Pith. sign in

REVIEW 3 major objections 4 minor 62 references

Quasiclassical circuit-theory of contiguous disordered multiband superconductors

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that Josephson current in disordered multiband junctions with coexisting spin-density-wave order is fixed by bulk Green's functions and transmission eigenvalues alone, derived from the quasiclassical equations by Yip's…

desk verdict A useful and mostly sound extension of quasiclassical circuit theory to multiband SC+SDW junctions, but the central current formula deserves a fuller derivation before I would build on it. read the letter →

arxiv 1908.07953 v2 pith:NYWDTEBI submitted 2019-08-21 cond-mat.str-el

Ammar A. Kirmani , Maxim Dzero , Alex Levchenko This is my paper · ORCID
classification cond-mat.str-el PACS 74.45.+c74.50.+r74.20.Rp
keywords quasiclassicaltheoryofsuperconductivityEilenbergerequationsJosephsonjunctionsmultibandsuperconductorss±-wavepairingspin-density-waveordercircuitAndreevreflectiondisorderin
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 the standard microscopic route to superconducting junctions—the Yip auxiliary-solution method applied to Eilenberger equations with Zaitsev boundary conditions—automatically reproduces the connector rules of Nazarov circuit theory. It then works out a concrete case: a contact between two disordered two-band superconductors with coexisting $s^{\pm}$-wave pairing and spin-density-wave order. The payoff is a compact current-phase relation in which the supercurrent is fixed by bulk Green's functions and the distribution of transmission eigenvalues alone, and is suppressed as $J_c \sim \Delta^2/(eR_N M)$ when the magnetic order dominates. If this is right, the current-phase relationship of such junctions is universal in the same sense as the standard Ambegaokar-Baratoff and Kulik-Omelyanchouk results, and the same machinery extends to other correlated-electron proximity circuits.

What carries the argument

The load-bearing device is Yip's auxiliary-solution construction: instead of solving the Eilenberger equation with the nonlinear Zaitsev boundary conditions directly, one solves two auxiliary (divergent) propagators on each side of the interface, combines them into the bounded physical solution via Eq. (A3), and finds that the interface matching conditions take the rational circuit-theory form of Eq. (A5). The essential objects are the bulk quasiclassical Green's functions ($g_z$, $f_z$, $s_z$, ...) and the transmission-eigenvalue distribution $\rho(D)$, with the transparency $D$ entering only through the denominators. This machinery converts the full boundary-value problem into a one-parameter average over transmissions and yields the central trace formula Eq. (27) and its compact current-phase consequences, Eqs. (43) and (45).

What would settle it

Numerically solve the full Eilenberger equations retaining the phase-gradient term $\partial\chi/\partial x$ for a junction whose electrodes carry a spatially varying phase and compare the resulting Josephson current with Eq. (27); a significant change would falsify the claim that the term can be ignored. A complementary experiment is to measure the current-phase relation of a disordered $s^{\pm}$-wave junction with strong spin-density-wave order and check whether the critical current follows $\Delta^2/(eR_N M)$ rather than the clean BCS scaling.

Watch

Extended reading notes

Core claim

The central claim is that the auxiliary-solution construction converts a nonlinear boundary-value problem into circuit-theory form: the interface values of the quasiclassical propagators are rational functions of the bulk left and right propagators with the transparency $D$ as the only parameter, exactly as in Nazarov's boundary rules. In the coexistence model the Josephson current is therefore a transmission-eigenvalue average of bulk quasiclassical propagators, Eq. (43), which in the symmetric step-function limit reduces to $J(\chi) = c_p \frac{\Delta^2}{eR_N} \frac{\sin\chi}{\sqrt{M^2+\Delta^2\cos^2(\chi/2)}}$, Eq. (45). The paper further shows numerically that the superconducting order parameter varies only in a narrow neighbourhood of the interface while the spin-density-wave order varies on a longer scale, justifying the step-function approximation used to derive these universal current-phase relations. Disorder and magnetization enter through the anomalous Green's functions and smear the sharp features of the voltage-dependent critical current compared with a clean BCS contact.

