Josephson Diode Effect from Nonequilibrium Current in a Superconducting Interferometer

Alex Levchenko; Daniel Shaffer; Jaglul Hasan; Mikhail Titov; Songci Li

REVIEW 3 major objections 5 minor 82 references

Josephson Diode Effect from Nonequilibrium Current in a Superconducting Interferometer

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper shows that a dissipative current in the normal part of a symmetric superconducting interferometer can produce a Josephson diode effect, including perfect and above-perfect diode efficiency, without any built-in inversion…

desk verdict A clean symmetry argument for a dissipative-current Josephson diode, but the 'supra-perfect' claim rests on an uncontrolled αL expansion; worth refereeing after fixing that and Eq. (18). read the letter →

arxiv 2505.00085 v2 pith:3LTKZ7SC submitted 2025-04-30 cond-mat.supr-con

Daniel Shaffer , Songci Li , Jaglul Hasan , Mikhail Titov , Alex Levchenko This is my paper · ORCID

classification cond-mat.supr-con

keywords JosephsondiodeeffectnonequilibriumsuperconductivityAndreevinterferometersupercurrent-phaserelationanomalousefficiencydissipativecurrentKeldyshGreen'sfunctions

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

This paper claims that a dissipative current flowing through the normal-metal wire of an otherwise fully symmetric superconducting interferometer is enough to generate a Josephson diode effect: the maximum supercurrent in one direction need not equal that in the opposite direction. This matters because a dc Josephson diode was previously thought to require simultaneous breaking of time-reversal and inversion symmetry, typically via magnetic fields and spin-orbit coupling or geometric asymmetry. Here the asymmetry comes entirely from the nonequilibrium electron-hole imbalance that the bias current injects into the normal region, and the diode efficiency can be tuned by voltage, temperature, and the spacing of the superconducting contacts. In the long-junction limit the calculation predicts perfect ($|\eta|=1$) and supra-perfect ($|\eta|>1$) diode behavior, in which the supercurrent is effectively one-way.

What carries the argument

The central object is the nonequilibrium quasiclassical Keldysh Green's function for a diffusive normal wire governed by the Usadel equation, with low-transparency superconducting contacts described by Kupriyanov-Lukichev boundary conditions. The key output is the electron-hole asymmetric part of the distribution function, $h_\sigma$, whose spatial derivative carries the dissipative current; the proximity effect produces discontinuities in $dh_\sigma/dx$ at the contacts, and those discontinuities generate the two nonequilibrium supercurrent contributions. Uncorrelated Andreev reflections produce the phase-independent term $J_{s,0}^{(\mathrm{neq})}$ (the diode term), while correlated crossed Andreev reflections produce the $\cos\varphi$ term $J_{s,2}^{(\mathrm{neq})}$; the ratio of $J_{s,0}^{(\mathrm{neq})}$ to the phase-coherent amplitude $\sqrt{(J_{s,1}^{(\mathrm{eq})})^2+(J_{s,2}^{(\mathrm{neq})})^2}$ is the diode efficiency.

What would settle it

Measure the positive and negative critical currents of a diffusive Andreev interferometer while sweeping the DC voltage between the normal terminals at fixed phase bias. The central claim predicts that the current-phase relation shifts vertically in proportion to the voltage, that one critical current vanishes at a finite bias, and that the diode efficiency grows as the spacing between the superconducting contacts shrinks relative to the wire length; a null result on any one of those scalings would refute it.

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Extended reading notes

Core claim

The central claim is that the current-phase relation of the Andreev interferometer takes the form $J_s(\varphi)=J_{s,0}^{(\mathrm{neq})}+J_{s,1}^{(\mathrm{eq})}\sin\varphi+J_{s,2}^{(\mathrm{neq})}\cos\varphi$, with the constant term $J_{s,0}^{(\mathrm{neq})}$ generated by the dissipative current in the normal wire and odd under reversal of the applied bias. That constant term shifts the sinusoid vertically, so the positive and negative critical currents have unequal magnitudes; the diode efficiency $\eta=J_{s,0}^{(\mathrm{neq})}/\sqrt{(J_{s,1}^{(\mathrm{eq})})^2+(J_{s,2}^{(\mathrm{neq})})^2}$ is therefore nonzero without any structural inversion asymmetry. As the distance $l$ between the NS interfaces shrinks relative to the wire length $L$, the constant term decays only linearly while the phase-coherent coefficients decay quadratically, so $\eta$ passes through one and can exceed it; at the same time one critical current vanishes and changes sign, leaving a window in which a non-dissipative supercurrent exists in only one direction.

