Analytical solution of coupled self--consistency and linearised Usadel equations for the dirty superconductors at $T_c$ and with the proximity effect

N.G. Pugach; S.S. Seidov

REVIEW 2 major objections 3 minor 27 references

Analytical solution of coupled self--consistency and linearised Usadel equations for the dirty superconductors at $T_c$ and with the proximity effect

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The authors reduce the coupled linearised Usadel and self-consistency equations to a matrix eigenvalue problem whose characteristic equation fixes the critical temperature and whose eigenvector fixes the spatial profiles of the order…

desk verdict A useful reformulation of a classic method, but the printed matrix elements are inconsistent due to missing normalization factors; fix that and the approach likely works. read the letter →

arxiv 2506.20557 v4 pith:AMO6JTSR submitted 2025-06-25 cond-mat.supr-con

S.S. Seidov , N.G. Pugach This is my paper · ORCID

classification cond-mat.supr-con

keywords Usadelequationproximityeffectcriticaltemperatureself-consistencyeigenvalueproblemdirtysuperconductoranomalousGreenfunctionorderparameter

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 tries to prove that the two equations governing a dirty (diffusive) superconducting film at its transition temperature—the linearised Usadel equation for the anomalous Green function and the BCS self-consistency equation for the order parameter—can be solved together without numerical iteration. The authors show that the anomalous Green function can be written as an integral of the order parameter through a separable Green's-function kernel, and that substituting this into the self-consistency equation turns the pair into a single homogeneous Fredholm integral equation. Projecting onto cosine eigenfunctions converts that equation into the matrix eigenvalue problem $c_m = U \sum_n K_{mn} c_n$. The characteristic equation $\det(\hat{K} - 1/U)=0$ then gives the critical temperature $T_c$, and the eigenvector supplies the spatial profiles of the order parameter and of the anomalous Green function. This matters because $T_c$ and those profiles are the quantities needed to design superconducting spin valves and other proximity devices, and the reduction replaces a costly iterative procedure with a finite matrix problem.

What carries the argument

The central object is the matrix $\hat{K}$ with elements $K_{mn}=\int_0^{\omega_D}\tanh(\varepsilon/2T_c)\,d\varepsilon \int_0^l \psi_m(x)\,\Re\,\varphi_n(x,\varepsilon)\,dx$, which combines the Green's function of the operator $\partial_x^2-k^2(\varepsilon)$ with the boundary-condition functions $\lambda^\pm_n(\varepsilon)$. Its diagonal part $K^0_{mn}$ follows from the cosine eigenfunctions $\psi_n(x)=\cos(\pi n x/l)$ and gives the bulk BCS contribution, while $K^1_{mn}$ carries the boundary-induced corrections. The eigenvalue condition $\det(\hat{K}-1/U)=0$ does the work of the original coupled differential-integral system: it selects the critical temperature and the eigenvector that fixes the spatial form of $\Delta(x)$ and $f_s(x)$.

What would settle it

Measure $T_c$ of a dirty superconducting film of known thickness $l$ and diffusion constant $D$ in contact with identical normal-metal layers, as a function of $l$, and compare the data with the roots of $\det(\hat{K}-1/U)=0$ truncated to a few modes. If the measured $T_c(l)$ disagrees with the eigenvalue prediction by more than the linearisation error while a full numerical solution of the coupled nonlinear equations (1) and (2) matches the data, the reduction to the matrix problem is not the right description.

Watch

Extended reading notes

Core claim

The central claim is that the coupled system (1)–(3) reduces, at $T=T_c$, to a linear eigenvalue problem for a matrix $\hat{K}$ whose elements are written out explicitly. The solution of the linearised Usadel equation is expressed as $f_s(x)=\sum_n \varphi_n(x,\varepsilon)\int_0^l \psi_n(x')\Delta(x')\,dx'$, with $\psi_n(x)=\cos(\pi n x/l)$; inserting this into the self-consistency equation and projecting onto the $\psi_n$ gives $c_m=U\sum_n K_{mn}c_n$. The matrix $\hat{K}$ splits into a diagonal part $K^0_{mn}$ that reproduces the bulk BCS self-consistency equation at zeroth order and a boundary part $K^1_{mn}$ built from the coefficients $\lambda^\pm_n(\varepsilon)$ that enforce the linearised proximity boundary conditions. Nonzero solutions exist only when $\det(\hat{K}-1/U)=0$, which determines $T_c$; the eigenvector components $c_n$ determine $\Delta(x)$ and hence $f_s(x)$. For weak boundary influence the same structure supplies a perturbation theory around the homogeneous BCS solution, with the first-order correction to the critical temperature given by $K^0_{00}+K^1_{00}=1/U$.