Load-bearing premise

The whole calculation rests on assuming that the superconducting phase is essentially uniform inside each electrode so that the phase-gradient term generated by the unitary transformation can be dropped; if the phase actually varies across the junction, the central trace formula and every Josephson result built on it miss additional contributions.

Editorial extensions

If this is right

  • For short junctions between arbitrary superconductors, the Josephson current is fixed by bulk quasiclassical Green's functions and the transmission distribution, with no additional interface fitting parameters.
  • In the regime where spin-density-wave order dominates, $M \gg \Delta$, the critical current scales as $\Delta^2/(eR_N M)$; this suppression is a concrete, measurable signature inside the magnetic phase.
  • The self-consistent profiles justify treating the order parameters as step functions, so current-phase relations computed from bulk values describe the full junction.
  • Disorder-induced interband scattering broadens the singular features in $I_c(V)$, so junction spectroscopy of such materials cannot be interpreted with clean BCS formulas.
  • The known universal Josephson results—Ambegaokar-Baratoff, Kulik-Omelyanchouk, Kupriyanov-Lukichev, and chaotic-cavity limits—all reappear as special cases of the same transmission averaging.

Reading between the lines

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

  • A natural extension, not carried out in the paper, is to apply the same auxiliary-solution machinery to junctions with other coexisting orders (nematic, charge-density wave) and to three-band models where nematic order can appear.
  • The phrase in the paper justifying the dropped phase-gradient term ('one does expect the phase to vary substantially') is likely a typo for 'does not expect'; explicitly retaining $\partial\chi/\partial x$ in a numerical solution would test whether the central trace formula survives when the phase varies inside the electrodes.
  • The predicted $J_c \sim \Delta^2/(eR_N M)$ suppression offers a diagnostic: in a disordered iron-pnictide junction, the critical current should dip inside the SDW-dominated part of the phase diagram, tracking the magnetic order parameter.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a quasiclassical Eilenberger theory for Josephson junctions between disordered multiband superconductors with coexisting s± superconducting and spin-density-wave order. The authors use the Yip auxiliary-solution method to satisfy the nonlinear Zaitsev boundary conditions, solve the Eilenberger equations self-consistently for the spatial profiles of the superconducting and magnetic order parameters, and compute the local density of states and the Josephson current as functions of phase and voltage. They then connect their trace formula to Nazarov circuit theory, give averaged expressions in terms of transmission-eigenvalue distributions, and present a compact current-phase relation for the SDW-coexisting case. The central claim is that the Josephson physics of such junctions is determined by bulk quasiclassical propagators plus the transmission distribution, with a universal suppression Jc ~ Δ²/(e RN M) when magnetic order dominates.

Significance. If the central trace formula and its circuit-theory interpretation are correct, this is a useful and fairly general result: it extends Nazarov's circuit theory beyond single-band systems to multiband superconductors with competing orders, and it provides a common framework that reproduces known limits such as Ambegaokar-Baratoff and Kulik-Omelyanchuk. The paper's strengths include a fully self-consistent numerical solution of the Eilenberger equations with nonlinear boundary conditions, explicit phase-diagram and density-of-states computations, and a clear demonstration of how transmission averaging reproduces several previously known current-phase relations. The main weakness is that the central trace formula is asserted rather than derived, and the approximation under which the spatial phase gradient is neglected is not quantified; these issues carry over into all Josephson-current results and the final averaged formulas.

