference of Fermi momenta for the majority and minority carriers. Note that in scenario (i) the propagation of minority carriers is suppressed leading to an overall exponential decay of the supercurrent with w. This is to be contrasted with the normal transport through the junction.

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Fig. 3: Normal and superconducting current of Nb/Ni/Nb junction.
figure 3

a First Fourier harmonic J1 of the supercurrent density [Eq. (3), black circles] and it is fit [Eq. (6), solid black line] as a function of Ni layer thickness, w, calculated for the Nb(110)/Ni(111)/Nb(110) junctions shown in Fig. 1c. Green semitransparent curves correspond to \({j}_{1}^{{{{\rm{fit}}}}}({{{{\bf{k}}}}}_{\parallel })\) for individual k∥. b Normal state conductance per unit of area as a function of w for the majority (G↑) and minority (G↓) spins [Eq. (13)] as well as G = G↑ + G↓. G does not depend on w and is approximated by the single value 〈G〉. c–e Supercurrent as a function of phase difference ϕ for c 4 atomic layers of Ni (strong 0-junction regime), d 8 layers (intermediate regime), and e 13 layers (strong π-junction regime). In the 0- and π-junction regimes, the J1 component dominates. In the intermediate regime, higher-order terms may prevail.

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In order to capture a realistic band structure, we develop a microscopic theory for the supercurrent in realistic MJJs. We use a combination of density functional theory (DFT) and Bogoliubov-de Gennes (BdG) model to investigate the 0–π transition in realistic material stacks of Nb/Ni/Nb junctions in the clean limit. This method allows one to predict and explain key properties of MJJs such as the period and decay of the critical current oscillations with the ferromagnet thickness.

Results

Qualitative discussion and main results

In this section, we describe basic concepts for the supercurrent flow in MJJs and summarize our results. Our main qualitative conclusions are supported by microscopic calculations for Nb/Ni/Nb MJJs. Ni appears to have a fairly long mean free path lMFP ≈ 60 Å (see estimations in Supplementary Note 1), which is comparable or larger than the typical thickness of the ferromagnet used in recent experiments22,23. Therefore, the motion of quasiparticles in Nb/Ni/Nb junction is quasi-ballistic, and our method is applicable to this system. Most of this paper focuses on clean Nb/Ni/Nb junctions.

First it is illuminating to consider a toy model, a one-dimensional SFS junction, and calculate the supercurrent in such a system for different Fermi energies, see Fig. 2a. Using the results of ref. 45, one finds that both majority and minority spin bands are both occupied in the limit Vex/EF ≪ 1 [i.e., scenario (ii) in Fig. 2a] the supercurrent does not decay with ferromagnet thickness at zero temperature,

$$I(\phi )=\frac{2e{{\Delta }}}{\hslash }\left\{\begin{array}{ll}\cos \delta \varphi \sin \frac{\phi }{2},&0 \,<\, \phi \,<\, \pi -2\delta \varphi ,\\ -\sin \delta \varphi \cos \frac{\phi }{2},&\pi -2\delta \varphi \,<\, \phi \,<\, \pi +2\delta \varphi ,\\ -\cos \delta \varphi \sin \frac{\phi }{2},&\pi +2\delta \varphi \,<\, \phi \,<\, 2\pi .\end{array}\right.$$
(1)

Here perfect interface transparency \({{{\mathcal{T}}}}\) is assumed, \({{{\mathcal{T}}}}\approx 1\). The phase offset δφ originates from the Fermi momentum difference of a quasi-particle and a quasi-hole forming Andreev bound state in the junction, see Fig. 2b, and is given by \(\delta \varphi =| {k}_{{{{\rm{F}}}}}^{\uparrow }-{k}_{{{{\rm{F}}}}}^{\downarrow }| w\approx {V}_{{{{\rm{ex}}}}}w/\hslash {v}_{{{{\rm{F}}}}}\). The 0- and π-junction regimes can be clearly identified at δφ = 0 and δφ = π/2, respectively. At the intermediate values 0 < δφ < π/2, the CPR is anharmonic which is a generic feature at 0–π transition as shown below. In the low transparency regime, \({{{\mathcal{T}}}}\ll 1\), one would expect qualitatively similar results with the maximal critical current being suppressed \({I}_{{{{\rm{c}}}}} \sim (e{{\Delta }}/\hslash ){{{\mathcal{T}}}}\) but still independent of the ferromagnet thickness, w.

In the case of large exchange splitting Vex/EF ≫ 1 [scenario (i) in Fig. 2a], the minority band may become unoccupied. Given that minority carriers are gapped out and their propagation through the junction is suppressed, the supercurrent decays exponentially with w45

$$I(\phi )\approx \frac{2e{{\Delta }}}{\hslash }\exp (-\kappa w)\left[1-\frac{{E}_{{{{\rm{F}}}}}}{8{V}_{{{{\rm{ex}}}}}}{\sin }^{2}(kw)\right]\sin \phi .$$
(2)

Here, \(\kappa =\sqrt{2{m}^{* }({V}_{{{{\rm{ex}}}}}\ -\ {E}_{{{{\rm{F}}}}})}/\hslash\) and \(k=\sqrt{2{m}^{* }({V}_{{{{\rm{ex}}}}}\ +\ {E}_{{{{\rm{F}}}}})}/\hslash\) with m* being effective electron mass. One may notice the drastic difference between normal-state and superconducting transport in this case—the former is weakly affected (because majority and minority contributions are additive) whereas the supercurrent is strongly suppressed. In this case, the measurement of normal-state junction resistance does not necessarily predict the magnitude of the supercurrent through the junction.

We now generalize above results for the realistic three-dimensional (3D) geometry and material composition of the MJJ. In the clean limit, the supercurrent I(ϕ) in the short-junction limit (w much smaller than the coherence length of the superconductor) is obtained from the spectrum of the Andreev bound states εν(ϕ, k∥) localized in the junction3 which now also depends on the parallel momentum k∥. The supercurrent density J(ϕ) per junction area A is J(ϕ) = I(ϕ)/A. For the junction with periodic atomic structure in xy-plane the supercurrent density at zero temperature is given by

$$J(\phi )=-\frac{e}{\hslash }\int _{{{{\rm{BZ}}}}}\ \frac{d{{{{\bf{k}}}}}_{\parallel }}{{(2\pi )}^{2}}\ \mathop{\sum}\limits_{\nu > 0}\frac{\partial {\varepsilon }_{\nu }(\phi ,{{{{\bf{k}}}}}_{\parallel })}{\partial \phi },$$
(3)

