Transverse Josephson diode effect in tilted Dirac systems

W. Zeng

REVIEW 3 major objections 5 minor 62 references

Transverse Josephson diode effect in tilted Dirac systems

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A tilted-Dirac Josephson junction with valley-contrasting gaps develops a phase-driven transverse supercurrent with unequal critical currents, reaching a 100% diode quality factor while the longitudinal current stays reciprocal.

desk verdict A clean calculation of a transverse Josephson Hall current in an infinite-strip tilted Dirac junction, with a real but unproven device-level claim: the 100% quality factor is an interface property until someone models transverse leads. read the letter →

arxiv 2412.01394 v1 pith:QNRW5EQA submitted 2024-12-02 cond-mat.supr-con cond-mat.mes-hall

W. Zeng This is my paper · ORCID

classification cond-mat.supr-concond-mat.mes-hall PACS 74.50.+r

keywords transverseJosephsondiodeeffectHallcurrenttiltedDiracconevalley-dependentgapAndreevboundstatescurrent-phaserelationnonreciprocalsupercurrent8-Pmmnborophene

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

Most proposals for Josephson diodes rectify supercurrents along the junction axis, so the input and output signals share one path. This paper predicts a transverse version: in a short, ballistic superconductor–normal–superconductor junction built on a tilted Dirac material with valley-contrasting gaps, breaking time-reversal symmetry produces an interface-bound supercurrent that flows perpendicular to the junction, is driven by the superconducting phase difference, and has a nonsinusoidal current-phase relation. Because the positive and negative transverse critical currents are unequal, the junction acts as a transverse Josephson diode, and the paper shows its quality factor can reach $100\%$ — total one-way character — when the valley-gap asymmetry $\kappa$ is tuned. The longitudinal Josephson current, by contrast, stays sinusoidal and reciprocal, so the diode behavior is genuinely transverse. If the prediction survives being connected to real leads, this geometry would decouple the output signal path from the input and open a new axis of control for nonreciprocal superconducting devices.

What carries the argument

The load-bearing object is the transverse current formula obtained from McMillan's Green's function in the heavy-doping limit, $$j^\ell_y = e k_B T \sum_{k_y, \omega_n>0} \frac{\$\Delta$}{\sqrt{\$omega_n^{2}$ + \$\Delta$^2}} \frac{v_y \tan\$\theta$ + \ell v_t \sec\$\theta$}{v_x} \operatorname{Re}\bigl[a^\ell_1(\varphi, i\omega_n) + a^\ell_2(\varphi, i\omega_n)\bigr] $e^{{2\Omega_n x/v_x}}$,$$ with $a^\ell_{1,2}$ the Andreev reflection amplitudes and $\theta = \arctan(v_y k_y/\mu_s)$. The prefactor splits into a skew-tunneling piece proportional to $v_y \tan\theta$ and a tilt-driven piece proportional to $\ell v_t \sec\theta$; in the heavy-doping regime their ratio is of order $|\mu/\mu_s|$, so the transverse Hall current is dominated by the tilt-induced transverse momentum of the Cooper pairs. The diode signature is packaged in the quality factor $Q_t = (j^+_{y,c} - |j^-_{y,c}|)/(j^+_{y,c} + |j^-_{y,c}|)$, and its physical origin is displayed in the Andreev bound states $E_{\mathrm{ABS}}(k_y, \varphi)$: the tilt skews the two valleys' spectra in opposite $k_y$ directions, and broken time-reversal symmetry splits the subgap group velocities ($v_+ \neq v_-$), so that for large $\kappa$ only one transverse propagation direction per valley survives.

What would settle it

Attach transverse normal leads to a finite-width version of the junction and measure the transverse current-phase relation while tuning $\kappa$: if the positive and negative transverse critical currents are equal once the transverse channel is electrically contacted, or if the contact's back-action on the Andreev amplitudes restores reciprocity, the 100% quality-factor claim fails. A numerical check on a finite-width strip, testing whether the interface current returns through edge channels rather than feeding a transverse reservoir, would likewise settle whether the effect is a device diode or an interface property.