major comments (3)
  1. [Section IV.B, Eq. (27)] The central current trace formula is introduced as the result of "somewhat lengthy calculation" but is not actually derived in the main text or in the appendices. Appendices A–D construct the auxiliary solutions and bulk propagators and state the matching coefficients, but they stop short of showing how these ingredients produce the trace formula Eq. (27). Because Eq. (27) is the starting point for all Josephson-current results in Figs. 6–7 and for Eqs. (42)–(45), the derivation must be supplied in a revised version, either in the main text or in a dedicated appendix with the key intermediate steps.
  2. [Section IV.B, Eqs. (25)–(27)] The unitary transformation Eq. (26) introduces a term proportional to ∂χ/∂x in the Eilenberger equation, and the paper drops this term with the sentence "This term, however, can be ignored for one does expect the phase to vary substantially across the junction." The sentence appears to contain a typo (presumably "does not expect"), but more importantly no estimate is given for the size of the neglected term. In a current-carrying Josephson junction the phase gradient in the electrodes is tied to the supercurrent and is not obviously negligible near a high-transparency interface; since Eq. (27) is obtained after this omission, this unquantified approximation is load-bearing. The authors should justify the short-junction limit explicitly, estimate (ℏvF/2)∂χ/∂x relative to Δ and M, or present a benchmark calculation that retains the phase-gradient term.
  3. [Section V, Eq. (45)] The compact current-phase relation Eq. (45) contains an unspecified prefactor cp described only as "of the order of one." Since Eq. (45) is presented as a closed-form result and is used for the scaling claim Jc ~ Δ²/(e RN M), leaving cp undetermined turns a supposedly quantitative formula into a parametric statement. The authors should evaluate cp explicitly from the transmission-average integral, or at least provide the integral representation from which cp follows, so that the result is reproducible and falsifiable.
minor comments (4)
  1. [Section IV.B, after Eq. (26)] The sentence "This term, however, can be ignored for one does expect the phase to vary substantially across the junction" appears to contain a typo; it should presumably read "does not expect." As written it asserts the opposite of the intended approximation.
  2. [Section IV.B, after Eq. (27)] The sentence "the quasiclassical functions which account for the magnetic order do not explicitly enter into the expression for the Josephson current" is misleading, because M enters explicitly in Eqs. (42)–(44) and also through the bulk propagators in Eq. (27). The intended statement should be rephrased, for example that M enters only through the bulk Green's functions and not through an additional interface-specific magnetic term.
  3. [Section V, Eq. (42)–(45)] The notation SSDWINISSDW is used without a definition or a diagram; a brief explanation of this junction type would help the reader, especially since it is the basis for Eq. (45).
  4. [Fig. 7] The two non-BCS curves in Fig. 7 are not accompanied by the disorder parameters Γ0 and Γπ used in the calculation; specifying these parameters would make the numerical results reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the Josephson and circuit-theory results are derived from the stated quasiclassical equations and boundary conditions, with independent checks against Ambegaokar-Baratoff and Kulik-Omelyanchuk limits.

full rationale

The paper's central derivation is a forward calculation from the model Hamiltonian, the Eilenberger equations (12), the normalization condition (13), and the Zaitsev boundary conditions (14). The auxiliary-solution construction in Appendix A follows the external Yip method and is used to obtain the interface relations (A5), which are then identified with Nazarov circuit-theory boundary conditions. Equation (27) is presented as the result of a lengthy calculation from these ingredients, not as an assumed input, and the later compact formulas (42)-(45) follow by substituting bulk propagators and averaging over transmission distributions. The reproduction of known limits such as Ambegaokar-Baratoff (33) and Kulik-Omelyanchuk (37) provides an independent benchmark for the method. The self-citations to prior work by the same authors (Refs. 34, 36) concern the bulk two-band coexistence model and phase diagram, but the Josephson and circuit-theory results are not imported from those papers; the bulk equations are rederived in the manuscript. The most load-bearing approximation, the dropping of the ∂χ/∂x term after the unitary transformation (26), is unquantified and is justified with an apparent typo ('can be ignored for one does expect the phase to vary substantially across the junction' instead of 'does not expect'). This is a correctness and completeness concern rather than a circularity: it is not an equivalence-by-definition, a fitted parameter disguised as a prediction, or a self-citation chain, so it does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper's central results rest on the standard quasiclassical framework plus the domain assumptions listed above. No experimental data are fitted; the only undetermined numerical constant is cp in Eq. (45). No new physical entities are introduced.

