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REVIEW 3 major objections 5 minor 63 references

Equal-spin and oblique-spin crossed Andreev reflections in ferromagnet/Ising superconductor/ferromagnet junction

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read In a ferromagnet/Ising superconductor/ferromagnet junction, crossed Andreev reflection can be equal-spin, splitting Cooper pairs into same-spin electrons.

desk verdict A clean BdG calculation with a genuinely new nonlocal spin channel; qualitative results are sound, but the robustness of the quantitative CAR claims is under-tested because the appendix only checks local AR. read the letter →

arxiv 2509.04840 v1 pith:7RJGHN3U submitted 2025-09-05 cond-mat.supr-con

Wei-Tao Lu , Qing-Feng Sun , Qiang Cheng This is my paper · ORCID
classification cond-mat.supr-con PACS 74.45.+c74.78.-w
keywords crossedAndreevreflectionequal-spintripletpairingIsingsuperconductorspin-valleylockingBogoliubov-deGennesmagnetoanisotropyhalf-metalCooperpairsplitting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies a ferromagnet/Ising superconductor/ferromagnet junction and claims that the superconductor's Ising spin-orbit coupling, together with equal-spin-triplet pairing, enables crossed Andreev reflection in which an incident electron is emitted as a hole with the same spin rather than the opposite spin. This equal-spin CAR is strongest for half-metallic ferromagnets and double-band Ising superconductors, oscillates with chemical potential, and shows magnetoanisotropy with period $\pi$, vanishing when the magnetization is out-of-plane and peaking in-plane. When the two ferromagnets' magnetizations are not aligned, an oblique-spin CAR appears with period $2\pi$, where the electron and hole spins are neither parallel nor antiparallel. These channels give electrically and magnetically controllable spin- and valley-polarized Cooper pair splitting.

What carries the argument

The load-bearing machinery is the Bogoliubov-de Gennes Hamiltonian of the junction with an Ising spin-orbit term $\beta\sigma_z$ in the superconductor and magnetization vectors $\mathbf{M}_{L,R}$ in the ferromagnets. The spin-flip scattering induced by the Ising spin-orbit coupling allows an in-plane-polarized electron to combine with a same-spin electron from the other lead into an equal-spin-triplet Cooper pair, transmitting a same-spin hole. The magnetization angles $\theta_{L,R}$ control the amplitude of this spin-flip process, producing the $\pi$-periodic anisotropy for parallel magnetizations and the $2\pi$-periodic oblique-spin response when the magnetizations are misaligned.

What would settle it

Measure nonlocal conductance in an F/ISC/F junction with half-metallic contacts and in-plane magnetization: the paper predicts a strong equal-spin CAR that vanishes when the magnetization is rotated out-of-plane and returns with period $\pi$. Absence of this $\pi$-periodic nonlocal signal in a clean device, or an observed $2\pi$ period in the parallel-magnetization geometry, would rule out the claimed equal-spin channel.

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Extended reading notes

Core claim

The central claim is that the Ising spin-orbit coupling in the superconductor mixes spin channels so that an electron from one ferromagnetic lead can pair with a same-spin electron from the other lead, producing a same-spin hole in a nonlocal Andreev process. With identical magnetization directions, this equal-spin CAR is nonzero only when the magnetization has an in-plane component; its angular dependence is $\pi$-periodic and symmetric about the in-plane direction, and in half-metallic leads it is the only surviving nonlocal channel because normal (opposite-spin) CAR is completely suppressed. When the magnetization directions of the two leads differ, a new oblique-spin CAR channel opens in which the spin of the transmitted hole is neither parallel nor antiparallel to the incident electron's spin, and the magnetoanisotropic period generally becomes $2\pi$. The oblique-spin channel is valley-sensitive: electrons from the two valleys contribute in different regions of chemical-potential and magnetization-angle space, so the junction can produce spin- and valley-polarized CAR.

Load-bearing premise

The results assume a one-dimensional junction with equal effective masses in the ferromagnet and superconductor, no spin-orbit coupling in the ferromagnets, and no interface barrier, and the robustness checks only cover local, not crossed, Andreev reflection.

