Introduction

S-wave singlet superconductivity and ferromagnetism are competing phases. Over the past half century considerable research has been undertaken in order to understand the interaction between these phenomena at superconductor/ferromagnet (S/F) interfaces1,2,3,4,5,6,7. A key experimental development was the demonstration of F-thickness-dependent oscillations in the Josephson critical current Ic in S/F/S junctions, first using weak ferromagnets (CuNi and PdNi8,9,10,11,12,13) and then strong ferromagnets (Fe, Co, Ni and NiFe14,15,16,17,18,19). This behaviour is a manifestation of the magnetic exchange field from F acting differentially on the spins of the singlet pairs, which induces oscillations in the superconducting order parameter in F superimposed on a rapid decay with a singlet coherence length of ξs < 3 nm10,15,17,19. The superconductivity in F can be detected via tunnelling density of states (Do S) measurements20,21 and point contract Andreev spectroscopy22,23. Furthermore, the magnetic exchange field from F induces a spin-splitting of the DoS in S close to the S/F interface24,25,26, which can potentially open triplet chanels in S materials over the length scale of ξF27,28.

Recently there is a focus on Josephson junctions with inhomogeneous barrier magnetism, involving misaligned F layers29,30,31,32,33,34 and/or rare earth magnets such as Ho or Gd35,36, in order to transform singlet pairs into spin-aligned triplet pairs3,6,37. Triplet pairs are spin-polarized and stable in a magnetic exchange field and decay in Fs over length scales exceeding ξs3,5. However, the relatively large (total) magnetic barrier thickness in triplet junctions introduces significant flux which, in combination with magnetic inhomogeneity, creates a complex dependence of Ic on external magnetic field H38,39.

A complication for junctions with magnetically inhomogeneous rare earths such as Ho (or Er) relates to the fact that the magnetic ordering and local magnetic susceptibility χ depends on a competition between Ruderman–Kittel–Kasuya–Yosida (RKKY) coupling between localized moments and shape anisotropy40. Let us take Ho an example. In single crystals the moments align into an antiferromagnetic spiral below 133 K made up of F-ordered basal planes with moments in successive planes rotated 30° relative to each other due to the RKKY coupling41,42. Below 20 K the moments in Ho tilt out-of-plane although this is not observed in thin film due to strain43. The antiferromagnetic spiral has a zero net magnetic moment but applying magnetic fields parallel to the basal planes44,45 induces an irreversible transition to a ferromagnetic state. In epitaxial thin-films, similar properties are reproduced although the antiferromagnetic spiral can remain stable over a wide field range46. In textured or polycrystalline thin films the antiferromagnetic spiral can remain reversible even after applying magnetically saturating fields47. At the edges of Ho, however, RKKY coupling is reduced which may favour easy magnetization alignment along edge regions. This translates to localized enhancements in χ at edges and thus an inhomogeneous magnetic induction in the junction.

In this paper we calculate the magnetic-field-dependence of the maximum Josephson critical current Ic in S/F/S junctions with a position- and magnetic-field-dependent-χ (Fig. 1). The model predicts anomalous Fraunhofer patterns due to spatial variations in χ and magnetic induction in which local minima in Ic(H) can be nonzero and non-periodic due to interference between magnetised and demagnetised regions.

Figure 1
figure 1

Magnetization process for an S/F/S Josephson junction with a position (x) and magnetic field (H) dependent magnetic susceptibility χ(x, H) and magnetization M(x, H). (a) For H = 0 the net barrier moment is zero everywhere but on increasing H (b–d), M increases faster at the junction edges and propagates inwards until the barrier moment saturates (H = Hs) (d). (e) Spatial variation of magnetic induction B and superconducting phase difference ϕ for 0 < H < Hs. The external field H is applied in the y direction. The variables (M, ϕ, H, B) are plotted on the z-axisand labelled.

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The S/F/S junction geometry under consideration is sketched in Fig. 1 which summarizes the magnetization process. We consider the case of a Josephson junction with a width L that is smaller than the Josephson penetration depth (which is usually the case for experiments), so the magnetic field H fully penetrates the barrier48. Following standard procedures (see e.g.49), we calculate the phase variation across the S/F/S barrier taking into account the contribution from the magnetic moment to the total flux through the junction during the magnetization process as summarised in Fig. 1(e). Applying H parallel to y causes the magnetization M along junction edges parallel to y to propagate inwards towards the junction centre until magnetic saturation H = Hs. The expansion of the magnetized region is assumed to be reversible with a width that depends on H and not magnetic field history. The propagation rate of the magnetized region is linear with H in our model and the position of the boundary between magnetized and demagnetized regions is a = L/2 – PH (where P is the propagation parameter and L the junction width). The magnetization is uniform in the y direction and position-dependent in the x direction with M(x) = χ(x)H. We note that for certain materials the propagation rate of the magnetized region with H may not be linear, but as a first approximation we choose a linear form here.