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Extended reading notes

Core claim

The paper's central claim is that a ballistic, short Josephson junction built from a tilted Dirac material with valley-dependent gaps carries a transverse supercurrent at its interfaces when time-reversal symmetry is broken by unequal valley masses. Expressed through Andreev reflection amplitudes in a Furusaki-Tsukada-like formula, this current $j_y(\varphi)$ is $2\pi$-periodic and nonsinusoidal, is nonzero even at zero phase difference, vanishes at $\kappa = 0$, and decays into the superconducting banks over the coherence length. The forward and backward transverse critical currents are unequal, so the junction acts as a transverse Josephson diode (TJDE) whose quality factor $Q_t$, evaluated with 8-Pmmn borophene parameters, reaches $\pm 100\%$ at tuned valley-gap asymmetry $\kappa$. The mechanism is the tilt-induced asymmetry of Andreev subgap modes: under broken time-reversal symmetry the two valleys' transverse group velocities satisfy $v_+ \neq v_-$, and beyond a critical $\kappa$ each valley has only one allowed transverse propagation direction. The longitudinal current remains a sinusoidal $\pi$-phase supercurrent with $j_x(\varphi) = -j_x(-\varphi)$, so the nonreciprocity is purely transverse.

Load-bearing premise

The transverse current is computed for an infinitely long strip that is translation-invariant along the transverse direction, with no transverse leads, edges, or bias in the model; the device-level promise that the effect decouples input from output requires this interface-bound equilibrium current to survive as an accessible transverse transport channel in a finite multiterminal device.

Editorial extensions

If this is right

The transverse interface current is nonzero only when both the tilt $v_t$ and the valley-gap asymmetry $\kappa$ are finite; it reverses sign with $\kappa$ and vanishes at $\kappa = 0$.
The transverse current-phase relation is $2\pi$-periodic and nonsinusoidal, with an anomalous contribution at $\varphi = 0$, and the forward versus backward critical currents differ, defining a transverse diode polarity $Q_t$ that flips sign with $\kappa$.
The quality factor $Q_t$ reaches $\pm 100\%$ at tuned valley-gap asymmetry and is larger at low temperature than near $T_c$.
The longitudinal Josephson current is a sinusoidal $\pi$-phase supercurrent with reciprocal critical currents, so no diode effect occurs along the junction axis.
The nonreciprocity originates in tilt-induced asymmetric Andreev subgap modes ($v_+ \neq v_-$) per valley, not in a Doppler shift of quasiparticle energies; at sufficiently large $\kappa$ each valley retains only a single transverse propagation direction.

Reading between the lines

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

My inference, not the paper's: the paper models an infinite strip, so the 100% quality factor is an equilibrium interface property; whether it survives in a finite multiterminal geometry with transverse contacts, where the contacts back-act on the Andreev amplitudes, is not computed and would decide the device claim.
My inference: because the transverse current grows with the tilt velocity $v_t$ and the mass imbalance, strain-tunable tilted Dirac materials or Floquet-driving schemes that generate the valley-dependent mass could switch the transverse diode on and off optically — a control knob the paper does not propose.
My inference: the short-junction, ballistic treatment leaves the behavior at junction length $L$ comparable to the coherence length, and with disorder, unaddressed; extending the calculation along those axes would test whether the transverse diode is a robust transport feature or a clean-limit phenomenon.

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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

Short version: this is a legitimate new calculation, not a new framework. The transverse Josephson diode effect in an infinite-strip tilted Dirac junction is computed cleanly; the symmetry checks pass; the Andreev-level picture explains the mechanism. But the 100% quality factor and the "complete decoupling" claim are properties of a boundary-free equilibrium interface current, not of a device with transverse leads. The stress-test concern holds up: Eq. (5) is evaluated at x=0 for a y-translation-invariant strip, and no transverse contacts, edges, or bias appear anywhere. So the advertised application is not derived.