where the k∥ integration is performed over the Brillouin zone (BZ) of the corresponding surface supercell of area, A, and the sum is carried over positive quasiparticle energies, εν(ϕ, k∥) > 0. Note that we use spin-resolved εν and therefore omit spin prefactor 2 in Eq. (3). The derivative is periodic in ϕ and can be represented as a Fourier series

$$-\frac{e}{\hslash }\ \frac{\partial {\varepsilon }_{\nu }(\phi ,{{{{\bf{k}}}}}_{\parallel })}{\partial \phi }=\mathop{\sum}\limits_{n\geqslant 1}{I}_{n\nu }({{{{\bf{k}}}}}_{\parallel })\sin [n\phi +\delta {\varphi }_{n\nu }({{{{\bf{k}}}}}_{\parallel })],$$

so that Eq. (3) can be written as

$$J(\phi )=\int _{{{{\rm{BZ}}}}}\ \frac{d{{{{\bf{k}}}}}_{\parallel }}{{(2\pi )}^{2}}\ \ \mathop{\sum}\limits_{\nu > 0}\ \mathop{\sum}\limits_{n\geqslant 1}{I}_{n\nu }\sin [n\phi +\delta {\varphi }_{n\nu }({{{{\bf{k}}}}}_{\parallel })].$$

As we will show below, away from 0-π transition the supercurrent is dominated by the first (n = 1) harmonic. Therefore, we focus henceforth on the first harmonic contribution and drop n index in the following discussion. Next, one may notice that the supercurrent amplitudes Iν(k∥) and phase offsets δφν(k∥) depend on the parallel momentum k∥. One may include this dependence and define an effective Fermi energy EF(k∥) that counts the energy corresponding to k∥ in each band of the ferromagnet from the bottom of the band. Depending on EF(k∥) and Vex, either scenario (i) or (ii) of Fig. 2a may be realized.

Thus, it is important to compare the exchange splitting with the bandwidth of d-character states in transition metals to make sure that a perturbation theory in Vex is justified. Assuming it is the case, one can estimate the phase offset δφν(k∥) by expanding in exchange splitting to find

$$\delta {\varphi }_{\nu }({{{{\bf{k}}}}}_{\parallel })\approx {V}_{{{{\rm{ex}}}}}w/\hslash {v}_{z\nu }({{{{\bf{k}}}}}_{\parallel }).$$
(4)

In general, the dependence of vzν on k∥ is complicated, especially in spd-transition metals. The combination of complicated amplitude and phase offset dependence on k∥ leads to a non-trivial supercurrent dependence on the ferromagnet thickness w. As shown in Fig. 3a, the critical current decays with w for the Nb(110)/Ni(111)/Nb(110) junctions. We analyze the details of the decay and perform an exponential fit in Section “Details of the supercurrent”. At small thicknesses, below 30 Å, this decay originates from the evanescent modes corresponding to gapped out minority or majority carriers which cannot propagate through the junction, see Table 1. This is the mechanism (i) discussed above. At larger thicknesses (i.e., w ≳ 50 Å) a loss of synchronization between different modes due to the dispersion of vzν(k∥) becomes important. This second mechanism (ii) has been previously discussed in the literature6,12 under assumptions of a single spherical Fermi surface and a small uniform exchange splitting Vex in the magnetic region. Within these assumptions, one finds that critical current should decay algebraically with the thickness w of a magnetic layer12. However, as we show below most of these assumptions do not apply to realistic SFS junctions involving transition metals. Thus, in order to understand CPR in realistic MJJs, one needs to use accurate ab initio methods, which capture the physical effects described above.

Table 1 Minimum, maximum, average, and root mean square of the Fermi velocities vF for the majority and minority Fermi surfaces in bulk Ni.
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To make a connection between the decay seen in Fig. 3 and the mechanisms responsible for it, Fig. 4a presents the energy band structure of bulk Ni. It is probably the highest fidelity band structure available: it very closely reproduces ARPES data in both majority and minority spin bands, with exchange splitting Vex = 0.3 eV46, and should be an excellent predictor of the real Fermi surface and velocities in Ni.

Fig. 4: Band structure.
figure 4

a Electronic band structure of bulk Ni (solid lines) calculated from first principles including many-body effects, see ref. 46. It is the highest-fidelity available and is very close to ARPES data (diamonds) in both majority (red) and minority (green) spin bands, with exchange splitting Vex = 0.3 eV. b Majority (solid line) and minority (dashed line) Fermi surfaces of bulk Ni in the k∥ plane (with kz = 0) corresponding to the (111) plane direction used for the stacking of the Nb/Ni/Nb junctions, as discussed in the text. Axes correspond to two Γ-L lines: the Fermi surface has a threefold symmetry in the entire plane. Majority band 6↑ is depicted as a solid red-blue line with color interpolating between red and blue depending on the Fermi velocity ℏ−1∂E/∂k, which ranges between 3 × 105 m/s (red) and 6 × 105 m/s (blue). Minority bands are shown by dashed lines: 3↓ (blue), 4↓ (cyan), 5↓ (green), and 6↓ (red).

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Here, we use this potential to analyze the bulk Ni Fermi surface [Fig. 4b] and Fermi velocities (Table 1). Counting from the bottom s-band, bands of Ni d character are bands 2 to 6. These bands are nearly full: only majority band 6↑ and minority bands 3↓–6↓ cross the Fermi level EF. Only band 6 has both majority and minority carriers at the Fermi surface. Bands 6↑ and 6↓ have roughly the same shape and the energy splitting is approximately constant and equal to Vex (see the right panel of Fig. 4 in ref. 46). Beyond this, however, the correspondence between the Ni band structure and a simple parabolic band structure deviates substantially, in two ways that critically affect the analysis. First, the Fermi velocity, \({v}_{{{{\rm{F}}}}}={\hslash }^{-1}(\partial E/\partial k){| }_{k = {k}_{{{{\rm{F}}}}}}\), is not fixed even for a single pocket: it varies in band 6↑ by a factor of 2 [see Fig. 4 and Table 1]. Accordingly, the splitting \({k}_{{{{\rm{F}}}}}^{\uparrow }-{k}_{{{{\rm{F}}}}}^{\downarrow }\) between bands 6↑ and 6↓ varies by a factor of two as expected from the twofold variation in vF. Second, bands 3↓–5↓ have no majority counterpart at EF, indicating that the wave number of bands 3↑–5↑ is complex. This is the origin for the exponential decay in scenario (i) in Fig. 2a as noted above: a large portion of Andreev levels is carried by Cooper pairs made of single-particle wave functions with one or both of \({k}_{{{{\rm{F}}}}}^{\uparrow }\) and \({k}_{{{{\rm{F}}}}}^{\downarrow }\) having an imaginary component. The magnitude of \({{{\rm{Im}}}}\ k\) depends on the particular mode and k∥ leading to a distribution of decay exponents. The slowest decay in each of these evanescent modes can be estimated from the distance ΔE of the closest approach to EF and the effective mass m*/me, using \({\hslash }^{2}{k}_{\min }^{2}/2{m}^{* }={{\Delta }}E\) and decay \(\kappa =2\pi /{{{\rm{Im}}}}({k}_{\min })\), see Table 1. (m*/me is found to be highly anisotropic, so only the effective transport mass \({m}^{* }=3\ {[1/{m}_{1}^{* }+1/{m}_{2}^{* }+1/{m}_{3}^{* }]}^{-1}\) is shown.) κ is only a rough measure of the evanescent mode decay for a given band.