Load-bearing premise

The derivation treats the boundary parameters $\alpha_0$ and $\alpha_l$ in $f'_s(0)=\alpha_0 f_s(0)$ and $f'_s(l)=\alpha_l f_s(l)$ as real constants that do not depend on energy, position, or the order parameter; if a real contact (for example a ferromagnet or a spin-orbit-coupled layer) makes them energy-dependent or complex, the simple eigenfunction reduction and the explicit matrix $\hat{K}$ no longer apply.

Editorial extensions

If this is right

For any proximity contact that can be described by linearised boundary conditions of the form $f'_s(0)=\alpha_0 f_s(0)$, $f'_s(l)=\alpha_l f_s(l)$, the critical temperature follows from $\det(\hat{K}-1/U)=0$ instead of an iterative numerical loop.
The eigenvector of $\hat{K}$ gives the full spatial profile of the order parameter $\Delta(x)$ and of the anomalous Green function $f_s(x)$ at $T_c$, not just the transition temperature.
For weak boundary effects, the bulk BCS self-consistency equation is the zeroth-order limit, and the proximity effect is a first-order perturbation: $K^1_{00}$ shifts $T_c$ and $c_n\neq 0$ for $n>0$ makes the order parameter nonuniform.
Because the matrix elements depend on film length, diffusion constant, Debye frequency, and the boundary coefficients, the method offers a cheap parameter scan for designing superconducting spin valves and similar devices.

Reading between the lines

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

The separable-kernel structure suggests that only a few low-order cosine modes are needed for thin films, so truncating $\hat{K}$ to a small matrix may already capture the $T_c$ shift; the paper does not test this convergence explicitly, but it follows from the growing denominators $K^0_{nn}-K^0_{00}$.
If a real contact makes the boundary coefficients energy-dependent, one could keep $\lambda^\pm_n(\varepsilon)$ inside the energy integral and build an enlarged kernel instead of a single matrix; the paper's construction does not cover that case, but its Green's-function expansion shows where the generalisation enters.
The same eigenfunction expansion could be applied componentwise to spin-triplet anomalous Green functions when boundary conditions couple spin channels, pointing toward the long-range triplet proximity effect that the introduction lists as a motivation.
The explicit analytic matrix makes it possible to ask how $T_c$ responds to small changes in each physical parameter at once, which is a faster route to device optimisation than rerunning a full numerical solver for every parameter set.

Share X Bluesky LinkedIn Reddit HN

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

Dear colleague,

Here's my take on Seidov & Pugach (arXiv:2506.20557). The paper reduces the coupled linearised Usadel equation and self-consistency equation at T_c to a homogeneous Fredholm eigenvalue problem, and then to a matrix eigenvalue problem. That reduction is the classic de Gennes kernel method, and the authors are explicit about that heritage. What's genuinely new is the explicit construction of the matrix K including boundary-condition terms, plus a perturbative scheme around the homogeneous BCS limit when the interface parameters are small. That is a modest but legitimate extension, and it could be useful for quick estimates in superconducting spin valve design.

The paper does several things well. The derivation is transparent, the homogeneous limit correctly reproduces the bulk BCS self-consistency equation (19), and no parameters are fitted. The authors also correctly note that the general case still requires a numerical eigenvalue problem, which is simpler than full iterative solution, though they don't benchmark that claim.

The soft spot is real, and it's load-bearing. The eigenfunctions in Eq. (7) are written as cos(πnx/l), which are not orthonormal on [0,l]; the normalization factors N_0=l and N_n=l/2 are never stated. Every matrix element computed via Eq. (15) therefore carries implicit 1/N_n factors that are missing. The result is that the same matrix element K^0_00 takes at least three different values depending on whether one uses (7)+(15), the printed (16), or the homogeneous limit (19). The characteristic equation |K−1/U|=0 is thus not well-defined as written, and any predicted T_c or eigenvector from the printed formulas is suspect. This is not a cosmetic issue; it changes the eigenvalue.

The good news is that the approach appears salvageable: if one uses the properly orthonormal basis φ_0=1/√l, φ_n=√(2/l)cos(πnx/l), the l-factors cancel and (19) is recovered. So the mathematical structure is likely right, but the manuscript needs a careful revision of the normalization convention, plus a re-derivation of Eqs. (15)–(16) and (20)–(22).

