Abstract
The development of superconducting memory and logic based on magnetic Josephson junctions relies on an understanding of junction properties and, in particular, the dependence of critical current on external magnetic flux (i.e. Fraunhofer patterns). With the rapid development of Josephson junctions with various forms of inhomogeneous barrier magnetism, Fraunhofer patterns are increasingly complex. In this paper we model Fraunhofer patterns for magnetic Josephson junctions in which the barrier magnetic susceptibility is position and externalmagneticfield dependent. The model predicts anomalous Fraunhofer patterns in which local minima in the Josephson critical current can be nonzero and nonperiodic with external magnetic flux due to an interference effect between magnetised and demagnetised regions.
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Introduction
Swave 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) interfaces^{1,2,3,4,5,6,7}. A key experimental development was the demonstration of Fthicknessdependent oscillations in the Josephson critical current I_{c} in S/F/S junctions, first using weak ferromagnets (CuNi and PdNi^{8,9,10,11,12,13}) and then strong ferromagnets (Fe, Co, Ni and NiFe^{14,15,16,17,18,19}). 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 ξ_{s} < 3 nm^{10,15,17,19}. The superconductivity in F can be detected via tunnelling density of states (Do S) measurements^{20,21} and point contract Andreev spectroscopy^{22,23}. Furthermore, the magnetic exchange field from F induces a spinsplitting of the DoS in S close to the S/F interface^{24,25,26}, which can potentially open triplet chanels in S materials over the length scale of ξ_{F}^{27,28}.
Recently there is a focus on Josephson junctions with inhomogeneous barrier magnetism, involving misaligned F layers^{29,30,31,32,33,34} and/or rare earth magnets such as Ho or Gd^{35,36}, in order to transform singlet pairs into spinaligned triplet pairs^{3,6,37}. Triplet pairs are spinpolarized and stable in a magnetic exchange field and decay in Fs over length scales exceeding ξ_{s}^{3,5}. 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 I_{c} on external magnetic field H^{38,39}.
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 anisotropy^{40}. Let us take Ho an example. In single crystals the moments align into an antiferromagnetic spiral below 133 K made up of Fordered basal planes with moments in successive planes rotated 30° relative to each other due to the RKKY coupling^{41,42}. Below 20 K the moments in Ho tilt outofplane although this is not observed in thin film due to strain^{43}. The antiferromagnetic spiral has a zero net magnetic moment but applying magnetic fields parallel to the basal planes^{44,45} induces an irreversible transition to a ferromagnetic state. In epitaxial thinfilms, similar properties are reproduced although the antiferromagnetic spiral can remain stable over a wide field range^{46}. In textured or polycrystalline thin films the antiferromagnetic spiral can remain reversible even after applying magnetically saturating fields^{47}. 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 magneticfielddependence of the maximum Josephson critical current I_{c} in S/F/S junctions with a position and magneticfielddependentχ (Fig. 1). The model predicts anomalous Fraunhofer patterns due to spatial variations in χ and magnetic induction in which local minima in I_{c}(H) can be nonzero and nonperiodic due to interference between magnetised and demagnetised regions.
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 barrier^{48}. 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 = H_{s}. 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 positiondependent 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 nonuniform. The line integral of B(x) across the junction gives the spatial gradient of the superconducting phase
where Φ_{0} 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
where φ_{0} 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 positiondependent current density j(x) in the magnetized and demagnetized regions are
where j_{c} 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 πstate^{50,51} and hence the directions of j_{c} and j_{m,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 φ_{0} = π/2 which yields
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 I_{c}
where \({I}_{c0}=Lw{j}_{c}\) is the H = 0 total critical current of the junction and \(l=0.5pf\) with \(p=\frac{{{\rm{\Phi }}}_{0}}{\bar{d}{L}^{2}}P\). Calculating the integrals analytically we obtain
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 I_{cm} is the total critical current in the magnetized state and \({I}_{cm}=Lw{j}_{cm}=LwQ{j}_{c}\). The shape of I_{c}(f) is thus determined by Q, p and q and its magnitude by j_{c} and the junction area. In Fig. 2 we have plotted example I_{c}(f) patterns.
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, I_{c}(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 I_{c}(f) deviate from a nonmagnetic junction with nonperiodic behaviour. Furthermore, the magnitudes of I_{c} at local minima are not always 0 and I_{c} 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 I_{c}(f) behaviour with periodic minima and peaks in I_{c}(f) with peak heights decreasing inversely with f.
The parameters q, p and Q, influence I_{c}(f) in different ways. For Q close to 1 the j_{c} in the magnetized and demagnetized regions closely match, but in the magnetized regions the superconducting phase oscillates faster. In the magnetized regions I_{c} quadratically decreases with f for small H (f ≪ 1) and the central peak is rounded, resembling a sinctype function. For Q far from 1 or negative, j_{c} differs in the magnetized and demagnetized regions. For small H (f ≪ 1), I_{c} is mainly determined by the propagation of the magnetised region and, because the demagnetised region shrinks linearly, I_{c} 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 I_{c}(f) for multiple sets of parameters. The main effect of q on I_{c}(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.
The influence of p on I_{c}(f) is most significant for p < 0.2. In this range, small changes in p affect the shape of I_{c}(f) significantly: multiple minima in I_{c}(f) combine and some minima split into two minima forming a maximum (Fig. 3). For Q < 0, there is a minimummaximum pair forming just below p < 0.2 (exact value depends on q and Q). For p > 0.2, the shape of I_{c}(f) weakly depends on p since the magnetized regions propagate rapidly with H and the magnetized regions dominate I_{c}(f).
Conclusions
We have presented a generalised model to predict the behaviour of I_{C}(H) Fraunhofer patterns in magnetic Josephson junctions with a nonuniform magnetic susceptibility that peaks at junction edges. An analytical expression for I_{c}(H) is derived and key parameters which describe the shape of I_{c}(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 I_{c}(H) behaviour magnetically complex Josephson junctions with simultaneous zero and Pi states.
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Acknowledgements
B.B. acknowledges funding from the Cambridge Trust and St John’s College Cambridge. B.B. and J.W. A.R. acknowledge funding from the Royal Society and the EPSRC through an International Network Grant and Programme Grant (No. EP/P026311/1 and No. EP/N017242/1). A.B. thanks The Leverhulme Trust for supporting his visiting Professorship in Cambridge University.
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The project was conceived and directed by J.W.R. B.B. and A.B. performed the calculations and the analysis with support from J.W.R. and S.K. The manuscript was written by B.B. and J.W.R. and all authors reviewed it.
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Börcsök, B., Komori, S., Buzdin, A.I. et al. Fraunhofer patterns in magnetic Josephson junctions with nonuniform magnetic susceptibility. Sci Rep 9, 5616 (2019). https://doi.org/10.1038/s41598019417643
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DOI: https://doi.org/10.1038/s41598019417643
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