free parameters (1)
  • cp in Eq. (45) = unspecified, claimed O(1)
    Appears as the prefactor of the SDW-modulated Josephson current. Its value and derivation are absent, and no single constant reproduces the M=0 elliptic-function limit Eq. (39) over the full phase range, so it likely conceals some phase- and energy-dependent content.
assumptions (6)
  • domain assumption Zaitsev boundary conditions with a single transparency D and no trajectory interference are valid for the interfaces considered.
    Used to match quasiclassical functions at x=0; the paper explicitly states that interference between quasiparticle paths on the two sides is ignored.
  • domain assumption Self-consistent Born approximation for disorder is adequate in the coexisting SC+SDW state.
    All disorder effects enter through Γ0 and Γπ self-energies; vertex corrections and localization effects are not assessed.
  • domain assumption The six-component ansatz Eq. (16) closes the Eilenberger equations for the s± plus SDW model.
    Projection onto these components and normalization Eqs. (22)-(24) are shown, but completeness of the basis for arbitrary interface trajectories is assumed.
  • domain assumption The Yip auxiliary-solution procedure requires that order parameters recover their bulk values away from the interface.
    Stated in Appendix A as the only assumption of the method; used to construct the bounded physical solution.
  • domain assumption The spatial phase gradient term after the unitary transformation (26) can be neglected.
    Needed to reduce the Eilenberger equation to real order parameters; the manuscript's wording is garbled and the condition is not quantified.
  • domain assumption Padé analytic continuation reliably converts Matsubara results to real-frequency LDOS.
    Used for Figs. 4-5 without specifying the approximation order or error.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quasiclassical circuit-theory of contiguous disordered multiband superconductors." pith.science (2026). https://pith.science/paper/NYWDTEBI

@misc{pith2026190807953,
  author       = {Pith},
  title        = {Pith review of: Quasiclassical circuit-theory of contiguous disordered multiband superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYWDTEBI}},
  note         = {Machine review of arXiv:1908.07953}
}
abstract

We consider a general problem of a Josephson contact between two multiband superconductors with coexisting superconducting and magnetic phases. As a particular example, we use the quasiclassical theory of superconductivity to study the properties of a Josephson contact between two disordered $s^{\pm}$-wave superconductors allowing for the coexistence between superconductivity and spin-density-wave orders. The intra- and inter-band scattering effects of disorder are treated within the self-consistent Born approximation. We calculate the spatial profile of the corresponding order parameters on both sides of the interface assuming that the interface has finite reflection coefficient and use our results to evaluate the local density of states at the interface as well as critical supercurrent through the junction as a function of phase or applied voltage. Our methods are particularly well suited for describing spatially inhomogeneous states of iron-based superconductors where controlled structural disorder can be created by an electron irradiation. We reveal the connection between our theory and the circuit-theory of Andreev reflection and extend it to superconducting junctions of arbitrary nature. Lastly, we outline directions for further developments in the context of proximity circuits of correlated electron systems.

Figures

Figures reproduced from arXiv: 1908.07953 by the authors.

Figure 1
Figure 1. FIG. 1: Phase diagram obtained by numerical solution of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Schematic representation of the Josephson contact [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (color online) Results for the spatial dependence of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Panel (a): local density of states in the bulk plot [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Josephson current through the junction as a function [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Plot of the real and imaginary parts of the Josephson [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Schematic plot illustrating the procedure of finding [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 52 canonical work pages

  1. [1]

    This completes the proof of the method based on find- ing auxiliary solutions

    Similarly, anticommutator in the numerator of the first equation in (D5), can be made diagonal by setting cl 1 = cl(Mr∆l−Ml∆r)λl n ωrn(∆2 l +M2 l )−ωln(∆r∆l +MrMl) (D7) and changing sign of cl in the second equation in (D5). This completes the proof of the method based on find- ing auxiliary solutions. The most interesting property of this method is that in...

    work page

  2. [2]

    A. A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Elsevier Science Publishers, 1988)

    work page 1988

  3. [3]

    A. N. Svidzinsky, Spatially inhomogeneous problems in the theory of superconductivity (Science, Moscow, 1982)

    work page 1982

  4. [4]

    Blatter, M

    G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994)

    1994

  5. [5]

    B. D. Josephson, Rev. Mod. Phys. 36, 216 (1964)

    work page 1964

  6. [6]

    A. A. Golubov, M. Y. Kupriyanov, and E. Il’ichev, Rev. Mod. Phys. 76, 411 (2004)

    2004

  7. [7]

    Eilenberger, Zeitschrift f¨ ur Physik214, 195 (1968)

    G. Eilenberger, Zeitschrift f¨ ur Physik214, 195 (1968)

    work page 1968

  8. [8]

    K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970)

    1970

Show all 62 references
  1. [9]

    C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997)

    1997

  2. [10]