What is genuinely good: the symmetry bookkeeping is careful and internally consistent. The transverse current vanishes when time-reversal symmetry is restored (κ=0) and when the tilt is zero; the longitudinal current stays reciprocal, with symmetric critical currents; and jy(φ=0) is odd in κ. The decomposition into skew-tunneling and tilt-induced parts is physically transparent, and the ABS analysis in Fig. 4 gives a clear mechanistic story: broken TRS plus tilt makes the transverse group velocities unequal, which is exactly the condition for a nonreciprocal transverse supercurrent. No parameters are fitted to produce the diode effect; the asymmetry is a computed output.

Soft spots, in order of importance. First, the device-level framing overreaches. A Josephson diode normally means rectified transport between terminals. Here the asymmetry is between the maxima of a local equilibrium current-phase relation along +y and −y in an infinite strip. To call it a diode and claim input-output path decoupling, you need to show this current survives as the current between real transverse leads. That is a missing calculation, not a minor omission, because contacts would break the translational invariance that the whole derivation relies on. Second, the 100% quality factor appears at tuned gap parameters and at the interface position x=0; realistic finite-width and finite-temperature effects could reduce it, though the temperature dependence shown is at least honest. Third, the supplementary algebra is compressed; I could follow the route, but not every step is independently checkable from the text, and the numerical implementation is unspecified.

I do not think any of this makes the central physics wrong. The model is standard, the symmetry constraints are respected, and the qualitative prediction of a tilt-assisted transverse supercurrent under broken TRS is plausible. But the paper should either supply a finite-width multiterminal calculation or explicitly restrict its claims to an interface Hall supercurrent.

My recommendation: send it to referees. It deserves serious review, and the revision request should center on the device claim, not on the core calculation.

Referee Report

3 major / 5 minor

Summary. This paper proposes a transverse Josephson diode effect (TJDE) in ballistic short SNS junctions based on tilted Dirac materials with valley-dependent gaps. Using a McMillan Green's-function construction and a Furusaki-Tsukada-type formula, the author derives the transverse current density j_y at the NS interface (Eq. (5)) together with the longitudinal Josephson current j_x (Eq. (6)). For parameters representing 8-Pmmn borophene, the transverse current-phase relation is nonsinusoidal and yields unequal positive and negative critical currents, leading to a quality factor Q_t (Eq. (7)) that can reach 100% in magnitude; the longitudinal CPR remains odd with j_x(phi)=-j_x(-phi), so no longitudinal diode effect appears. The ABS analysis attributes the TJDE to tilt-induced asymmetric transverse subgap modes under broken time-reversal symmetry (valley-dependent gaps kappa != 0).

Significance. The strength of the paper is its symmetry bookkeeping and the use of a standard ballistic transport framework. The author explicitly checks the limiting behaviors: j_y vanishes at kappa=0 (time-reversal preserving) and at v_t=0, and the longitudinal current remains reciprocal; these are nontrivial internal consistency checks that are satisfied. The qualitative explanation in terms of asymmetric Andreev subgap modes is physically transparent. The result, if correct, constitutes a new transverse variant of the Josephson diode effect and could be of interest for nonreciprocal superconducting circuits. The paper does not fit parameters to the target effect; the model parameters are taken from the borophene literature and kappa is scanned, so the prediction is not circular. The main caveat is that the calculation is for an infinite strip and computes a local equilibrium current density; the device-level claims go beyond the model.

major comments (3)

