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arxiv: 2603.01269 · v1 · pith:PP3J653Anew · submitted 2026-03-01 · ❄️ cond-mat.mes-hall

Sub-Sharvin conductance and Josephson effect in graphene

Adam Rycerz This is my paper

Pith reviewed 2026-05-21 11:45 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords grapheneJosephson junctionsuperconductor-graphene-superconductorDirac-Bogoliubov-de Gennessub-Sharvin conductanceSharvin conductancepotential profileballistic transport
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0 comments X

The pith

Tuning the electrostatic potential in graphene Josephson junctions from rectangular to parabolic raises IcRN toward the ballistic limit in the unipolar regime.

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

The paper numerically solves the Dirac-Bogoliubov-de Gennes equation for short superconductor-graphene-superconductor junctions while varying the longitudinal electrostatic potential profile via gate electrodes. It finds that in the unipolar regime where the chemical potential sits above the barrier top, the product IcRN evolves continuously from the graphene-specific range of 2.1-2.4 e/Δ0 toward the ballistic value of π e/Δ0. At the same time the normal-state conductance rises from the sub-Sharvin value of roughly one-quarter the Sharvin conductance to the full Sharvin conductance G_Sharvin = g0 |μ| W / (π ħ vF). In the tripolar regime with negative chemical potential both conductance and critical current are suppressed by potential smoothing, yet IcRN stays near the graphene-specific values.

Core claim

Numerical analysis shows that smoothing the potential from rectangular to parabolic causes IcRN to increase gradually toward the ballistic bound while normal conductance approaches the Sharvin limit when μ > 0; when μ < 0 both quantities decrease but IcRN remains in the graphene range of roughly 2.1-2.4 e/Δ0, with the current-phase relation skewness also examined.

What carries the argument

Numerical solution of the Dirac-Bogoliubov-de Gennes equation for continuously tunable rectangular-to-parabolic potential profiles that control Cooper-pair tunneling.

If this is right

  • In the unipolar regime, gate-controlled potential smoothing can continuously interpolate between graphene-specific and ballistic Josephson transport.
  • Normal-state conductance can be increased from sub-Sharvin to full Sharvin value by the same potential tuning.
  • In the tripolar regime the product IcRN is robust against potential smoothing even though absolute currents drop.
  • The current-phase relation skewness varies with the potential profile shape.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Gate engineering of potential curvature offers a route to tunable Josephson devices without changing junction length or doping.
  • Similar potential-profile effects may appear in other Dirac-fermion Josephson systems such as topological insulators or transition-metal dichalcogenides.
  • The sub-Sharvin to Sharvin crossover in normal conductance provides an independent experimental signature of the same potential tuning.

Load-bearing premise

The chosen grid discretization and boundary conditions for the Dirac-Bogoliubov-de Gennes equation reproduce the continuum physics of the parabolic potential without significant numerical artifacts.

What would settle it

An experiment that measures IcRN staying fixed near 2.1-2.4 e/Δ0 rather than rising toward π e/Δ0 when the potential is made progressively more parabolic in a unipolar graphene junction would falsify the main result.

Figures

Figures reproduced from arXiv: 2603.01269 by Adam Rycerz.

Figure 1
Figure 1. Figure 1: Top: Schematic of a graphene strip of width [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Current-phase relation for S-g-S Josephson junction in the case of rectangular potential barrier and infinitely-doped leads, corre [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Normal-state conductance 1/RN (top) and critical current Ic (bottom) for the system of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: It easy to see that the characteristics of such [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 4
Figure 4. Figure 4: Product IcRN for the data shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Skewness of the current-phase relation S displayed versus the chemical potential for the same system parameters as in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Product IcRN displayed versus skewness of the current-phase relation (datapoints). Each dataset contains nine datapoints corre￾sponding to m = 2, 4, . . . , 256, and ∞, for a fixed value of the chemical potential, µ = ±0.1 eV, or ±0.2 eV (see the legend). Remaining system parameters are same as in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

Titov and Beenakker [Phys. Rev. B 74, 041401(R) (2006)] found, by solving the Dirac-Bogoliubov-De-Gennes equation, that the product of critical current and normal-state resistance for superconductor-graphene-superconductor (S-g-S) Josephson junction takes values (for a short junction and zero temperature) between $I_cR_N\approx{}2.1$ and $I_cR_N\approx{}2.4$ in units of $e/\Delta_0$, where $\Delta_0$ is the superconducting gap. These values are notably higher than the tunnelling bound ($\pi/2$), but lower than the ballistic bound ($\pi$). Here we analyze numerically the tunneling of Cooper pairs through S-g-S junctions in which the longitudinal electrostatic potential profile is tuned, within gates electrodes, from a rectangular to a parabolic one. In the unipolar regime (i.e., when the chemical potential is above the top of a barrier, $\mu>0$), it is found that $I_cR_N$ gradually evolves from the graphene-specific to the ballistic value. At the same time, the normal-state conductance increases from the sub-Sharvin value of $1/R_N\approx(\pi/4)\,G_{\rm Sharvin}$ towards to the Sharvin value $G_{\rm Sharvin}=g_0|\mu|W/(\pi\hbar{}v_F)$, with the conductance quantum $g_0=4e^2/h$, the junction width $W$, and the Fermi velocity in graphene $v_F$. In contrast, in the tripolar regime ($\mu<0$), both normal-state conductance and the critical current are suppressed when smoothing the potential; however, $I_c{}R_N$ remains close to the graphene-specific range, even for a parabolic potential. The skewness of the current-phase relation is also discussed.