A spatial variation in M(x) means that the magnetic induction B(x) is also non-uniform. The line integral of B(x) across the junction gives the spatial gradient of the superconducting phase

$$\frac{\partial \phi }{\partial x}=\frac{2\pi {\int }^{}Bdz}{{{\rm{\Phi }}}_{0}}=\frac{2\pi [\bar{d}{\rm{H}}+4\pi \chi dH]}{{{\rm{\Phi }}}_{0}},$$
(1)

where Φ0 is the flux quanta [h/(2e) ≈ 2.06 × 10−15 Wb] and \(\bar{d}=d+2\lambda \) is the effective junction thickness. Hence, φ(x) in the magnetized (a < |x| < L/2) and demagnetized (\(|x| < a\)) regions is given by

$$\phi (x)=\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{x}{L}+{\phi }_{0},|x| < a\,{\rm{w}}{\rm{h}}{\rm{e}}{\rm{r}}{\rm{e}}\,{\rm{\Phi }}=HL\bar{d},$$
(2)
$$\phi (x)=\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{(x-a)}{L}(1+4\pi {\chi }_{0}\frac{d}{\bar{d}})+\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{a}{L}+{\phi }_{0},\,a < |x| < L/2$$
(3)

where φ0 is a constant that is set to give the maximum total critical current through the junction. The second term in equation (3) ensures φ(x) is continuous. The spatial variation of the magnetic parameters and the superconducting phase difference are sketched in Fig. 1(e).

The position-dependent current density j(x) in the magnetized and demagnetized regions are

$$j(x)={j}_{c}\ast \,\sin [\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{x}{L}+{\phi }_{0}],\,{\rm{f}}{\rm{o}}{\rm{r}}\,|x| < a$$
(4)
$$j(x)={j}_{c}\ast Q\ast \,\sin [\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{(x-a)}{L}(1+4\pi {\chi }_{0}\frac{d}{\bar{d}})+\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{a}{L}+{\phi }_{0}],\,{\rm{f}}{\rm{o}}{\rm{r}}\,a < |x| < \frac{L}{2}$$
(5)

where jc is the maximum critical current density in the demagnetized region and Q is the ratio of the critical current densities in the magnetized and demagnetized regions - i.e. \(Q={j}_{m,c}/{j}_{c}.\) The net exchange field in the magnetised regions can favour a transition to a π-state50,51 and hence the directions of jc and jm,c can be opposite to each other meaning Q can be negative. The total critical current through the junction is thus \(I={\int }_{\frac{w}{2}}^{\frac{w}{2}}{\int }_{-L/2}^{L/2}j(x)dxdy=w{\int }_{-L/2}^{L/2}j(x)dx\), where w is the junction width in the y direction. From symmetry, the maximum critical current is therefore achieved by setting φ0 = π/2 which yields

$$\begin{array}{c}{I}_{c}(f)=2w{\int }_{0}^{\frac{L}{2}}j(x)dx\,=\\ 2w{j}_{c}\{{\int }_{0}^{a}\sin [\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{x}{L}+\frac{\pi }{2}]dx+Q{\int }_{a}^{\frac{L}{2}}\sin [\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{(x-a)}{L}(1+4\pi {\chi }_{0}\frac{d}{\bar{d}})+\frac{2\pi {\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\frac{a}{L}+\frac{\pi }{2}]dx\}.\end{array}$$
(6)

To help illustrate the general features of our model, we introduce the following dimensionless parameters: the relative position of the boundary between the magnetised and demagnetised regions l = a/L, the effective permeability \(q=4\pi {\chi }_{0}\frac{d}{\bar{d}}+1\), and the normalised flux \(\,{f}=\frac{{\rm{\Phi }}}{{{\rm{\Phi }}}_{0}}\). Substituting these parameters into equation (6) gives the following expression for Ic

$${I}_{c}=2{I}_{c0}{\int }_{0}^{l}\cos [2\pi f\tilde{x}]d\tilde{x}+2{I}_{c0}Q{\int }_{l}^{\frac{1}{2}}\cos [2\pi f(\tilde{x}-l)q+2\pi fl]d\tilde{x},$$
(7)

where \({I}_{c0}=Lw{j}_{c}\) is the H = 0 total critical current of the junction and \(l=0.5-pf\) with \(p=\frac{{{\rm{\Phi }}}_{0}}{\bar{d}{L}^{2}}P\). Calculating the integrals analytically we obtain

$${I}_{c}=\frac{{I}_{c0}}{\pi f}\{\sin (2\pi f(1/2-pf))+\frac{Q}{q}[\sin (2\pi f(1/2+pf(q-1)))-\,\sin (2\pi f(1/2-pf))]\}$$
(8)
$$=\frac{{I}_{c0}}{\pi f}[\sin (\pi f(1-2pf))+2\frac{Q}{q}\,\sin (\pi {f}^{2}pq))\cos (\pi f(1+pf(q-2)))].$$
(9)

For p = 0, meaning the junction is demagnetised for all values of H, we recover the standard Fraunhofer relation \({I}_{c}(f)={I}_{c0}\frac{\sin \,\pi f}{\pi f}\). The solution takes the same form when the magnetised and the demagnetised regions are equivalent – i.e. Q = 1 and q = 1 for all values of p.