Let us now focus on the mechanism (ii) for the supercurrent decay, i.e., loss of synchronization between different transverse modes. This mechanism is well-known in diffusive systems where quasiparticle trajectory is random and thus the phase offset δφ accumulated along such a trajectory also gets randomized. Thus, upon averaging Eq. (3) over different disorder realizations, one ends up with exponentially decaying critical current6. Previously, such a suppression of the supercurrent with junction thickness, w, was often associated with impurity scattering in the ferromagnet. Here, we demonstrate that this dephasing mechanism can also appear in clean systems (where quasiparticle trajectory is well defined) due to the dispersion of the velocity vzν with an in-plane momentum k∥. Specifically, we find that, in the Nb/Ni/Nb junctions, this mechanism becomes relevant for junctions thicker than 50 Å, see Fig. 3. In SFS junctions the combination of disorder in the ferromagnet, interface scattering as well band-structure-induced dephasing ultimately determines the magnitude of the supercurrent. However, we believe that band structure effects provide an upper bound on the magnitude of the critical current as a function of w.

Next, we present numerical results which support the qualitative discussion presented above.

Choice of MJJ stack

It is illuminating to compare our numerical simulations for the supercurrent with the experimental measurements involving quasi-ballistic MJJs. As previously discussed, we believe that the Nb/Ni/Nb junctions represent a good model system for which experimental data are readily available15,22,23. The best-performing stacks consist of Nb(110)/Cu/Ni(111)/Cu/Nb(110). Cu spacer layers seem to be essential to get strong supercurrent, likely because it prevents intermixing of the Ni and Nb. Our model junction simulates this geometry, via supercells in the plane normal to the stack to account for the lattice mismatch (see Fig. 1 and Supplementary Note 2), though we do not include the Cu layers. We anticipate that Cu spacers will mainly affect transmission matrix elements rather than the dependence of the supercurrent on ferromagnet thickness, which is the main focus of this work. Furthermore, as discussed before, the Cu spacers will suppress the direct interaction between the ferromagnet and the superconductor and reduce the inverse proximity effect justifying Andreev approximation for the boundary conditions, see Eq. (14). Therefore, we consider only the simplified Nb/Ni/Nb stack and vary the number of layers (atomic planes) of Ni. In addition, we also consider the effect of different crystallographic orientations of the Ni planes and investigate Nb/Ni(110)/Nb junctions in Supplementary Note 3.

Band structure of Nb/Ni/Nb junctions

To make a Nb/Ni superlattice, the unit cells of the Nb and Ni regions in the plane normal to the interface must be coincident. This is complicated by the severe lattice constant mismatch, and also the incompatibility of the (110) and (111) atomic planes. It is necessary to construct superlattices with Nb(110) and Ni(111) both rotated to the z-axis and with lattice vectors in the plane coincident. A supercell with nearly coincident vectors was found (see Supplementary Note 2 for details). By applying a small shear strain to the Ni, the lattice vectors are made exactly coincident. Figure 5a shows the Nb(110) surface supercell and the Ni(111) surface supercell with equal lattice vectors used to match the Nb/Ni interfaces. Each atomic plane of Ni(111) contains 14 atoms and each atomic plane of Nb(110) consists of 10 atoms. The atomic structure of the Nb(110)/Ni(111)/Nb(110) for 5 layers (atomic planes) of Ni is shown in Fig. 1c.

Fig. 5: Magnetization of Nb/Ni/Nb junction.
figure 5

a Top view of the Nb(110) and Ni(111) surface supercells used to build the Nb/Ni interfaces and the Nb(110)/Ni(111)/Nb(110) stacks shown in Fig. 1c. The surface supercell is defined from two 2D vectors a1 = [10.8, 0] Å and a2 = [6.03, 7.11] Å with periodic boundary condition in the 2D plane. The corresponding reciprocal space defines the 2D k∥ vectors used in the calculations. Each atomic plane of Ni (Nb) contains 14 (10) atoms of Ni (Nb). b Magnetic moment profile of Nb(110)/Ni(111)/Nb(110) junctions for different thicknesses of Ni (from 3 to 9 layers). The value of the moment is an average over the moments of the 14 Ni atoms in each atomic plane. Note the magnetic dead layer at the Nb/Ni interfaces and that all moments vanish for the shortest junction made of 3 layers of Ni.

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Next, we performed self-consistent DFT calculations within the local density approximation (LDA) in order to obtain the relaxed structure and corresponding electronic structure. For the smallest structures, we performed a constrained optimization. Only the atoms in the planes closest to the Nb/Ni interfaces are allowed to relax to facilitate the stacking of arbitrarily large cells. The Nb/Ni interplanar spacing has also been optimized to minimize the total energy, see Supplementary Methods for more details.

Once the structure is determined, one can determine the normal-state thermodynamic and transport properties of the junction, e.g., calculate the magnetization profile and spin-resolved conductance through the junction as a function of Ni thickness. For transport calculations, we use a layer transport technique47 which employs the atomic spheres approximation (ASA). Careful checks were made of ASA band structures of elemental Nb and Ni, and also superlattices, to confirm that they are very similar to the full potential linear muffin-tin orbital (LMTO) DFT-LDA ones.

We find that the magnetic properties of Ni are sensitive to their local environment, indicative of the itinerant ferromagnetism. As shown in Fig. 5b, the magnetization profile is non-uniform in the junction with averaged magnetic moments per atom being suppressed near the Nb interface. For thickness larger than 4 layers, one recovers the bulk value of ~0.6μB in the middle layers, away from the Nb/Ni interfaces. The averaged moment drops down towards the edges and becomes considerably reduced down to ~0.1μB at the interface with Nb. The strong reduction of magnetism is exemplified for the short junction with three layers of Ni where the moments on the Ni atoms have completely vanished. Such a non-uniform magnetic moment dependence in Nb/Ni/Nb junctions affects superconducting properties of the SFS junctions in a non-trivial way. For example, Nb/Ni/Nb junctions thinner than four layers of Ni behave as essentially SNS junctions.