    Y. V. Nazarov, Phys. Rev. Lett. 73, 1420 (1994)

    1994

  3. [11]

    Y. V. Nazarov, Superlattices and Microstructures 25, 1221 (1999)

    1999

  4. [12]

    Argaman, Europhysics Letters (EPL) 38, 231 (1997)

    N. Argaman, Europhysics Letters (EPL) 38, 231 (1997)

    1997

  5. [13]

    Dorokhov, Solid State Communications 51, 381 (1984)

    O. Dorokhov, Solid State Communications 51, 381 (1984)

    1984

  6. [14]

    K. M. Schep and G. E. W. Bauer, Phys. Rev. Lett. 78, 3015 (1997)

    1997

  7. [15]

    Shibauchi, A

    T. Shibauchi, A. Carrington, and Y. Matsuda, Annual Review of Condensed Matter Physics 5, 113 (2014)

    2014

  8. [16]

    Chubukov, Annual Review of Condensed Matter Physics 3, 57 (2012)

    A. Chubukov, Annual Review of Condensed Matter Physics 3, 57 (2012)

    2012

  9. [17]

    A. Moor, A. F. Volkov, and K. B. Efetov, Phys. Rev. B 83, 134524 (2011)

    2011

  10. [18]

    Apostolov and A

    S. Apostolov and A. Levchenko, Phys. Rev. B 86, 224501 (2012)

    2012

  11. [19]

    Vakaryuk, V

    V. Vakaryuk, V. Stanev, W.-C. Lee, and A. Levchenko, Phys. Rev. Lett. 109, 227003 (2012)

    2012

  12. [20]

    Lin, Phys

    S.-Z. Lin, Phys. Rev. B 86, 014510 (2012)

    2012

  13. [21]

    V. G. Stanev and A. E. Koshelev, Phys. Rev. B 86, 174515 (2012)

    2012

  14. [22]

    A. E. Koshelev and V. Stanev, EPL (Europhysics Let- ters) 96, 27014 (2011)

    2011

  15. [23]

    A. A. Golubov and I. I. Mazin, Applied Physics Letters 102, 032601 (2013)

    2013

  16. [24]

    E. Berg, N. H. Lindner, and T. Pereg-Barnea, Phys. Rev. Lett. 106, 147003 (2011)

    2011

  17. [25]

    Y. S. Yerin and A. N. Omelyanchouk, Low Temperature Physics 36, 969 (2010)

    2010

  18. [26]

    Y. Ota, M. Machida, T. Koyama, and H. Matsumoto, Phys. Rev. Lett. 102, 237003 (2009)

    2009

  19. [27]

    Y. Ota, N. Nakai, H. Nakamura, M. Machida, D. Inotani, Y. Ohashi, T. Koyama, and H. Matsumoto, Phys. Rev. B 81, 214511 (2010)

    2010

  20. [28]

    W.-Q. Chen, F. Ma, Z.-Y. Lu, and F.-C. Zhang, Phys. Rev. Lett. 103, 207001 (2009)

    2009

  21. [29]

    Tsai, D.-X

    W.-F. Tsai, D.-X. Yao, B. A. Bernevig, and J. Hu, Phys. Rev. B 80, 012511 (2009)

    2009

  22. [30]

    Linder, I

    J. Linder, I. B. Sperstad, and A. Sudbø, Phys. Rev. B 80, 020503 (2009)

    2009

  23. [31]

    T. K. Ng and N. Nagaosa, EPL (Europhysics Letters) 87, 17003 (2009)

    2009

  24. [32]

    Yip, Jour

    S.-K. Yip, Jour. of Low Temp. Phys. 109, 547 (1997)

    1997

  25. [33]

    A. V. Zaitsev, Sov. Phys. JETP 59, 1015 (1984)

    1984

  26. [34]

    M. G. Vavilov and A. V. Chubukov, Phys. Rev. B 84, 214521 (2011)

    2011

  27. [35]

    Dzero, M

    M. Dzero, M. Khodas, A. D. Klironomos, M. G. Vavilov, and A. Levchenko, Phys. Rev. B 92, 144501 (2015)

    2015

  28. [36]

    A. A. Kirmani, A. Levchenko, and M. Dzero, unpublished (2019)

    2019

  29. [37]