Load-bearing premise

The calculation keeps only the lowest-order terms in the transparency of the superconductor-normal contacts, but the dimensionless combination of that transparency with the wire length is acknowledged to be of order one in the diffusive regime; if so, the neglected higher-order Andreev processes could change the current-phase relation and the predicted diode efficiency.

Editorial extensions

If this is right

The device acts as a voltage-tunable, switchable Josephson diode that needs no magnetic field, no spin-orbit coupling, and no asymmetric junction geometry.
Because $J_{s,0}^{(\mathrm{neq})}$ shifts the current-phase relation vertically, sweeping the bias voltage makes the negative critical current pass through zero and change sign, so strictly nonzero supercurrents are possible in only one direction.
The voltage scale required for perfect or supra-perfect diode behavior becomes smaller as $l/L$ decreases, so the effect can be engineered geometrically.
Replacing the normal wire by a superconducting wire with SIS interfaces gives a dissipation-free variant of the same mechanism, namely a four-terminal SIS'IS junction.
If nonequilibrium effects are at play in the observed supra-perfect bulk superconducting diode in twisted trilayer graphene, this mechanism offers a concrete interpretation of that behavior.

Reading between the lines

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

A direct experimental test is to measure the critical-current asymmetry of a diffusive Andreev interferometer as a function of DC bias on the normal terminals and of the contact spacing $l/L$; the predicted linear-in-voltage shift and the divergence of $\eta$ as $l/L\to0$ are specific enough to falsify the mechanism.
By the same logic, a pure thermal gradient should not produce this diode effect, since a heat current does not couple to the superconducting phase; this could be checked separately from the electrical-bias experiment.
If the second-order-in-$\alpha$ result survives higher-order corrections, the mechanism offers a generic route to field-free diodes in diffusive hybrid nanostructures, bypassing the need for spin-orbit-coupled materials.
Self-field effects from the bias current are neglected; incorporating them could either mask or enhance the predicted diode efficiency, depending on geometry.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The short take: this paper identifies a genuine and clean mechanism for a Josephson diode—a dissipative current in the normal wire of an otherwise symmetric interferometer acts as a time-reversal-odd polar axis, generating a constant term in the current-phase relation. That symmetry argument is sound, and the effect very likely exists. But the headline quantitative claim, that the diode efficiency can exceed 1 and diverge as ℓ→0, is not controlled by the calculation as it stands.

What is good: the decomposition J_s = J_0^neq + J_1^eq sinφ + J_2^neq cosφ, the identification of J_0^neq as the diode source, the explicit formula for η, and the observation that a heat current alone would not couple to φ (footnote 76). The paper is also honest about the prior work in [63,64], acknowledging that the CPR itself was already there; the new part is the interpretation and parameter survey.

The main soft spot is the truncation. The linearized Usadel equation requires |f| << 1, but near the interfaces f scales as αL (Eq. 10). The coefficients in Eq. (17) are computed to second order in α, and the paper itself notes after Eq. (17) that αL can be O(1) because L/l_tr >> 1—which means the small parameter is missing exactly in the regime they advertise. So the η → ∞ divergence as ℓ→0 is a leading-order artifact; higher-order Andreev processes add relative corrections of order αL and can saturate or shift the η = 1 line. The symmetry argument proves the effect is allowed, not that the supra-perfect regime exists as plotted.

Second, Eq. (18) is algebraically wrong: for |η|>1 the correct relation is η' = 1/η, not η sgn[1−|η|]. That is minor for the main claim but should be fixed. Third, Eq. (12) is quoted from Ref. [63] rather than rederived; acceptable, but it makes it harder to check the validity of the approximation.

Who is this for? The Josephson-diode and hybrid-superconductor community. It deserves a serious referee: the idea is good, the presentation is clear, and the flaw is addressable rather than fatal. The authors should either demonstrate control of the next-order terms or restrict the quantitative claims to αL << 1, and correct Eq. (18). I would conditional-accept after those revisions.