It is well-known that the LDA tends to overestimate local moments M in itinerant magnets48 because spin fluctuations reduce the average moment49, and underestimate M when local moments are very large46. For Ni, LDA yields M in good agreement with the experiment, but this is likely an artifact of an accidental cancellation of errors. Most important for transport is the exchange splitting Vex, which the LDA predicts to be 0.6 eV, about twice larger than the experimental value of 0.3 eV50. It is possible to reproduce both M and Vex at the same time, but a high-level theory, potentially including spin–orbit coupling, is needed to surmount both kinds of errors inherent in the LDA46,51. The high cost and poor scaling of such a theory are not practical for these junctions, so we elect to stay within the LDA and scale the self-consistently calculated Vex. This was the approach Karlsson and Aryasetiawan used to calculate the spin-wave spectra in Ni52. Scaling of Vex can be accomplished using different approaches, e.g., by adding some effective magnetic field to simulate the effect of spin fluctuations. Since Ni is a simple case with a nearly linear relation between M and Vex, the band structure hardly depends on the details in which the LDA potential is modified. Here, we first perform fully self-consistent calculations. Then, to construct the potential for transport properties, we rescale the spin component of the density by a constant factor, which we denote as M/M0. This enables parametric studies of transport as a function of Vex. M/M0 = 0.5 yields the observed Vex = 0.3 eV, and we use this scaling unless stated otherwise.

The conductance per unit of area, G/A, is shown in Fig. 3b. It is weakly dependent on the thickness, w, of the magnetic layer, as expected for a metallic system in the absence of disorder.

Superconducting transport

We now focus on superconducting properties. The dependence of the supercurrent on the phase difference ϕ for 5, 8, and 11 Ni layers is shown in Fig. 3c–e. One can present current-phase relation, J(ϕ) in a form of a Fourier series

$$J(\phi )=\mathop{\sum}\limits_{n\geqslant 1}{J}_{n}\sin (n\phi ).$$
(5)

In the 0-junction mode, the first term in the Fourier series dominates with J1 > 0 [see solid black line in Fig. 3c]. In the π-junction case [Fig. 3e], the supercurrent is also mostly defined by the first harmonic but with J1 < 0. Close to the 0–π transition J1 dies out so that the behavior is governed by higher Fourier harmonics53, e.g., for 8 Ni layers supercurrent has mostly second harmonic, J2, shown by the dashed black line in Fig. 3d.

Figure 6 shows the critical current density, \({J}_{{{{\rm{c}}}}}=\mathop{\max }\limits_{\phi }| J(\phi )|\), normalized by normal-state conductance, eJcA/GΔ, as a function of w (blue squares). Far from the 0–π transitions, the critical current coincides with the absolute value of the first Fourier harmonic, e∣J1∣A/GΔ (red circles). Since J1 contains a sign of the current and has better numerical stability than Jc, we use this quantity for the analysis. We exclude very thin junctions (three Ni layers or less) from the analysis since the magnetic properties are suppressed there. In particular, for the first data point in Fig. 6 corresponding to three layers of Ni the magnetization is completely suppressed [see Fig. 5b] and the ratio eJcA/GΔ ≈ 2.2 is significantly higher than the one for thicker Ni regions with non zero magnetic moments. We compare this value with the result for the short disordered SNS junction. The combination of analytical energy spectrum54 with Dorokhov distribution of channel transmissions55,56 leads to a ratio of 2.1 (horizontal dashed line in Fig. 6). We attribute this difference to the fact that there is no interfacial disorder in our model.

Fig. 6: Dependence of the supercurrent on Nb thickness.
figure 6

Comparison of critical current density, \(J_{\rm{c}}=\mathop{\max }\limits_{\phi }| J(\phi )|\) (blue squares), the absolute value of first Fourier component for supercurrent, ∣J1∣ [see Eq. (5), red circles], and its fitting \(|J_1^{\rm{fit}}|\) [Eq. (6), black curve]. All these quantities are “normalized” by normal-state conductance, G.

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The J1 dependence on w can be fit by the following expression

$${J}_{1}^{{{{\rm{fit}}}}}(w)={{{\Theta }}}_{J}\exp (-w/{\xi }_{J})\cos [\pi (w+{\delta }_{J})/{\lambda }_{J}],$$
(6)

where ΘJ = 3.20 A/μm2, ξJ = 41.1 Å, λJ = 23.2 Å, and δJ = − 2.75 Å are fitting parameters. We interpret ξJ as a decay length, λJ as the “half-period” of the oscillation in J1 as a function of w, and δJ as a measure of the suppressed magnetization in Ni layers near the Ni/Nb boundaries. \({J}_{1}^{{{{\rm{fit}}}}}(w)\) accurately fits the discrete points J1 as shown in Figs. 3a, 6, 7, and 10 by black solid line and black circles, accordingly.

Details of the supercurrent

In order to gain insight into the evolution of J1 with w, let us resolve contributions from different k∥. For this, we rewrite Eq. (3) as

$$J(\phi )=A\int _{{{{\rm{BZ}}}}}\frac{d{{{{\bf{k}}}}}_{\parallel }}{{(2\pi )}^{2}}\ j(\phi ,{{{{\bf{k}}}}}_{\parallel }),$$
(7a)
$$j(\phi ,{{{{\bf{k}}}}}_{\parallel })=-\frac{e}{\hslash }\ \frac{1}{A}\ \mathop{\sum}\limits_{\nu > 0}\frac{\partial {\varepsilon }_{\nu }(\phi ,{{{{\bf{k}}}}}_{\parallel })}{\partial \phi }.$$
(7b)

Here, A = 76.75 Å2 is the area of the surface supercell shown in Fig. 5a. Similar to Eq. (5), we denote the first ϕ-harmonic of j(ϕ, k∥) as j1(k∥). The evolution of the first Fourier harmonic of the supercurrent J1 and j1(k∥) as a function of w is shown in Fig. 7. Calculations were performed for two different sets of k∥ with 290 discrete k∥-points (full black circles) and 4142 k∥-points (empty black circles). One can see that both sets give the same result for J1, establishing that the k∥ integration is well converged. In Fig. 7, “errorbars” denote the standard deviation in j1(k∥) with respect to the k∥-summed average, J1(ϕ). Individual j1(k∥) is shown by semi-transparent horizontal dashes. The important observation is that while J1 decays with w, the dispersion in j1(k∥) does not change significantly.