    Dzero and A

    M. Dzero and A. Levchenko, Phys. Rev. B 98, 054501 (2018)

    2018

  30. [38]

    A. B. Vorontsov, M. G. Vavilov, and A. V. Chubukov, 14 Phys. Rev. B 81, 174538 (2010)

    2010

  31. [39]

    R. M. Fernandes and J. Schmalian, Phys. Rev. B 82, 014521 (2010)

    2010

  32. [40]

    Kieselmann, Phys

    G. Kieselmann, Phys. Rev. B 35, 6762 (1987)

    1987

  33. [41]

    Bruder, Phys

    C. Bruder, Phys. Rev. B 41, 4017 (1990)

    1990

  34. [42]

    ˇSmakov, I

    J. ˇSmakov, I. Martin, and A. V. Balatsky, Phys. Rev. B 64, 212506 (2001)

    2001

  35. [43]

    Naaman, W

    O. Naaman, W. Teizer, and R. C. Dynes, Phys. Rev. Lett. 87, 097004 (2001)

    2001

  36. [44]

    Suderlow, I

    H. Suderlow, I. Guillamon, G. Rodrigo, J, and S. Vieira, Supercond. Sci. Technol. 27, 063001 (2014)

    2014

  37. [45]

    M. H. Hamidian, S. D. Edkins, S. H. Joo, A. Kostin, H. Eisaki, S. Uchida, M. J. Lawler, E. A. Kim, A. P. Mackenzie, K. Fujita, et al., Nature 532, 343 EP (2016)

    2016

  38. [46]

    M. T. Randeria, B. E. Feldman, I. K. Drozdov, and A. Yazdani, Phys. Rev. B 93, 161115 (2016)

    2016

  39. [47]

    Graham and D

    M. Graham and D. K. Morr, Phys. Rev. Lett. 123, 017001 (2019)

    2019

  40. [48]

    I. O. Kulik and I. K. Yanson, The Josephson Effect in Su- perconducting Tunneling Structures (Israel Program for Scientific Translations, 1972)

    1972

  41. [49]

    C. W. J. Beenakker, Transport Phenomena in Mesoscopic Systems, edited by H. Fukuyama and T. Ando (Springer, Berlin, 1992) [arXiv: cond-mat/0406127] (2004)

    2004 arXiv

  42. [50]

    L. G. Aslamazov and A. I. Larkin, Sov. Phys. JETP 43, 698 (1976)

    1976

  43. [51]

    Ambegaokar and A

    V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10, 486 (1963)

    1963

  44. [52]

    I. O. Kulik and A. N. Omel’yanchuk, Sov. J. Low Temp. Phys. 4, 142 (1978)

    1978

  45. [53]

    I. O. Kulik and A. N. Omel’yanchuk, JETP Lett. 21, 96 (1975)

    1975

  46. [54]

    A. A. Golubov, M. Y. Kupriyanov, and V. F. Lukichev, Fizika Nizkikh Temperatur 10, 799 (1984)

    1984

  47. [55]

    Brinkman and A

    A. Brinkman and A. A. Golubov, Phys. Rev. B 61, 11297 (2000)

    2000

  48. [56]

    P. W. Brouwer and C. W. J. Beenakker, Phys. Rev. B 51, 7739 (1995)

    1995

  49. [57]

    C. M. Whisler, M. G. Vavilov, and A. Levchenko, Phys. Rev. B 97, 224515 (2018)

    2018

  50. [58]

    Eschrig, Phys

    M. Eschrig, Phys. Rev. B 61, 9061 (2000)

    2000

  51. [59]

    Tanaka, Y

    Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev. B 76, 054522 (2007)

    2007

  52. [60]

    Tanaka, Y

    Y. Tanaka, Y. Asano, and A. A. Golubov, Phys. Rev. B 77, 220504 (2008)

    2008

  53. [61]

    B. Lu, P. Burset, Y. Tanuma, A. A. Golubov, Y. Asano, and Y. Tanaka, Phys. Rev. B 94, 014504 (2016)

    2016

  54. [62]

    Burset, B

    P. Burset, B. Lu, S. Tamura, and Y. Tanaka, Phys. Rev. B 95, 224502 (2017)

    2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.