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 numerically solves the Dirac-BdG equation for S-g-S Josephson junctions in graphene, examining the effect of tuning the longitudinal electrostatic potential profile from rectangular to parabolic. In the unipolar regime (μ>0), it reports that IcRN evolves continuously from the graphene-specific range (≈2.1–2.4 e/Δ0) toward the ballistic value (π e/Δ0) while normal-state conductance rises from sub-Sharvin (≈(π/4)G_Sharvin) to the full Sharvin limit; in the tripolar regime (μ<0) both conductance and critical current are suppressed but IcRN remains near the graphene-specific values. The skewness of the current-phase relation is also discussed.

Significance. If the numerical trends are robust, the work establishes a tunable crossover between graphene-specific and conventional ballistic Josephson transport via electrostatic potential shaping, extending the Titov-Beenakker results to experimentally relevant smooth profiles and suggesting a route to engineer IcRN and conductance in graphene-based superconducting devices.

major comments (2)
  1. [Numerical implementation] Numerical implementation (Sec. II or equivalent): the discretization of the Dirac-BdG equation for parabolic potentials is presented without reported grid-size scaling, cutoff-energy checks, or explicit comparison of the rectangular limit under identical numerics to the known analytical graphene-specific IcRN. Because the central claim of a smooth sub-Sharvin-to-Sharvin crossover and corresponding IcRN evolution rests on faithful representation of continuum mode transmission and Andreev amplitudes, these convergence diagnostics are required to exclude discretization artifacts.
  2. [Unipolar-regime results] Unipolar-regime results (Sec. III or equivalent, figures showing IcRN vs. potential-shape parameter): the reported gradual increase in IcRN and conductance would be strengthened by an explicit overlay of the rectangular-potential data obtained with the same numerical scheme against the Titov-Beenakker analytic bounds before presenting the parabolic cases.
minor comments (2)
  1. [Abstract / conductance discussion] The definition of G_Sharvin = g0 |μ| W / (π ħ vF) is given in the abstract but should be restated with an equation number in the main text for immediate reference when discussing the sub-Sharvin to Sharvin transition.
  2. [Figure captions] Notation for the chemical potential sign convention (μ>0 unipolar vs. μ<0 tripolar) is clear but would benefit from a brief reminder in the figure captions that accompany the potential-profile sketches.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the numerical implementation and presentation of results. We address each major comment below and describe the revisions that will be incorporated in the next version of the manuscript.

read point-by-point responses

  1. Referee: [Numerical implementation] Numerical implementation (Sec. II or equivalent): the discretization of the Dirac-BdG equation for parabolic potentials is presented without reported grid-size scaling, cutoff-energy checks, or explicit comparison of the rectangular limit under identical numerics to the known analytical graphene-specific IcRN. Because the central claim of a smooth sub-Sharvin-to-Sharvin crossover and corresponding IcRN evolution rests on faithful representation of continuum mode transmission and Andreev amplitudes, these convergence diagnostics are required to exclude discretization artifacts.

    Authors: We agree that explicit convergence diagnostics strengthen the reliability of the numerical results. In the revised manuscript we will add a dedicated subsection (or appendix) reporting grid-size scaling tests, confirming that the chosen cutoff energy captures all relevant modes, and demonstrating that the rectangular-potential limit recovers the Titov-Beenakker analytic IcRN values (within numerical precision) when the same discretization scheme is applied. These additions will explicitly rule out discretization artifacts for the reported crossover. revision: yes

  2. Referee: [Unipolar-regime results] Unipolar-regime results (Sec. III or equivalent, figures showing IcRN vs. potential-shape parameter): the reported gradual increase in IcRN and conductance would be strengthened by an explicit overlay of the rectangular-potential data obtained with the same numerical scheme against the Titov-Beenakker analytic bounds before presenting the parabolic cases.

    Authors: We concur that an explicit baseline comparison improves clarity. In the revised figures of Section III we will include an overlay (or inset) of the rectangular-potential IcRN and conductance values obtained with the identical numerical code, plotted directly against the Titov-Beenakker analytic bounds before displaying the parabolic-profile data. This will make the continuous evolution from the graphene-specific regime to the ballistic limit more transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity: numerical solution of Dirac-BdG for new potential profiles is independent of inputs

full rationale

The paper's central results follow from direct numerical integration of the Dirac-Bogoliubov-de Gennes equation applied to previously unexamined parabolic longitudinal potentials, with the graphene-specific IcRN range (2.1–2.4 e/Δ0) taken verbatim from the external citation to Titov and Beenakker (2006) rather than being redefined or fitted within the present work. No parameter is adjusted to reproduce a target IcRN or conductance value, no self-citation supplies a uniqueness theorem or ansatz that forces the reported crossover, and the sub-Sharvin to Sharvin conductance shift is an output of the same continuum discretization applied to the new profiles. The derivation therefore remains self-contained against the external literature benchmark and does not reduce to any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the Dirac-Bogoliubov-de Gennes equation for graphene and on the numerical representation of gate-defined potential profiles; no new particles or forces are introduced.

free parameters (2)
  • chemical potential μ
    Used to define unipolar (μ>0) and tripolar (μ<0) regimes; its value relative to the barrier height controls the reported qualitative difference.
  • potential profile parameters
    The transition from rectangular to parabolic shape is controlled by gate voltages whose precise functional form is not specified in the abstract.
axioms (2)
  • domain assumption The Dirac-Bogoliubov-de Gennes equation accurately describes quasiparticle transport in graphene Josephson junctions.
    Invoked throughout the abstract as the equation solved numerically.
  • domain assumption Zero-temperature and short-junction limits apply as in the referenced 2006 work.
    The reported IcRN range is compared directly to the zero-temperature short-junction values of Titov and Beenakker.

pith-pipeline@v0.9.0 · 5883 in / 1561 out tokens · 41088 ms · 2026-05-21T11:45:18.084538+00:00 · methodology

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