Fig. 7: First Fourier harmonic, J1, of the supercurrent as a function of junction thickness, w.
figure 7

Full black circles correspond to J1 calculated with 290 k∥-points, empty circles (mostly superposed onto the full black circles) correspond to 4142 k∥-points. A solid black line is \({J}_{1}^{{{{\rm{fit}}}}}(w)\) [J1 fit given by Eq. (6)]. Green semitransparent dashes show \({J}_{1}^{{{{\rm{fit}}}}}(w)\) contributions [Eq. (7b)] for the individual k∥-points. The vertical “errorbars” correspond to the standard deviation of j1(k∥) with respect to J1. The standard deviation is the same for both sets of k∥-point indicating that these results are independent of the chosen discretization.

Full size image

Figure 8 shows color plots of j1(k∥) corresponding to local extrema of \({J}_{1}^{{{{\rm{fit}}}}}(w)\) (4, 13, 25, 36, 48, and 60 layers labeled in Fig. 7). For small w, one can see that most of all the k∥ contributions to J1 have the same sign, i.e., positive in the 0-junction regime and negative in the π-junction regime. In this regime, the decay is predominantly due to evanescent modes decaying into the junction. For larger w, the dephasing mechanism becomes important since the phase offset spread grows with w. One can observe the apparition of contributions of the opposite sign for w ≳ 50 Å. This dephasing mechanism is mainly due to the variation of the Fermi velocity with k∥, and becomes more important with increasing w.

Fig. 8: Colorplot of j1(k∥) with k∥ = (kx, ky).
figure 8

Each panel corresponds to the local extrema of \({J}_{1}^{{{{\rm{fit}}}}}(w)\) shown in Fig. 7. For 4 and 13 layers, all the k∥ channels contribute with the same sign. For 48 layers and larger, different k∥ channels lose synchronization and contribute to the total supercurrent with different signs.

Full size image

In order to study the distribution of the phase offsets and decay exponents for different modes, we fit the individual j1(k∥) using an expression analogous to Eq. (6). The set of the resulting fitted curves \({j}_{1}^{{{{\rm{fit}}}}}(w,{{{{\bf{k}}}}}_{\parallel })\) for 4142 k∥-points are shown by the green semitransparent curves in Fig. 3a. Here, to minimize the numerical “noise,” j1(w, k∥) curves are smoothed over the 2D k∥ space using Gaussian filter with \({\sigma }_{{{{{\bf{k}}}}}_{\parallel }}=0.01\) Å−1 which is of the order of the Fermi wave vector in Nb. Thus, each data point j1(w, k∥) approximately corresponds to a transverse conducting channel. This fitting procedure works reasonably well, e.g., the relationship in Eq. (7a) holds if one replaces J(ϕ) by \({J}_{1}^{{{{\rm{fit}}}}}(w)\) and j(ϕ, k∥) by \({j}_{1}^{{{{\rm{fit}}}}}(w,{{{{\bf{k}}}}}_{\parallel })\) for w ≳ 10 Å.

The distribution of the fitting parameters for \({j}_{1}^{{{{\rm{fit}}}}}(w,{{{{\bf{k}}}}}_{\parallel })\) is shown in Fig. 9. Histograms for decay lengths and half-periods reveal a complicated picture describing different contributions to the supercurrent in real materials. First of all, in Fig. 9a one can see the distribution of the half-periods, λj, which is similar to a Gaussian distribution with a mean value 〈λj〉 = 23.2 Å and standard deviation 2.8 Å. The mean value 〈λj〉 is very close to λJ [see text after Eq. (6)] while the spread in λj leads to dephasing and is responsible for the exponential decay of J1 at large w. Indeed, it is well-known that the average of an oscillatory function with respect to a random fluctuating phase (described by a Gaussian distribution) results in an exponentially decaying function.

Fig. 9: Fitting parameters for model supercurrent.
figure 9

Analysis of the fitting parameters of \({j}_{1}^{{{{\rm{fit}}}}}(w)\) for individual k∥, shown in Fig. 3a by semitransparent green lines. Distribution of a half-periods, λj, and b decay lengths, ξj.

Full size image

In addition to the dephasing mechanism, the decay of the supercurrent originates from the evanescent modes. The histogram for decay lengths, ξj is shown in Fig. 9b. Here, small ξj corresponds to fast-decaying \({j}_{1}^{{{{\rm{fit}}}}}(w,{{{{\bf{k}}}}}_{\parallel })\), large ξj is responsible for non-decaying modes (i.e., modes with the decay exponents larger than the junction thickness). The right-skewed distribution of the decay exponents has a mean value of 〈ξj〉 = 108 Å which is much larger than ξJ in Eq. (6). The shoulder at small ξj presumably corresponds to the evanescent mode decay comprising of d bands 2 and 3, see Table 1, whereas the tail at large ξj originates predominantly from band 6. Overall, one can see that a fit with a single decay exponent, discussed in Eq. (6), is quite oversimplified for an Nb/Ni/Nb junction considered here.

Exchange splitting

We now turn to the discussion of the effect of exchange splitting energy on the supercurrent in MJJs. So far we have used M/M0 = 0.5, which yields the experimentally observed Vex = 0.3 eV. It is interesting to investigate how a ferromagnet with a different Vex (but otherwise the same band structure as Ni) would affect the w-dependence of J1. In Fig. 10, we show the results for parametric variations in M/M0. One can see in Fig. 10a that the half-period, λJ, and the decay length, ξJ, strongly depend on M. Here, black points correspond to M/M0 = 0.5, and the self-consistent calculations with no rescaling correspond to M/M0 = 1. In order to understand how λJ and ξJ depend on M, we perform the fitting procedure Eq. (6) for different magnetic moments and plot, in Fig. 10b, the supercurrent density as a function of the rescaled thickness, \((w+{\delta }_{J})/2{\tilde{\lambda }}_{J}\) with \({\tilde{\lambda }}_{J}=({M}_{0}/M)\ 11.6\) Å. Remarkably J1, as a function of the rescaled thickness, collapses to the same universal curve. The inset demonstrates that the fitted half-period, λJ, is proportional to 1/M showing that the oscillation period scales linearly with the inverse of Vex in this parameter range.

Fig. 10: Magnetic moments rescaling.
figure 10

J1 for different rescaling of the magnetic moments M/M0 as a function of a the thickness, w, and b the rescaled thickness, \((w+{\delta }_{J})/2{\tilde{\lambda }}_{J}\), where δJ is the fitting parameter in Eq. (6) and \({\tilde{\lambda }}_{J}=({M}_{0}/M)\ 11.6\) Å. J1 values are shown by circles; corresponding fittings \({J}_{1}^{{{{\rm{fit}}}}}(w)\) are shown by lines of the same color. (Inset) Half-period, λJ, fitted using Eq. (6) vs. \({\tilde{\lambda }}_{J}\).