A secondary limitation: the boundary condition (3) uses constant, energy-independent α0, αl. That is a stated modeling choice, not an error, but it means the 'general case' is less general than the abstract suggests; energy-dependent or complex boundary parameters would break the separable-kernel structure.

Who is this for? Researchers working on proximity-coupled superconducting thin films and spin valves who need a quick semi-analytical estimate of T_c and profile. With the normalization fixed and a worked example added, this would be a solid methods paper. As it stands, the central quantitative output cannot be trusted. I'd send it to peer review with a request for major revision, focused on the normalization and a concrete check against a known numerical case.

Best,

Referee Report

2 major / 3 minor

Summary. The manuscript treats a one-dimensional dirty superconducting film of length l at T_c in the presence of proximity contacts described by linearised boundary conditions. It expresses the solution of the linearised Usadel equation for the anomalous Green function f_s as an integral of the order parameter Δ(x) using an eigenfunction expansion of the Green function, substitutes this into the self-consistency equation, and obtains a homogeneous Fredholm equation that is reduced to a matrix eigenvalue problem c = U K c. The characteristic equation determines T_c, and the eigenvector determines the spatial profiles. The authors verify that in the limit of homogeneous boundary conditions the problem reduces to the bulk BCS self-consistency equation and derive first-order perturbative corrections for weak boundary influence.

Significance. The proposed reduction is appealing: it avoids a full numerical iteration of the coupled differential-integral equations, contains no fitted parameters, and yields an explicit matrix whose spectrum gives T_c and the order-parameter profile. If made internally consistent, this would be a useful tool for proximity-effect and superconducting-spintronics calculations. The main result, however, depends critically on the normalization of the basis functions and on the consistency of the printed matrix elements; these points must be fixed before the quantitative claims can be accepted.

major comments (2)

[Section IV, Eqs. (15), (16), and (19)] The matrix elements K^0_mn are not internally consistent. With the eigenfunctions defined in Eq. (7), ∫_0^l ψ_0^2 dx = l and ∫_0^l ψ_n^2 dx = l/2 for n ≥ 1, so the first term of Eq. (15) yields K^0_00 = l∫_0^{ω_D} [1/(2ε)] tanh(ε/2T_c) dε, whereas Eq. (19) uses K^0_00 = ∫_0^{ω_D} [1/(2ε)] tanh(ε/2T_c) dε, and Eq. (16), for n=0, gives ∫_0^{ω_D} [1/(2 ε l^2)] tanh(ε/2T_c) dε. These three expressions differ by factors of l, and because the characteristic equation |K − 1/U|=0 determines T_c, the predicted critical temperature and eigenvector depend on which expression is used. The reduction is likely salvageable by expanding in an orthonormal basis (φ_0=1/√l, φ_n=√(2/l) cos(πnx/l)) or by carrying all normalization factors explicitly, but as written the central quantitative output is not well defined.
[Section III.A, Eqs. (5)–(9)] The Green-function expansion is written without the normalization factors required by the unnormalized functions of Eq. (7). The resolution of unity for the set {cos(πnx/l)} is (1/l) + (2/l)∑_{n≥1} ... , so Eq. (5) should contain the inverse norms 1/N_n, with N_0=l and N_n=l/2 for n≥1. The same missing factors propagate into the coefficients λ±_n in Appendix A and into the matrix elements K_mn, so without an explicit normalization convention the reader cannot reproduce the expansion or the eigenvalue problem.

minor comments (3)

[Eq. (3)] The notation α0,l is confusing because it seems to denote two distinct coefficients; please write α_0 and α_l and state their dimension (inverse length).
[After Eq. (9)] The function φ_n(x, ε) is defined only as 'the function in curly brackets'; writing it out explicitly would help the reader implement the matrix elements.
[Section V] The claim that solving the eigenvalue problem is computationally simpler than the usual iterative procedure would be strengthened by a brief complexity estimate or a small numerical comparison with an existing method.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the reduction to an eigenvalue problem is a self-contained reformulation.

full rationale

The derivation is self-contained and non-circular. Equation (9) expresses fs as an integral transform of Δ; substituting it into the self-consistency equation (1) yields the homogeneous Fredholm equation (10), and multiplying by ψm and integrating gives the matrix eigenvalue problem (14) with K defined by (15). At no point is a parameter fitted to the target output, nor is the target quantity Tc or the spatial profile Δ(x) assumed as input. The homogeneous-limit check, Eq. (19), is a consistency test against the standard BCS self-consistency equation, not an assumed result. The cited prior works involving the authors (e.g., Refs. [8], [11]–[14]) appear as context or applications and are not load-bearing for the analytical reduction. The normalization inconsistency between Eqs. (16) and (19) visible to a correctness reviewer is a self-consistency error in the printed calculation, not a circular step: the eigenproblem is not defined by assuming its solution. Therefore no circular step is present.