At magnetic saturation f is 1/2p meaning equation (7) is only valid for \(|f|\) < 1/2p. For \(|f|\ge \) 1/2p, the barrier is magnetised with a high effective permeability q with \({I}_{c}(f)={I}_{cm}\frac{\sin (\pi fq)}{(\pi fq)}\), where Icm is the total critical current in the magnetized state and \({I}_{cm}=Lw{j}_{cm}=LwQ{j}_{c}\). The shape of Ic(f) is thus determined by Q, p and q and its magnitude by jc and the junction area. In Fig. 2 we have plotted example Ic(f) patterns.

Figure 2
figure 2

Ic(f) vs p and f for positive and negative Q. The blue curves show Ic(f) for q = 3, Q = −1 (a) and Q = 1 (b). The red curves show standard Fraunhofer patterns for a demagnetized junction (p = 0).

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When the susceptibilities in the magnetized and demagnetized regions are different, we observe an interference in the critical current. However, due to the movement of the boundary between the magnetized and demagnetized regions with field, Ic(f) is more complicated than simply the superposition of two sinc functions. Due to phase oscillations in the magnetised regions, the field position and number of local minima and maxima that appear in Ic(f) deviate from a non-magnetic junction with non-periodic behaviour. Furthermore, the magnitudes of Ic at local minima are not always 0 and Ic at local maxima do not decrease inversely with f as expected but can even increase. Once the barrier is fully magnetised (f > 1/2p), we recover standard Ic(f) behaviour with periodic minima and peaks in Ic(f) with peak heights decreasing inversely with f.

The parameters q, p and Q, influence Ic(f) in different ways. For Q close to 1 the jc in the magnetized and demagnetized regions closely match, but in the magnetized regions the superconducting phase oscillates faster. In the magnetized regions Ic quadratically decreases with f for small H (f ≪ 1) and the central peak is rounded, resembling a sinc-type function. For Q far from 1 or negative, jc differs in the magnetized and demagnetized regions. For small H (f ≪ 1), Ic is mainly determined by the propagation of the magnetised region and, because the demagnetised region shrinks linearly, Ic decreases linearly and the central peak is sharp. The difference in the shape of the central peak for Q = 1 and Q = −1 is demonstrated in Fig. 2.

The other two parameters p and q affect the position of the minima and maxima as illustrated in Fig. 3 which shows the position of the local minima and maxima of Ic(f) for multiple sets of parameters. The main effect of q on Ic(f) related to the spacing between minima and maxima. In general, higher values of q bring minima and maxima closer to the origin (H = 0) since the higher permeability in the junction causes the superconducting phase to oscillate faster with f in the magnetised regions. However, for Q close to 1, some pairs of minima converge towards each other and the maximum between them disappear.

Figure 3
figure 3

The positions of local maxima and minima in Ic(f). (a,b) show the positions of the minima (blue) and the maxima (red) of Ic(f) with increasing q. (c–f) illustrate the movement of the minima and the maxima as p changes for q  =  3 (c–d), and q  =  1.5 (e–f). The grey lines indicate the field values where the barrier is magnetised fully. In the fully magnetized regions (shaded grey), a standard Ic(f) Fraunhofer behaviour is observed.

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The influence of p on Ic(f) is most significant for p < 0.2. In this range, small changes in p affect the shape of Ic(f) significantly: multiple minima in Ic(f) combine and some minima split into two minima forming a maximum (Fig. 3). For Q < 0, there is a minimum-maximum pair forming just below p < 0.2 (exact value depends on q and Q). For p > 0.2, the shape of Ic(f) weakly depends on p since the magnetized regions propagate rapidly with H and the magnetized regions dominate Ic(f).

Conclusions

We have presented a generalised model to predict the behaviour of IC(H) Fraunhofer patterns in magnetic Josephson junctions with a non-uniform magnetic susceptibility that peaks at junction edges. An analytical expression for Ic(H) is derived and key parameters which describe the shape of Ic(H) are identified: the effective magnetic permeability q of the magnetised region; the propagation p of the magnetised region into the demagnetized region; and Q, the ratio of the local critical current density in the magnetized and demagnetized regions. The calculations can be easily applied to understand the Ic(H) behaviour magnetically complex Josephson junctions with simultaneous zero and Pi states.