Editorial extensions

If this is right

  • With half-metallic leads and parallel in-plane magnetizations, normal crossed Andreev reflection is fully suppressed, leaving equal-spin CAR as the sole nonlocal transport channel.
  • Rotating the magnetization by $\pi/2$ switches equal-spin CAR on and off, providing a magnetic switch for Cooper pair splitting.
  • Double-band Ising superconductors strongly enhance equal-spin CAR while weakening equal-spin local Andreev reflection, so gating between single- and double-band regimes selects the dominant process.
  • When magnetizations are non-parallel, oblique-spin CAR carries spin- and valley-polarized current that can be tuned by chemical potentials and magnetization angles, giving controllable spin/valley entanglement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same spin-flip equal-spin-triplet mechanism should also generate equal-spin supercurrents and 0-$\pi$ transitions in Ising-superconductor Josephson junctions, extending the paper's transport picture to phase-coherent devices.
  • In a wider two-dimensional junction, transverse momentum should add oblique-spin CAR at finite angles, possibly enriching the polarization control but also introducing decoherence channels to test.
  • A direct experimental test is nonlocal conductance in a half-metal/ISC/half-metal stack with in-plane magnetization; observing the predicted $\pi$-periodic on/off switching would confirm the equal-spin mechanism.
  • The valley selectivity of oblique-spin CAR suggests the junction could act as a valley filter, where the magnetization direction selects Cooper pairs from one valley.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies subgap transport in a one-dimensional ferromagnet/Ising superconductor/ferromagnet (F/ISC/F) junction described by a two-band k·p model with Ising spin-orbit coupling and spin-singlet pairing. The authors solve the Bogoliubov-de Gennes equations by mode matching and compute local Andreev reflection (LAR) and crossed Andreev reflection (CAR) coefficients for both spins and both valleys. They find that when the two ferromagnets have identical magnetization directions, an in-plane magnetization enables equal-spin LAR and equal-spin CAR, with a magnetoanisotropic period π and strong enhancement in the half-metal regime and for double-band Ising superconductors, while the normal CAR is suppressed. When the two magnetization directions differ, they identify an oblique-spin CAR in which the reflected hole spin is neither parallel nor antiparallel to the incident electron spin, and they report a 2π magnetoanisotropic period for this case. The paper further shows that the spin and valley composition of the CAR can be controlled by the chemical potentials and the magnetization directions, which they propose as a route to spin- and valley-polarized Cooper pair splitting.

Significance. If the central claims hold, the paper provides a concrete proposal for electrically and magnetically controlled spin- and valley-polarized Cooper pair splitting in Ising superconductors, extending the known equal-spin Andreev reflection of Ref. [29] to the nonlocal crossed geometry. The strengths of the manuscript are its clean, parameter-free BdG formulation; explicit scattering wave functions and flux-normalized coefficients; consistent symmetry checks (e.g., TA(θ)=TA(π±θ), valley degeneracies, TA(θL,θR)=TA(θR,θL)); and an appendix that tests some robustness aspects. The calculation is reproducible from the given equations and contains no fitted parameters. The main weakness is the gap between the idealized 1D model and the 2D TMD platform, particularly for the nonlocal equal-spin CAR, which is the headline result and whose robustness to mass mismatch, interface barriers, and transverse momentum is not directly tested.