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

The derivation relies on standard BCS/Usadel domain assumptions and on a linear, constant-coefficient Robin boundary condition. No free parameters are fitted; U, ω_D, D, l, α0, α_l are inputs. The central assumption is that the boundary influence can be encoded in two constants independent of energy; the Green's function expansion also assumes an orthonormal mode basis, but the normalisation is not specified in the text.

assumptions (5)

domain assumption The linearised Usadel equation (2) is valid near T_c in the dirty limit.
Standard quasiclassical approximation; stated in the introduction and used throughout.
domain assumption The self-consistency equation (1) with hard cutoff ω_D and real-part spectral integration is the BCS gap equation for the film.
Follows from BCS theory; cited to Tinkham [16].
domain assumption The proximity effect is fully described by linear boundary conditions f'(0)=α0 f(0) and f'(l)=α_l f(l) with constant α0, α_l.
Eq. (3); this is a modelling restriction. Energy-dependent or non-linear boundary conditions would break the derivation.
standard math The eigenfunction expansion (5)-(7) of the Green's function converges and the ψ_n form a complete basis; the normalisation of ψ_n is used ambiguously.
Self-adjoint Sturm-Liouville expansion; however, the text never states whether ψ_n are normalised, which creates the inconsistency between Eqs. (16) and (19).
domain assumption For weak boundary coupling, the dominant eigenvalue belongs to the homogeneous n=0 mode, and perturbation theory in K1 is valid.
Section IV.B assumes α0,l are small and the unperturbed solution is the bulk BCS one; no quantitative condition for validity is given.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Analytical solution of coupled self--consistency and linearised Usadel equations for the dirty superconductors at $T_c$ and with the proximity effect." pith.science (2026). https://pith.science/paper/AMO6JTSR

@misc{pith2026250620557,
  author       = {Pith},
  title        = {Pith review of: Analytical solution of coupled self--consistency and linearised Usadel equations for the dirty superconductors at $T_c$ and with the proximity effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMO6JTSR}},
  note         = {Machine review of arXiv:2506.20557}
}

read the original abstract

In this manuscript we consider a superconducting film in the vicinity of the critical temperature and presence of the proximity effect. We analytically solve the corresponding linearised Usadel equation and the self-consistency equation, defining the critical temperature. This is a system of coupled differential and integral equations for the anomalous Green function and the order parameter of the superconductor. The proximity effect defines the boundary conditions. The formal solution of the system is found for the general case of the linearised boundary conditions defined by the proximity effect, reducing the set of equations to an eigenvalue problem. The latter defines the critical temperature of the superconducting phase transition and the spatial distributions of the anomalous Green function and the superconducting order parameter.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

Boundary Effects in Superconductors,

P. G. De Gennes, “Boundary Effects in Superconductors,”Reviews of Modern Physics, vol. 36, pp. 225–237, Jan. 1964

work page 1964
[2]

Proximity effects in superconductor-ferromagnet heterostructures,

A. I. Buzdin, “Proximity effects in superconductor-ferromagnet heterostructures,”Reviews of Modern Physics, vol. 77, pp. 935–976, Sept. 2005

work page 2005
[3]

Spin-polarized supercurrents for spintronics: a review of current progress,

M. Eschrig, “Spin-polarized supercurrents for spintronics: a review of current progress,”Reports on Progress in Physics, vol. 78, p. 104501, Oct. 2015

work page 2015
[4]

Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures,

F. S. Bergeret, A. F. Volkov, and K. B. Efetov, “Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures,”Reviews of Modern Physics, vol. 77, pp. 1321–1373, Nov. 2005

work page 2005
[5]

General boundary conditions for quasiclassical theory of superconductivity in the diffusive limit: application to strongly spin-polarized systems,

M. Eschrig, A. Cottet, W. Belzig, and J. Linder, “General boundary conditions for quasiclassical theory of superconductivity in the diffusive limit: application to strongly spin-polarized systems,” New Journal of Physics, vol. 17, p. 083037, Aug. 2015

work page 2015
[6]

