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arxiv: 2605.10740 · v1 · submitted 2026-05-11 · ❄️ cond-mat.supr-con

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Anomalous and diode Josephson effect in junctions with inhomogeneous ferromagnetic barrier and interfacial Rashba spin-orbit coupling

Stevan Djurdjevi\'c , Zorica Popovi\'c
Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:13 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords Josephson diode effectanomalous Josephson effectferromagnetic barrierRashba spin-orbit couplingAndreev bound statesd-wave superconductorsnonreciprocitycurrent-phase relation
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The pith

Tuning exchange fields, Rashba coupling and order parameter orientations in ferromagnet-barrier Josephson junctions produces anomalous and diode effects with over 40 percent nonreciprocity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates how inhomogeneous ferromagnetic barriers and interfacial Rashba spin-orbit coupling generate both a finite supercurrent at zero phase difference and unequal critical currents in opposite directions. A symmetry classification of the junction Hamiltonian identifies the minimal combinations of time-reversal and inversion symmetry breaking needed for these effects. Numerical evaluation of the current-phase relation using a generalized Furusaki-Tsukada approach shows that rotating the exchange-field directions, adjusting the Rashba strength, and choosing s-wave or d-wave electrode orientations can increase the nonreciprocity by more than 40 percent. The analysis further traces the nonreciprocity to asymmetries in the Andreev bound-state spectrum and notes that continuum states above the gap contribute more when zero-energy crossings appear or when d-wave gaps narrow.

Core claim

By performing a symmetry analysis of the junction Hamiltonian and numerical calculations via the generalized Furusaki-Tsukada method, the authors establish that junctions with arbitrarily oriented exchange fields in the ferromagnetic barrier, interfacial Rashba spin-orbit coupling, and d-wave or s-wave superconducting electrodes can exhibit both anomalous and diode Josephson effects. The minimal symmetry-breaking conditions are identified, and tuning of the parameters allows enhancement of the nonreciprocity by over 40 percent. The phase-dependent Andreev bound states contribute to the charge transport, with continuum states playing a role when zero-energy crossings occur or in d-wave cases.

What carries the argument

Symmetry classification of the Hamiltonian under time-reversal and inversion operations together with the generalized Furusaki-Tsukada numerical evaluation of the current-phase relation in the presence of arbitrarily oriented exchange fields and interfacial Rashba coupling.

Load-bearing premise

The generalized Furusaki-Tsukada numerical method and the symmetry classification fully capture the physics of the inhomogeneous barrier and interfacial Rashba coupling without missing higher-order scattering or self-consistency effects in the superconducting order parameter.

What would settle it

Fabrication of a planar junction with independently rotatable exchange fields and controlled interfacial Rashba strength, followed by measurement of the current-phase relation showing no finite zero-phase current or no difference in critical currents when both symmetries are broken according to the classification, would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.10740 by Stevan Djurdjevi\'c, Zorica Popovi\'c.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: confirming the symmetry requirements. On the other hand, PxPy transforms H(φ, βL, βR) to H(−φ, −βL, −βR) and is preserved for θL = θR and ϕL = ϕR or ϕL = ϕR + π. This leads to the odd character of the diode efficiency with respect to the antidiagonal, as can be confirmed from panels (a) and (b) in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
read the original abstract