Referee Report

3 major / 5 minor

Summary. The manuscript studies a diffusive Andreev interferometer consisting of a normal-metal wire connected to two superconducting reservoirs through low-transparency interfaces and to two normal reservoirs biased by a voltage. Using the quasiclassical Keldysh-Usadel formalism and closely following Ref. [63], the authors derive the current-phase relation J_s(phi) = J_s,0^(neq) + J_s,1^(eq) sin(phi) + J_s,2^(neq) cos(phi), where the constant and cosine terms arise from the nonequilibrium electron-hole distribution induced by the dissipative normal current. From this CPR they define the diode efficiency eta = J_s,0^(neq)/sqrt((J_s,1^(eq))^2 + (J_s,2^(neq))^2) and find regimes where eta = 1 and |eta| > 1, which they call a supra-perfect Josephson diode effect, with eta diverging as the geometry parameter ell = l/L goes to zero. The effect is presented as a new route to the Josephson diode effect that requires no structural inversion symmetry breaking.

Significance. The conceptual message is significant: if the calculation is controlled, the paper provides an explicit microscopic mechanism by which a purely dissipative normal current produces a DC Josephson diode effect in a geometrically symmetric junction, with a simple analytic CPR and an intuitive interpretation in terms of uncorrelated Andreev reflections. The paper also gives concrete, falsifiable predictions: the diode efficiency is voltage-tunable, independent of alpha at leading order, and grows as ell tends to zero. These features would be of genuine interest to the mesoscopic superconductivity community. However, the quantitative claim of supra-perfect efficiency currently rests on a perturbative calculation whose small parameter is not demonstrated in the parameter range of interest, and the central coefficient integrals are quoted rather than derived. The symmetry argument establishes that a diode effect is allowed, but the headline quantitative claim is not yet fully supported.

major comments (3)

[Section III, after Eq. (17)] The expansion is truncated at second order in the interface parameter alpha, but the manuscript itself notes that alpha L = T_B (L/l_tr) can be of order unity even in the tunneling limit T_B << 1. Since the linearized Usadel equation (9) requires |f| << 1, and Eq. (10) gives f(±d/2) ~ alpha L times a geometry factor, the calculation has no demonstrated small parameter for ell not close to zero. For ell -> 0 the geometry factor suppresses f locally, but the perturbative series in alpha L is not shown to remain controlled, and corrections of relative order (alpha L)^2 would generically affect all three coefficients in Eq. (17). The statements that eta -> infinity as ell -> 0 and that supra-perfect JDE occurs for any fixed low voltage are therefore not established. The authors should either restrict the claim to a parameter regime where the expansion parameter is explicitly small and show |eta| > 1 there, or extend the calculation to higher order and demonstrate convergence.
[Section III, Eqs. (12) and (15)] The central formulas — the derivative discontinuity in Eq. (12) and the coefficient integrals in Eq. (15) — are quoted from Ref. [63] without derivation, and the text does not state the precise conditions under which terms beyond second order in alpha and beyond the leading low-temperature, low-voltage asymptotics may be neglected. Because these equations are the quantitative basis for the diode efficiency, the paper should provide a derivation outline or an appendix, including the small parameters and any assumptions about the relation between d, l, and L.
[Section III, Eq. (18)] The identity eta' = eta sgn(1 - |eta|) is incorrect. For |eta| > 1, J_c+ and J_c- have the same sign, and one obtains eta' = 1/eta, not eta sgn(1 - |eta|). For example, J_c+ = 2 and J_c- = 1 gives eta = 3 and eta' = 1/3, whereas the written formula returns -3. This is a secondary issue because it affects only the redefined efficiency, but it should be corrected.

minor comments (5)

[Eq. (10a)] The notation e^{i chi2/1} appears to be a typo; it should presumably be e^{i chi2} or e^{i chi2/2}.
[Fig. 1 and Eq. (7)] The length d in Eq. (7) is never defined in relation to l and L; Fig. 1 uses l/2 as the distance from the NS interface, and the text later uses ell = l/L, so the geometry should be clarified.
[Eq. (12)] In Eq. (12), the left-hand side delta(dh_sigma/dx) has units of inverse length while the right-hand side L h_sigma Re[F1 + F2 cos phi] has units of length; please check the prefactor.
[Before Eq. (16)] The phrase 'both nonequilibrium terms are anomalous' is confusing: J_s,2^(neq) is the anomalous (phase-shifting) term, while J_s,0^(neq) is a phase-independent term; the term 'anomalous' is usually reserved for the former.
[Fig. 2 caption] The statement that eta is independent of alpha is true at the truncated order but should be qualified, since higher-order corrections would in general introduce an alpha dependence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the CPR and diode coefficients are computed from a standard quasiclassical formalism; the target diode effect is not an input or fitted output.