major comments (3)
  1. [Sec. III.B and Appendix (Eq. 14, Figs. 13-14)] The robustness of the headline nonlocal results is not established. The text in Section III states that the equal masses, zero interface potential, and zero SOC in the ferromagnets 'should have no substantial effect ... (see the appendix)', but the appendix only tests mass mismatch and interface potential for local Andreev reflection in a single F/ISC junction, not for crossed Andreev reflection in the F/ISC/F junction. The equal-spin CAR amplitude is a coherent product of left-interface conversion, propagation through the ISC, and right-interface conversion, so changes in the wave-vector matching caused by a mass ratio mS/mF or a barrier potential V can alter the interference phases and hence the nonlocal CAR. I request either explicit CAR calculations for finite mS/mF and V, or a reformulation of the claims as restricted to the idealized junction.
  2. [Sec. III.B, Fig. 5(b) and the paragraph following Fig. 5] The claim that equal-spin CAR is 'dramatically enhanced' and shows a 'remarkable peak near µF = M' is tied to the 1D density-of-states singularity, as the text itself states that the peak arises because the density of states of the F leads is singular at µF = M. In a 2D TMD junction with translational invariance along the transverse direction, k_y is conserved and the transport coefficients must be integrated over k_y; the 1D van Hove singularity is then replaced by a smooth 2D density of states, which will broaden or suppress this peak and weaken the quantitative enhancement in the half-metal regime. The paper does not provide a k_y-integrated calculation or an estimate of the effect of transverse modes. This is a load-bearing gap for the quantitative central claim.
  3. [Sec. III.C, Figs. 10-11] The contrast between the π period for equal-spin CAR and the 2π period for oblique-spin CAR should be framed more carefully. When θL=θR is varied and both magnetizations are rotated together, the numerical results show TA(θ)=TA(π+θ). When θR is held fixed and only θL is varied, the system is no longer invariant under θL→θL+π, so the 2π period is simply the reflection symmetry TA(θL)=TA(−θL) for the chosen parameter path, not a new fundamental period. The text should state this distinction explicitly; otherwise the phrase 'the magnetoanisotropic period generally becomes 2π instead of π' could be misread as a contradiction with the earlier π-period result.
minor comments (5)
  1. [Sec. II, Eq. (6)] The notation Ee(h)=+(−) ... is ambiguous: the reader must infer that the first sign is for electrons and holes and the second sign for spin parallel/antiparallel to the magnetization. Please rewrite this equation with explicit labels.
  2. [Sec. II, after Eq. (2)] The solutions g1...g4 for H^S_+ are explicit, but the corresponding changes for H^S_- are only described in words ('the wave vectors ... should become'). An explicit equation or table for the H^S_- wave vectors would aid reproducibility.
  3. [Sec. III.B, near Fig. 4] The sentence containing 'oppositing to the case' has a typo. Also, the notation K→ and K← should be defined carefully: the arrows denote spin direction in the in-plane basis, while ↑/↓ denote the out-of-plane spin basis, and this distinction is not stated explicitly.
  4. [Appendix, Figs. 13-14] The claim in Section III that the idealized junction 'should have no substantial effect' is too strong given the appendix's own results: Fig. 14 shows RA falling below 0.12 for E<Δ for large mass ratios or interface potentials, which is a substantial quantitative suppression. Please rephrase the statement in the main text to match the appendix results.
  5. [References] Reference [58] is cited as an arXiv preprint without a journal reference; if a published version exists, it should be cited instead.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: equal-spin and oblique-spin CAR are computed outputs of a BdG scattering problem; self-citations are corroborative, while the appendix robustness gap is a limitation, not a circular step.

full rationale

The transport coefficients are obtained by solving the stated BdG equations with boundary conditions, not by fitting or by renaming an input. The Hamiltonian in Eqs. (1)-(5) contains only the Ising band splitting ±βσ_z and a spin-singlet pairing term Δiσ_y; no equal-spin-triplet pairing term is inserted by hand. The scattering amplitudes in Eqs. (8)-(11) are determined by matching ψ and ψ′ at both interfaces, so the appearance of equal-spin CAR at θ = π/2 in the half-metal regime and of oblique-spin CAR for θ_L ≠ θ_R is a genuine output of the calculation. The paper's single-interface result in Fig. 2 independently reproduces the equal-spin AR reported in the self-cited Ref. [29], making that self-citation corroborative rather than load-bearing. The statement that Ising SOC generates spin-triplet correlations is attributed to the external Ref. [6], and no uniqueness theorem or ansatz is imported from the authors' prior work. Remaining self-citations are contextual. The one notable gap is that the appendix tests robustness to mass mismatch and interface potential only for LAR (Eq. (14), Figs. 13-14), while the main text asserts that these effects should not substantially affect the discussed CAR; this is an unsupported extrapolation and a robustness concern, but it is not a circular reduction because the CAR results themselves are derived, not assumed. No equation reduces to an input by construction, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The calculation uses a standard BdG scattering model with nominal parameters (M=30, β=10 in units of Δ) that are not fitted to the predicted effects. The main auxiliary assumptions are the minimal Ising superconductor model, the idealized 1D junction, and the neglect of interface barriers; robustness of these simplifications is checked only for LAR in the Appendix.