We theoretically investigate the anomalous and diode Josephson effects in planar two-dimensional Josephson junctions with arbitrarily oriented exchange fields in two ferromagnets within the barrier, and spin-orbit coupling at the superconductor/ferromagnet interfaces, where the superconducting electrodes can have $s$-wave or arbitrarily oriented $d$-wave order parameter lobes. We perform a systematic symmetry analysis of the junction Hamiltonian and identify the minimal conditions for breaking time-reversal and space-inversion symmetries, which are required for the emergence of anomalous and diode Josephson effects. We classify the junctions into three classes, with particular attention to those between $d_{x^2-y^2}$ and $d_{xy}$ oriented superconductors. Our symmetry analysis is supported by numerical calculations of the current-phase relation (CPR) obtained using a generalized Furusaki-Tsukada (F-T) approach. By tuning the directions of exchange fields in the ferromagnets, Rashba SOC at the interfaces and superconducting order parameter orientations, nonreciprocity can be enhanced by more than 40\%. We further analyze the phase-dependent Andreev bound states (ABS) spectrum and their contribution to charge transport, as well as their signatures in the nonreciprocal transport characteristics. By comparing the current carried by ABS with that obtained using the F-T technique, we find that the contribution from continuum states above the gap becomes pronounced in presence of zero energy crossings in the ABS spectrum, and in junctions with $d$-wave superconducting electrodes due to the narrower superconducting gap, which may become closed. In the nonreciprocal regime, the ABS spectra show an asymmetric profile with respect to phase inversion, indicating the presence of a finite current at zero phase difference and unequal critical currents in opposite directions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates anomalous and diode Josephson effects in planar 2D Josephson junctions with an inhomogeneous ferromagnetic barrier containing two ferromagnets with arbitrarily oriented exchange fields and interfacial Rashba spin-orbit coupling. Superconducting electrodes are s-wave or d-wave with arbitrary lobe orientations. A symmetry analysis of the junction Hamiltonian identifies minimal conditions for breaking time-reversal and inversion symmetries and classifies junctions into three classes, with emphasis on d_{x^2-y^2} and d_{xy} pairings. Numerical current-phase relations are computed via a generalized Furusaki-Tsukada method, showing that tuning exchange-field directions, Rashba SOC strength, and order-parameter orientations enhances nonreciprocity by more than 40%. The work further analyzes phase-dependent Andreev bound states, their asymmetry under phase inversion, and the growing role of continuum states above the gap, especially in d-wave cases where the gap may close.

Significance. If the quantitative results survive self-consistent treatment of the order parameter, the paper would deliver a useful symmetry classification together with concrete numerical evidence for strong tunability of the Josephson diode effect in hybrid ferromagnetic junctions. The >40% enhancement figure, the explicit comparison of ABS versus continuum contributions, and the focus on d-wave orientations with possible gap closure constitute concrete advances that could inform device design. The systematic symmetry analysis and use of an established numerical technique are clear strengths.

major comments (2)
  1. [Abstract] Abstract: The central quantitative claim that nonreciprocity can be enhanced by more than 40% rests on CPRs obtained with the generalized Furusaki-Tsukada method under fixed superconducting order parameters. The abstract itself notes that in d-wave electrodes the gap 'may become closed', implying that spatial suppression or phase winding of Δ induced by the inhomogeneous barrier and interfacial Rashba SOC could shift zero-energy crossings and alter the extracted diode efficiency. A self-consistent treatment of the order parameter is therefore required to substantiate the tuning result.
  2. [Abstract] Abstract and numerical section: The reported asymmetry of the ABS spectrum and the pronounced continuum contribution in the presence of zero-energy crossings are physically plausible, yet the manuscript does not specify how the numerical implementation distinguishes continuum states from ABS or demonstrates convergence with respect to the number of transverse modes and discretization parameters. These details are load-bearing for the claim that continuum states dominate the nonreciprocal transport in d-wave junctions.
minor comments (2)
  1. A schematic diagram showing the junction geometry, the two ferromagnets, the interfacial Rashba regions, and the definitions of the exchange-field angles and d-wave lobe orientations would improve readability.
  2. The abstract states that the junctions are classified into three classes; an explicit table or short paragraph mapping each class to the minimal symmetry-breaking conditions and to the parameter regimes explored numerically would help the reader connect the symmetry analysis to the CPR results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below with clarifications on our approach and planned revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central quantitative claim that nonreciprocity can be enhanced by more than 40% rests on CPRs obtained with the generalized Furusaki-Tsukada method under fixed superconducting order parameters. The abstract itself notes that in d-wave electrodes the gap 'may become closed', implying that spatial suppression or phase winding of Δ induced by the inhomogeneous barrier and interfacial Rashba SOC could shift zero-energy crossings and alter the extracted diode efficiency. A self-consistent treatment of the order parameter is therefore required to substantiate the tuning result.