Full size image

Note that deviations from the linear regimes become significant for M/M0 ≳ 1.7. For the clarity of the data, we do not show the results for M/M0 > 1 in Fig. 10b.

Different orientations of Ni plane

We now discuss the difference in crystal orientation in Nb/Ni/Nb junctions. We have performed calculations for Nb/Ni/Nb junctions built from stacking the Ni atomic planes in the (110) orientation instead of the (111) orientation. The results are given in Supplementary Note 3. Qualitatively, the same physics holds for both stacks built from (111) and (110) Ni planes. However, our calculations show that the actual value for the period of oscillation and for the current decay depend crucially on the details of the electronic structure of the junctions, such as the relative crystal orientation.

Spin–orbit coupling

Finally, we considered the effect of spin-orbit coupling in SFS junctions. Spin–orbit coupling leads to the mixing of the minority and majority channels and may change current–phase relationship. The interplay between Zeeman splitting and spin-orbit coupling have been discussed in ref. 45; the regime of interest is the Zeeman-field-dominated regime considered there. Indeed, we find that SOC in Ni is much smaller than the exchange splitting because of the low atomic number of Ni. As we show in Supplementary Note 4, the SOC in Ni-based MJJ considered here does not change the qualitative picture described above but rather leads to small quantitative changes to the Josephson current.

Discussion

In this paper, we identified two generic mechanisms for the decay of the supercurrent with junction thickness: (i) exchange-splitting induced gap opening for minority or majority carriers and (ii) dephasing between different modes due to the significant quasiparticle velocity dispersion with the transverse momentum. It was previously believed that disorder in the ferromagnet is mainly responsible for the supercurrent decay in SFS junctions. In the present work, we have shown that band structure effects also contribute to the critical current suppression and thus provide an upper bound for the supercurrent in an ideal (i.e., disorder-free) structure.

We found that the Nb/Ni/Nb junction is a suitable system for comparison with the simulations because of the long mean free path in Ni relative to the junction thickness and the quasi-ballistic nature of quasiparticle propagation in the ferromagnet. We have found good agreement with published experimental data for the half-period of the critical current oscillations: λJ ≈ 23 Å [see Eq. (6) and text after it] vs. ≈ 26 Å in the experiment, ref. 22. We have also found that the critical current decays exponentially with the ferromagnet thickness w. This is to be contrasted with previously assumed algebraic decay based on results for the clean SFS junctions using a simple parabolic-like band structure. We believe that in measured Nb/Ni/Nb junctions with w ≲ 50 Å the mechanism (i) is likely to be responsible for the supercurrent decay. This finding is crucial for the material and geometry optimization of MJJs and superconducting magnetic spin valves.

Understanding the interplay of band structure effects and disorder in MJJs is an interesting open problem. We believe that interfacial disorder due to, for example, the surface roughness will mix different k∥ modes and will lead to a larger spread of half-periods. This, in turn, will further enhance the dephasing mechanism (ii) of the supercurrent decay discussed here. Strong disorder in the bulk (i.e., mean free path much smaller than junction thickness w) would lead to the diffusive motion of quasiparticles in the ferromagnet which is a significant departure from the quasi-ballistic junction limit considered here. We think that bulk disorder would induce even more dephasing between different modes because phase offsets in this case will depend on different random trajectories of minority and majority carriers. We, therefore, believe that bulk disorder will lead to even stronger decay of the supercurrent with junction thickness, w.

Methods

We develop a numerical method to perform realistic simulations of MJJs using a combination of first-principles DFT and BdG calculations. The former is used to obtain the normal-state properties (e.g., band structure, Fermi velocities, and magnetization) and to calculate the normal scattering matrices through the inhomogeneous 3D realistic junctions. As a next step, we take superconductivity into account and calculate supercurrent through the stack assuming the short junction limit.

Normal transport: ab initio description

To calculate the normal scattering matrix we use the Questaal package for electronic structure calculations based on the LMTO method57. It calculates the full non-linear, i.e., nonequilibrium, transport properties of an infinite system describing a central (C) region cladded by two semi-infinite left (L) and right (R) leads47,58, as represented below

$$\overbrace{\ldots | {{{\mathcal{L}}}}| {{{\mathcal{L}}}}}^{\begin{array}{c}{{{\rm{L}}}}\end{array}}\ | \ \overbrace{{{{{\rm{PL}}}}}_{0}| {{{{\rm{PL}}}}}_{1}| \ldots | {{{{\rm{PL}}}}}_{L-1}}^{\begin{array}{c}{{{\rm{C}}}}\end{array}}\ | \ \overbrace{{{{\mathcal{R}}}}| {{{\mathcal{R}}}}| \ldots }^{\begin{array}{c}{{{\rm{R}}}}\end{array}}$$

The LCR system [Fig. 1c] can be partitioned into an infinite stack of principal layers (PLs) which interact only with their nearest neighbors. This is possible because the screened LMTO structure constants are short-ranged59. In the present case, the C region consists of the ferromagnet, plus two layers of Nb at the LC and CR interfaces respectively. This is the range over which the perturbation from C significantly modifies the potential in the L or R region. To construct the Nb/Ni/Nb stack, coincident site lattices for Ni and Nb must be found (details of how this was accomplished are given in Supplementary Note 2). Planes of coincident site lattices are stacked to form the Nb/Ni/Nb structures. Figure 5a shows the Nb and Ni planes we used, which are denoted here as “surface supercells.”

The electronic current flows along the z-direction, perpendicular to the PLs lying in the xy-plane (transverse direction), see Fig. 1c. Periodic boundary conditions are used within each PL. The corresponding reciprocal space defines the two-dimensional (2D) k∥ vectors, i.e., the transverse modes, used in the calculations. The k∥ mesh is discretized and integrals over k∥ are performed numerically.

The electronic structure of the C region can be separated from L and R regions through self-energies, ΣL and ΣR, that modify the Hamiltonian of the C region. They are most easily calculated if the potential of each PL in the L or R region is identical all through the bulk region. This is the reason for adding a few Nb layers folded into the C region. Thus the periodically repeating unit cells in the L and R regions can be safely assumed to have the potential of the bulk crystal. To construct the self-energies, the potentials of the PL in an infinite stack are needed. These potentials are functions only of the PL in their own region and may be calculated in several ways. ΣL and ΣR are obtained from “surface” Green’s function (a fictitious system that consists of a semi-infinite stack of PL, each with the same potential). Note that the potential of the C region is calculated self-consistently. This is important, as the local moments of Ni are small at the boundary layers, and build up gradually, see Fig. 5b.