[Abstract and 'Transverse Josephson diode effect' section (after Eq. (5))] The abstract states that the TJDE 'can completely decouple the input signal path from the output,' and Fig. 3 presents a 100% quality factor. However, the model is an infinite strip that is translationally invariant along y, and j_y is computed as the equilibrium current density at the NS interface x=0 (Eq. (5) and the discussion after Eq. (3)). No transverse leads, edges, or bias enter the calculation. A transverse current in an infinite strip is not by itself a transport current in a multiterminal device; a finite-width sample would require edge states/return currents and a concrete way to collect the transverse signal. Unless the author provides a finite-width multiterminal calculation or explicitly restricts the claim to the interface current density per unit width, the device-level decoupling claim is unsupported.
[Eq. (7) and Fig. 3] The quality factor Q_t is defined from j_y(phi) at a single point x=0. The transverse current decays exponentially into the superconducting regions (stated in the TJDE section), so the total transverse current through a finite junction is not computed. To make Q_t an observable, the author should define the measurable quantity (e.g., the transverse current per unit length integrated over the junction region, or the current collected by transverse terminals) and demonstrate that the diode efficiency is unchanged under that definition. As written, Q_t characterizes a local current density rather than a supercurrent in a circuit.
[Eq. (5) and Supplementary Material] The central formula (5) is not independently verifiable from the manuscript. The Andreev amplitudes a^ell_1 and a^ell_2 are not given; the SM constructs the Green's function and then jumps to Eq. (S31) after dropping normal-reflection terms, but the matching conditions leading to a^ell_{1,2} are not shown. Because all numerical results and the diode claim follow from these amplitudes, the derivation should be completed (or a computer algebra script should be provided) so that a reader can reproduce Eq. (5) and the CPRs in Figs. 2 and 3.

minor comments (5)

[ABS analysis, paragraph after Fig. 4(b)] The relation E_ABS(ky,phi) = -E_ABS(ky,phi) is a typo; as written it forces E_ABS = 0. Please state the intended symmetry, e.g., E_ABS(ky,phi) = -E_ABS(-ky,-phi) for the particle-hole symmetry.
[Eqs. (3) and (S14)] The notation nu_0 and nu_z is undefined. Please define the Pauli matrices acting in Nambu space.
[Model and formalism, after the Heisenberg equation] The 'source current density operator' j_s is not defined. Please provide its explicit form or clarify the decomposition of the total current so that the statement that j_y is contributed only by the electronic term can be checked.
[Fig. 2 caption] The caption uses 'mu L / hbar v_F = 4.2'; please specify that L is the junction length and mu is the normal-region chemical potential, and that this product sets the normal-region phase.
[Introduction and Model and formalism] Typos: 'In order to obtained' should be 'to obtain'; 'Uptonow' and 'somehighlyefficient' in the Introduction are missing spaces in the compiled text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: TJDE is a computed output of the model, with no fitted parameter or load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained. The transverse current is an output of the BdG calculation (Eqs. 3, 5) using the Furusaki-Tsukada formula, with the valley-dependent gaps and tilt entering only as model inputs; no parameter is fitted to the transverse critical-current asymmetry or to the quality factor Qt, which are computed diagnostics. The 100% quality factor arises from numerically found current-phase relations, not from the definition of Qt. The self-citations ([16], [21], [22], [56]) are background or standard-method citations: [16] motivates the possibility of transverse transport in tilted systems, and [56] is one of two references for the Andreev transfer-matrix construction; none is used as a load-bearing uniqueness theorem or as a substitute for the calculation. The finite-width device claim is a scope limitation: the model assumes translational invariance along y and evaluates an equilibrium interface current, so whether this current is measurable as a transverse terminal current in a multiterminal device is not demonstrated. That is a modeling gap, not a circular reduction. No step in the argument equates a predicted quantity with an input by definition.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces a named effect (transverse Josephson diode effect) but no new physical entities: no new particles, fields, forces, or conserved quantities. The mechanism uses already-known ingredients (tilted Dirac bands, valley-contrasting gaps, proximity s-wave pairing, McMillan/Furusaki-Tsukada formalism). The two free control parameters that the 100% claim depends on are the valley-gap asymmetry κ and the heavy-doping chemical potential μs, with temperature scanned.

free parameters (3)