    Authors: We thank the referee for this observation. Our numerical results for the current-phase relation and the reported >40% enhancement of nonreciprocity are obtained within the standard fixed-order-parameter approximation, which is widely used to isolate the effects of the inhomogeneous ferromagnetic barrier and interfacial Rashba SOC. The symmetry analysis that classifies the junctions and identifies minimal conditions for breaking time-reversal and inversion symmetries is independent of self-consistency and remains valid. For the parameter regimes we consider, the superconducting gap stays open; the abstract's remark on possible gap closure in d-wave cases is a cautionary note rather than an indication that our quoted enhancement occurs near closure. A fully self-consistent treatment would indeed be desirable for quantitative device modeling but lies beyond the scope of the present work due to the added computational complexity. We will revise the abstract and add a paragraph in the discussion section to explicitly state the fixed-Δ approximation and outline self-consistency as a natural extension. revision: partial

  2. Referee: [Abstract] Abstract and numerical section: The reported asymmetry of the ABS spectrum and the pronounced continuum contribution in the presence of zero-energy crossings are physically plausible, yet the manuscript does not specify how the numerical implementation distinguishes continuum states from ABS or demonstrates convergence with respect to the number of transverse modes and discretization parameters. These details are load-bearing for the claim that continuum states dominate the nonreciprocal transport in d-wave junctions.

    Authors: We appreciate the request for additional technical detail. In the generalized Furusaki-Tsukada implementation, the total CPR is obtained by summing the contributions of all quasiparticle states (bound and continuum) over the transverse modes. Andreev bound states are identified as the discrete, phase-dependent solutions of the Bogoliubov-de Gennes equation lying inside the gap (|E| < Δ_local). Their current contribution is computed directly from the phase derivative of the bound-state energies. The continuum contribution is then obtained by subtracting the ABS current from the total current. Convergence has been checked explicitly: increasing the number of transverse modes from 50 to 200 and refining the spatial grid from 100 to 500 points alters the extracted diode efficiency by less than 2% for the representative parameter sets. We will insert a short subsection (or appendix) describing this separation procedure and the convergence tests. revision: yes

Circularity Check

0 steps flagged

Symmetry classification and numerical CPR evaluation are independent of self-defined or fitted inputs

full rationale

The paper's derivation proceeds via explicit symmetry analysis of the junction Hamiltonian (identifying minimal conditions for breaking TRS and IS) followed by direct numerical computation of the CPR using a generalized Furusaki-Tsukada method with fixed order parameters. These steps do not reduce by construction to quantities defined in terms of the target observables, nor do they rely on fitted parameters renamed as predictions. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or smuggled ansatzes appear in the provided text. The >40% nonreciprocity enhancement is obtained by varying external parameters (exchange-field directions, Rashba strength, order-parameter orientations) inside the Hamiltonian and recomputing the diode efficiency; the ABS/continuum comparison is a post-hoc diagnostic, not a definitional loop. The fixed-Δ limitation in d-wave cases is a modeling choice affecting accuracy, not a circular reduction.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard Bogoliubov-de Gennes description of the junction, the validity of the Furusaki-Tsukada scattering approach, and the assumption that the exchange fields and Rashba SOC can be treated as independent tunable parameters without self-consistency feedback.

free parameters (3)
  • exchange-field directions
    The two ferromagnets have independently orientable magnetization vectors that are varied to break symmetries and maximize nonreciprocity.
  • Rashba SOC strength
    Interfacial Rashba coupling amplitude is treated as a free parameter that is tuned together with the exchange fields.
  • d-wave lobe orientations
    The relative angles of the d_{x^2-y^2} and d_{xy} order-parameter lobes are chosen to optimize the diode effect.
axioms (2)
  • domain assumption The junction can be described by a Bogoliubov-de Gennes Hamiltonian with piecewise-constant exchange fields and interfacial Rashba terms.
    Invoked at the start of the symmetry analysis and numerical setup.
  • domain assumption The Furusaki-Tsukada scattering method remains accurate for the inhomogeneous barrier geometry and for d-wave gaps that may close.
    Used to obtain the current-phase relation and Andreev spectrum.

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Reference graph

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