With the potential in hand, the normal-state transport can be calculated using scattering formalism58,60. For this, knowing the retarded Green’s function, \({{{{\mathcal{G}}}}}^{{{{\rm{r}}}}}\), of the junction is sufficient. However, this is not the case for the Josephson current: the individual eigenfunctions are required. Within a Green’s function framework, \({{{{\mathcal{G}}}}}^{{{{\rm{r}}}}}\) must be organized by normal modes which correspond to the eigenstates of the L and R leads for a prescribed energy E. In the PL representation, the Hamiltonian has been discretized into the linear combinations of the LMTO basis functions, and the normal modes are represented as eigenvectors of these basis functions. The Schrödinger equation becomes a differential equation in the PL basis functions61. Eigenvectors are calculated by solving a quadratic eigenvalue problem61,62, whose eigenvalues correspond to \(\exp (\pm i{k}_{z,n}a)\), where a is the thickness of the PL. The wavenumber kz,n of the normal mode n can be complex, but to correspond to a propagating mode kz,n must be real. By solving the equation as a function of the energy E, one gets all the eigenvalues and eigenvectors which provide the information needed to construct the self-energies ΣL and ΣR (for each k∥ and each spin σ). Note that, in the mode basis, the imaginary part of the self-energies is proportional to the (band) velocity of the modes, and is diagonal for non-degenerate modes62,63.

The retarded Green’s function \({{{{\mathcal{G}}}}}^{{{{\rm{r}}}}}\) of the C region (connected to the L and R leads) can be written as a matrix in the normal mode basis,

$$\begin{array}{l}{{{{\mathcal{G}}}}}_{\sigma }^{{{{\rm{r}}}}}(E,{{{{\bf{k}}}}}_{\parallel })=\left\{{[{{{{\mathcal{G}}}}}_{{{{\rm{C}}}},\sigma }^{{{{\rm{r}}}}}(E,{{{{\bf{k}}}}}_{\parallel })]}^{-1}\right.\\ \qquad\qquad\qquad{\left.-{{{\Sigma }}}_{{{{\rm{L}}}},\sigma }(E,{{{{\bf{k}}}}}_{\parallel })-{{{\Sigma }}}_{{{{\rm{R}}}},\sigma }(E,{{{{\bf{k}}}}}_{\parallel })\right\}}^{-1},\end{array}$$
(8)

where \({{{{\mathcal{G}}}}}_{{{{\rm{C}}}}}^{{{{\rm{r}}}}}\) is the Green’s function of the isolated C region. On this basis, \({{{{\mathcal{G}}}}}^{{{{\rm{r}}}}}\) is decomposed into four blocks

$${{{{\mathcal{G}}}}}_{\sigma }^{{{{\rm{r}}}}}(E,{{{{\bf{k}}}}}_{\parallel })=\left[\begin{array}{ll}{{{{\mathcal{G}}}}}_{{{{\rm{LL}}}},\sigma }^{{{{\rm{r}}}}}(E,{{{{\bf{k}}}}}_{\parallel })&{{{{\mathcal{G}}}}}_{{{{\rm{LR}}}},\sigma }^{{{{\rm{r}}}}}(E,{{{{\bf{k}}}}}_{\parallel })\\ {{{{\mathcal{G}}}}}_{{{{\rm{RL}}}},\sigma }^{{{{\rm{r}}}}}(E,{{{{\bf{k}}}}}_{\parallel })&{{{{\mathcal{G}}}}}_{{{{\rm{RR}}}},\sigma }^{{{{\rm{r}}}}}(E,{{{{\bf{k}}}}}_{\parallel })\end{array}\right]\ ,$$
(9)

upon projecting onto the propagating modes of the L and R leads. These four quantities and the mode velocities completely determine the normal state transport properties of the junctions.

The transmission matrices are defined by the off-diagonal parts of Eq. (9). More specifically, the transmission coefficients \({[{t}_{{{{\rm{LR}}}},\sigma }]}_{nm}\), connecting L and R regions, are given by60

$$\begin{array}{l}{[{t}_{{{{\rm{LR}}}},\sigma }]}_{nm}(E,{{{{\bf{k}}}}}_{\parallel })=i\sqrt{| {[{v}_{{{{\rm{L}}}},\sigma }]}_{n}(E,{{{{\bf{k}}}}}_{\parallel })| }\\ \qquad\qquad\qquad\qquad\times {[{{{{\mathcal{G}}}}}_{{{{\rm{LR}}}},\sigma }^{{{{\rm{r}}}}}]}_{nm}(E,{{{{\bf{k}}}}}_{\parallel })\ \sqrt{| {[{v}_{{{{\rm{R}}}},\sigma }]}_{m}(E,{{{{\bf{k}}}}}_{\parallel })| },\end{array}$$
(10)

where \({[{v}_{{{{\rm{L}}}},\sigma }]}_{n}\) and \({[{v}_{{{{\rm{R}}}},\sigma }]}_{m}\) are the velocity matrix elements for propagating modes n and m in the L and R leads, respectively. The transmission matrix tRL can be obtained from Eq. (10) by replacing L ↔ R. The reflection coefficients are given by the diagonal blocks of Eq. (9). For instance, on the L side

$$\begin{array}{l}{[{r}_{{{{\rm{LL}}}},\sigma }]}_{nn^{\prime} }(E,{{{{\bf{k}}}}}_{\parallel })=i\sqrt{| {[{v}_{{{{\rm{L}}}},\sigma }]}_{n}(E,{{{{\bf{k}}}}}_{\parallel })| }\\ \qquad\qquad\qquad\qquad\times {[{{{{\mathcal{G}}}}}_{{{{\rm{LL}}}},\sigma }^{{{{\rm{r}}}}}]}_{nn^{\prime} }(E,{{{{\bf{k}}}}}_{\parallel })\ \sqrt{| {[{v}_{{{{\rm{L}}}},\sigma }]}_{n^{\prime} }(E,{{{{\bf{k}}}}}_{\parallel })| }-{\delta }_{nn^{\prime} }.\end{array}$$
(11)

The reflection matrix rRR on the R side is obtained from Eq. (11) by replacing L ↔ R.