assumptions (4)
  • domain assumption Minimal two-band k·p model with Ising SOC (H± = ℏ²k²/2m - μ ± βσz) and s-wave pairing Δ iσy describes the superconducting state of TMD monolayers.
    Used in Sec. II, Eqs. (1) and (2), following refs. [6,10]; the equal-spin-triplet correlations emerge from the Ising SOC.
  • domain assumption Junction transport is treated in one dimension with plane-wave matching; masses in F and ISC are equal, F has no SOC, and the interface potential is zero.
    Stated in Sec. II and in the parameter paragraph of Sec. III; partially relaxed only for LAR in the Appendix.
  • standard math Reflectivity and transmissivity defined via probability currents (Eqs. 9, 11-13) correctly give the LAR and CAR coefficients.
    Standard Blonder-Tinkham-Klapwijk-type current conservation; consistent with refs. [21,22,24,29].
  • domain assumption Magnetization directions of the two F leads can be set independently and rotated without affecting the Ising superconductor.
    Assumed in Secs. IIIB and IIIC; motivated by the Meissner effect and experimental stacks in refs. [57,58].

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Pith. "Pith review of Equal-spin and oblique-spin crossed Andreev reflections in ferromagnet/Ising superconductor/ferromagnet junction." pith.science (2026). https://pith.science/paper/7RJGHN3U

@misc{pith2026250904840,
  author       = {Pith},
  title        = {Pith review of: Equal-spin and oblique-spin crossed Andreev reflections in ferromagnet/Ising superconductor/ferromagnet junction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7RJGHN3U}},
  note         = {Machine review of arXiv:2509.04840}
}
read the original abstract

We study the subgap transport through a ferromagnet/Ising superconductor/ferromagnet (F/ISC/F) junction by solving the Bogoliubov-de Gennes equations. It is found that the crossed Andreev reflection (CAR) and local Andreev reflection (LAR) strongly depend on the spin polarized F, the magnetization direction, and the Ising superconducting phase. For the same magnetization directions of the two F leads, the equal-spin CAR could take place due to spin-flip mechanism induced by the Ising spin-orbit coupling and equal-spin-triplet pairing. Both equal-spin CAR and equal-spin LAR exhibit a remarkable magnetoanisotropy with period \pi and show oscillatory behavior with chemical potential. The equal-spin CAR is more prominent for half-metal F and double-band ISC while the normal CAR is completely suppressed. When the magnetization directions of the two F leads are different, the oblique-spin CAR occurs and its magnetoanisotropic period generally becomes 2\pi instead of \pi. In the oblique-spin CAR process, the spins of the electron and hole are neither parallel nor antiparallel. Furthermore, the property of oblique-spin CAR is very sensitive to the spin and valley degrees of freedom. The spin and valley polarized CAR can be achieved and controlled by the chemical potentials and the magnetization directions.

Figures

Figures reproduced from arXiv: 2509.04840 by the authors.

Figure 1
Figure 1. FIG. 1. (a) The schematic diagram for the F/ISC/F junction. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (c) shows the dependence of LAR on the magne￾tization direction. For half-metal F with µF < M, RA is devoted by the equal-spin AR and the normal AR process is completely suppressed, thus, RA = 0 at θL = 0, π. On the contrary, RA reaches its maximum at θL = π/2, 3π/2 because the Cooper pairs have the spin-triplet compo￾nents | ⇔i and | ⇒i with spin pointing to in-plane direc￾tions. The magnetoanisotropy of RA is very… view at source ↗
Figure 3
Figure 3. displays the contour plot for LAR coefficient RA and CAR coefficient TA as a function of the potentials µF and µS when (a)(b) θ = 0 and (c)(d) π/2. One may find that RA is zero at θ = 0 for half-metal Fs with µF < M [see [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 6
Figure 6. Figure 6: discusses the magnetoanisotropy phenomenon of the LAR and the CAR, the change of which are not monotonic with θ. From [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. LAR coefficient [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: exhibits the CAR devoted by (a) K →, (b) −K →, (c) K ←, and (d) −K ← electrons when θL = π/2 and θR = 0. The total TA by the four kinds of electron is the result in [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. LAR coefficient [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. LAR coefficient [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. LAR coefficient [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. LAR coefficient [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. LAR coefficient [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]

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