full rationale

The paper's derivation chain is: linearized Usadel equation (9), boundary conditions (7), solution for f^R (10), distribution functions (11), discontinuity of the derivative of the distribution function (12), supercurrent expression (14), coefficient integrals (15), and finally the diode efficiency eta in (16) with the low-energy asymptotics (17). None of these steps assumes the existence or magnitude of the diode effect. The coefficients J_s,0^(neq), J_s,1^(eq), and J_s,2^(neq) are integrals over the Green's functions and nonequilibrium distribution functions; the diode efficiency eta is then defined from the extrema of the current-phase relation, not fitted. The paper explicitly quotes the supercurrent calculation from Ref. [63], which is coauthored by one of the present authors, but that reference is a separate published calculation focused on thermopower oscillations and does not itself assert the Josephson diode effect. The present paper's contribution is the identification and analysis of the constant term J_s,0^(neq) as a diode term, which is a legitimate corollary of prior independent work rather than a self-citation chain that forces the conclusion. No parameter is fitted to the target quantity, and the supra-perfect diode regime follows analytically from the scaling J_s,0^(neq) ∝ ell, J_s,1^(eq), J_s,2^(neq) ∝ ell^2 in the limit ell -> 0. The paper's own caveat that alpha L need not be small in the tunneling limit is a validity concern about the second-order truncation, not a circularity. Therefore the derivation is self-contained and its central claim is not equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No constants are fitted to data; V, T, l/L, alpha, and L are control parameters. The load-bearing assumptions are the linearized weak-proximity description, the rigid boundary conditions, the zeroth-order distribution function, and the neglect of inelastic scattering. No new particles or fields are introduced.

assumptions (6)

standard math Keldysh-Usadel quasiclassical Green's function formalism with normalization condition Eq. (2).
The central calculation is built on this formalism; it is unproved background assumed from quasiclassical superconductivity theory.
domain assumption Weak proximity effect: the Usadel equation is linearized in the anomalous Green's function f_R, Eq. (9).
The paper assumes low-transparency interfaces justify dropping quadratic terms, but does not establish that alpha L is a small parameter.
domain assumption Rigid Kupriyanov-Lukichev boundary conditions at the NS interfaces, Eq. (7), with superconducting reservoirs described by Eq. (6), including abrupt gap drop and no self-consistency.
This idealizes the interfaces; the paper states these are standard and neglects self-field and self-consistency corrections.
domain assumption The distribution function h_sigma in the wire is evaluated in the absence of proximity effect, Eq. (11), and only the leading derivative discontinuities at the interfaces are retained, Eq. (12).
The diode effect comes entirely from these corrections; a full self-consistent distribution could modify the magnitude.
domain assumption Charge conservation in the superconducting arm fixes the electron-hole part of the distribution, h_mu = 0 in the absence of a thermal gradient.
Quoted from Ref. [63]; constrains the dissipative current while allowing the voltage bias to enter through h_mu_T.
domain assumption Inelastic scattering in the normal wire is neglected, as stated after Eq. (1).
A real wire has electron-phonon and electron-electron collisions that relax the nonequilibrium distribution h_sigma; the predicted effect could be reduced.

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Cite this review

Pith. "Pith review of Josephson Diode Effect from Nonequilibrium Current in a Superconducting Interferometer." pith.science (2026). https://pith.science/paper/3LTKZ7SC

@misc{pith2026250500085,
  author       = {Pith},
  title        = {Pith review of: Josephson Diode Effect from Nonequilibrium Current in a Superconducting Interferometer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LTKZ7SC}},
  note         = {Machine review of arXiv:2505.00085}
}

read the original abstract

We investigate the Josephson diode effect in a superconducting interferometer under nonequilibrium conditions. In contrast to its thermodynamic counterpart, which requires the simultaneous breaking of time-reversal and inversion symmetry, we demonstrate that a diode-like asymmetry of the critical current can emerge solely due to a dissipative current in the normal region of an otherwise symmetric Josephson junction. This effect is driven entirely by the nonequilibrium conditions, without the need for additional inversion symmetry breaking. Using the standard quasiclassical Keldysh Green's function formalism, we explicitly calculate the diode coefficient from the supercurrent-phase relation of the interferometer. Remarkably, within certain ranges of control parameters, such as applied voltage, temperature, and the geometric aspect ratio of the device, the diode coefficient can exceed its nominal perfect value.

Figures

Figures reproduced from arXiv: 2505.00085 by the authors.

**Figure 2.** FIG. 2. (a-b) Plots of the Josephson diode efficiency coefficient [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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