We define the normal scattering matrix as

$$S=\left[\begin{array}{llll}{r}_{{{{\rm{LL}}}},\uparrow }&0&{t}_{{{{\rm{LR}}}},\uparrow }&0\\ 0&{r}_{{{{\rm{LL}}}},\downarrow }&0&{t}_{{{{\rm{LR}}}},\downarrow }\\ {t}_{{{{\rm{RL}}}},\uparrow }&0&{r}_{{{{\rm{RR}}}},\uparrow }&0\\ 0&{t}_{{{{\rm{RL}}}},\downarrow }&0&{r}_{{{{\rm{RR}}}},\downarrow }\end{array}\right]\ ,$$
(12)

where we omit E- and k∥-dependence for brevity.

The linear-response normal conductance, Gσ, per spin is given by

$$\frac{{G}_{\sigma }}{A}=\frac{{e}^{2}}{h}\int _{{{{\rm{BZ}}}}}\ \frac{d{{{{\bf{k}}}}}_{\parallel }}{{(2\pi )}^{2}}\ \mathop{\sum}\limits_{n,m}{| {[{t}_{{{{\rm{LR}}}},\sigma }]}_{nm}({E}_{{{{\rm{F}}}}},{{{{\bf{k}}}}}_{\parallel })| }^{2}.$$
(13)

It is calculated at the Fermi energy, E = EF. The total conductance is given by G = G↑ + G↓. Figure 3b shows that G, G↑, and G↓ are almost independent of the junction thickness, w.

Superconducting transport: a scattering matrix approach

Equation (12) is the normal-state scattering matrix for metal–ferromagnet–metal (NFN) structure taking into account reflection at both NF interfaces. To account for superconductivity, we introduce a step-like superconducting pairing potential Δ = 3.1 meV and use the Andreev approximation to account for electron–hole scattering processes3,9. This approach combines the details of the atomic structure of Nb/Ni/Nb and superconductivity within the mean-field approximation.

The direct contact between S and F layers, as shown in Fig. 1a, would lead to an interaction between them. The back-action of the ferromagnet on the superconductor (i.e., the inverse proximity effect) results in a spatial dependence of the pairing potential near the SF interfaces64,65,66. In the case of a clean SFS junction model, the self-consistent BdG calculations have been discussed in refs. 67,68,69,70,71. In typical experimental systems the superconductor is disordered, so the disorder effect on pairing potential needs to be considered, see, e.g., ref. 72. Furthermore, in recent experiments S and F layers are separated by an intermediate spacer layer, which significantly reduces this inverse proximity effect. Thus, to understand the inverse proximity effect in realistic SFS devices one would need to include both of the above mentioned ingredients in the model as well as to take into account the inhomogeneous magnetization in the ferromagnet (see e.g. Fig. 5b) which is outside the scope of this paper. For the sake of clarity, we focus here on a realistic band structure in the ferromagnet and its effect on the supercurrent in the SFS structures. Henceforth, we also neglect the orbital effects of the fringe magnetic field, created by the ferromagnet.

Since Δ ≪ EF, spin-resolved Andreev reflection at SNL and NRS interfaces is described by

$${r}_{{{{\rm{A}}}}}(\phi )=\left[\begin{array}{llll}0&{{{\bf{1}}}}\ {e}^{i\phi /2}&0&0\\ {{{\bf{1}}}}\ {e}^{i\phi /2}&0&0&0\\ 0&0&0&{{{\bf{1}}}}\ {e}^{-i\phi /2}\\ 0&0&{{{\bf{1}}}}\ {e}^{-i\phi /2}&0\end{array}\right]\ ,$$
(14)

where ϕ is the phase difference between left and right superconducting leads and 1 is the identity matrix. In the short junction limit, the main contribution to the supercurrent comes from Andreev bound states localized in the junction having energy εν. The energy spectrum of Andreev states can be obtained using the following equation54,

$$\alpha (\varepsilon )\ \left[\ \begin{array}{ll}0&\ {r}_{{{{\rm{A}}}}}^{* }(\phi )\\ {r}_{{{{\rm{A}}}}}(\phi )\ \ &0\end{array}\ \right]\ \ \left[\ \begin{array}{ll}S({E}_{{{{\rm{F}}}}}\ +\ \varepsilon ,{{{{\bf{k}}}}}_{\parallel })\ \ \ &0\\ 0&\ \ {S}^{* }({E}_{{{{\rm{F}}}}}\ -\ \varepsilon ,{{{{\bf{k}}}}}_{\parallel })\end{array}\ \right]\ {{{\Psi }}}^{{{{\rm{in}}}}}\ =\ {{{\Psi }}}^{{{{\rm{in}}}}},$$
(15)

where \(\alpha (\varepsilon )=\sqrt{1-{\varepsilon }^{2}/{{{\Delta }}}^{2}}+i\varepsilon /{{\Delta }}\). The vector \({{{\Psi }}}^{{{{\rm{in}}}}}=\left[{\psi }_{{{{\rm{e}}}}\uparrow }^{{{{\rm{L}}}}\to },\right.\)\({\psi }_{{{{\rm{e}}}}\downarrow }^{{{{\rm{L}}}}\to },\,\)\({\psi }_{{{{\rm{e}}}}\uparrow }^{{{{\rm{R}}}}\leftarrow },\,\)\({\psi }_{{{{\rm{e}}}}\downarrow }^{{{{\rm{R}}}}\leftarrow },\,\)\({\psi }_{{{{\rm{h}}}}\uparrow }^{{{{\rm{L}}}}\to },\,\)\({\psi }_{{{{\rm{h}}}}\downarrow }^{{{{\rm{L}}}}\to },\,\)\({\psi }_{{{{\rm{h}}}}\uparrow }^{{{{\rm{R}}}}\leftarrow },\,\)\({\left.{\psi }_{{{{\rm{h}}}}\downarrow }^{{{{\rm{R}}}}\leftarrow }\right]}^{{{{\rm{T}}}}}\) corresponds to the electron- and hole-like (e/h) waves in NL and NR regions incident on the F region from the left (→) and from the right (←).

Simulations show that S is weakly-dependent on E in the range [EF − Δ, EF + Δ]. Therefore, we expand S(E, k∥) in E − EF and keep only the leading term, i.e., S(E, k∥) ≈ S(EF, k∥) = S(k∥). Using this approximation, one can simplify the quantization condition (15) and reduce it to the matrix eigenvalue problem (see details in ref. 73). This approach allows one to reliably calculate the Andreev bound states spectrum, εν(ϕ, k∥). The zero-temperature supercurrent, J, through the junction is given by Eq. 3. Figure 3c–e shows J as a function of a phase difference, ϕ. Figure 3a shows the first Fourier harmonic of the supercurrent as a function of the junction thickness, w.