κ = (λ+ - λ−)/Δ0 (valley-gap asymmetry) = scanned; Q_t reaches ±1 at specific |κ| values (Fig. 3a)
The valley-dependent gaps λ± are treated as tunable (hydrogenation/strain or Floquet fields, per Refs. [31-33]). The headline 100% quality factor is realized only at particular κ values, so κ is a control parameter the central claim depends on.
μs (superconducting-region chemical potential) = 200 Δ0
Chosen to justify the heavy-doping limit |μs| >> {|μ|, Δ} so that normal-reflection contributions to the transverse current are dropped (|μ/μs| ≈ 0.02). The transverse current formula (Eq. 5) and hence Q_t rest on this approximation.
T (temperature, in units of Tc) = scanned from 0.02 to 0.5 Tc
Temperature controls the pairing amplitude through Δ(T) = Δ0 tanh(1.74 sqrt(Tc/T - 1)) and strongly affects Q_t (Fig. 3b). It is a scan parameter, not fitted to data.

assumptions (5)

domain assumption Proximity-induced s-wave pairing with valley-singlet Cooper pairs; the BdG Hamiltonian decouples into two valley sectors
Invoked in the Model section after Eq. (2). If the proximity pairing were valley-dependent or finite-momentum, the transverse current and the diode efficiency would change.
domain assumption Heavy-doping limit in the superconducting regions, |μs| >> {|μ|, Δ}, so normal-reflection contributions to the transverse current are negligible
Used to pass from Eq. (S26) to Eqs. (S27) and (5). The entire transverse current formula and the quality factor Q_t depend on dropping the b-coefficient terms.
domain assumption Short junction limit L << ξ0
The paper states 'We consider the short junction L << ξ0' in the Model section; the CPR and the Andreev level analysis are computed in this limit.
domain assumption Translational invariance along y (ky conserved) in an infinite strip, with no transverse leads or edges in the model
Used to write the Green's function as G(x,x')e^{iky(y-y')} and to define the transverse current per unit length. The claimed device application (decoupled input/output) requires transverse contacts that are absent from the model.
standard math McMillan Green's function construction and the Furusaki-Tsukada current formula correctly give equilibrium Josephson currents
The central calculational method, cited to Refs. [34,37,38]. It is a well-established formalism in the field.

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Cite this review

Pith. "Pith review of Transverse Josephson diode effect in tilted Dirac systems." pith.science (2026). https://pith.science/paper/QNRW5EQA

@misc{pith2026241201394,
  author       = {Pith},
  title        = {Pith review of: Transverse Josephson diode effect in tilted Dirac systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QNRW5EQA}},
  note         = {Machine review of arXiv:2412.01394}
}

abstract

We theoretically study the transverse charge transport in Josephson junctions based on the tilted Dirac materials with valley-dependent gaps. It is shown that a finite tilt-assisted transverse Josephson Hall current is present under broken time-reversal symmetry. This transverse current is driven by the superconducting phase difference across the junction and exhibits a nonsinusoidal current-phase relation, leading to the transverse Josephson diode effect (TJDE), where the critical currents flowing oppositely along the transverse direction are asymmetric. Compared to the conventional longitudinal Josephson diode effect, the predicted TJDE supports a fully polarized diode efficiency with a $100\%$ quality factor and can completely decouple the input signal path from the output, suggesting potential applications for nonreciprocal superconducting devices.

Figures

Figures reproduced from arXiv: 2412.01394 by the authors.

**Figure 2.** FIG. 2. [(a), (b)] Current-phase relation for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** (a), we present the quality factor of the TJDE as a function of κ. The quality factor exhibits the symmetric relation Qt(κ) = −Qt(−κ). This symmetric behavior of Qt is similar to that of the longitudinal Josephson diode effect predicted on the surface of topological insulators [6], where the broken TRS is induced by the external magnetic field. Remarkably, the quality factor can even reach 100% by tuning the valley-… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Andreev level [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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