Introduction to topological superconductivity and majorana fermions,

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and majorana fermions,”Semiconductor Science and Technology, vol. 27, p. 124003, nov 2012. 8

work page 2012
[7]

Majorana Qubits,

F. Hassler, “Majorana Qubits,” 2014

work page 2014
[8]

Density of States in the Heterostructure Ferromagnetic Insulator-Superconductor-Ferromagnetic Insulator,

D. V. Seleznev, S. S. Seidov, N. G. Pugach, D. G. Bezymiannykh, S. I. Mukhin, and B. G. L’vov, “Density of States in the Heterostructure Ferromagnetic Insulator-Superconductor-Ferromagnetic Insulator,”Journal of Superconductivity and Novel Magnetism, vol. 38, p. 9, Feb. 2025

work page 2025

Show all 27 references

[9]

Superconducting spintronics,

J. Linder and J. W. A. Robinson, “Superconducting spintronics,”Nature Physics, vol. 11, pp. 307– 315, Apr. 2015

2015
[10]

Peculiarities of performance of the spin valve for the superconducting current,

P. V. Leksin, A. A. Kamashev, N. N. Garif’yanov, I. A. Garifullin, Y. V. Fominov, J. Schumann, C. Hess, V. Kataev, and B. B¨ uchner, “Peculiarities of performance of the spin valve for the superconducting current,”JETP Letters, vol. 97, pp. 478–482, June 2013

2013
[11]

Boosting the superconducting spin valve effect in a metallic superconductor/ferromagnet heterostructure,

P. V. Leksin, A. A. Kamashev, J. Schumann, V. E. Kataev, J. Thomas, B. B¨ uchner, and I. A. Gar- ifullin, “Boosting the superconducting spin valve effect in a metallic superconductor/ferromagnet heterostructure,”Nano Research, vol. 9, pp. 1005–1011, Apr. 2016

2016
[12]

Superconducting spin-valve effect in heterostructures with ferromagnetic Heusler alloy layers,

A. A. Kamashev, N. N. Garif’yanov, A. A. Validov, J. Schumann, V. Kataev, B. B¨ uchner, Y. V. Fominov, and I. A. Garifullin, “Superconducting spin-valve effect in heterostructures with ferromagnetic Heusler alloy layers,”Physical Review B, vol. 100, p. 134511, Oct. 2019

2019
[13]

Record electron self-cooling in cold-electron bolome- ters with a hybrid superconductor-ferromagnetic nanoabsorber and traps,

A. V. Gordeeva, A. L. Pankratov, N. G. Pugach, A. S. Vasenko, V. O. Zbrozhek, A. V. Blago- datkin, D. A. Pimanov, and L. S. Kuzmin, “Record electron self-cooling in cold-electron bolome- ters with a hybrid superconductor-ferromagnetic nanoabsorber and traps,”Scientific Reports...

2020
[14]

Ferro- magnetic planar Josephson junction with transparent interfaces: aφjunction proposal,

D. M. Heim, N. G. Pugach, M. Y. Kupriyanov, E. Goldobin, D. Koelle, and R. Kleiner, “Ferro- magnetic planar Josephson junction with transparent interfaces: aφjunction proposal,”Journal of Physics: Condensed Matter, vol. 25, p. 215701, May 2013

2013
[15]

Generalized Diffusion Equation for Superconducting Alloys,

K. D. Usadel, “Generalized Diffusion Equation for Superconducting Alloys,”Physical Review Letters, vol. 25, pp. 507–509, Aug. 1970

1970
[16]

Tinkham,Introduction to Superconductivity: Second Edition

M. Tinkham,Introduction to Superconductivity: Second Edition. Mineola, NY: Dover Publica- tions, 2015

2015
[17]

Comparison of several methods for determining the critical temper- ature of a superconducting transition in ferromagnet/superconductor heterostructures,

V. Tumanov and Y. Proshin, “Comparison of several methods for determining the critical temper- ature of a superconducting transition in ferromagnet/superconductor heterostructures,”Magnetic resonance in solids, vol. 26, no. 1, 2024

2024
[18]

Nonmonotonic critical temperature in superconductor/ferromagnet bilayers,

Y. V. Fominov, N. M. Chtchelkatchev, and A. A. Golubov, “Nonmonotonic critical temperature in superconductor/ferromagnet bilayers,”Physical Review B, vol. 66, p. 014507, June 2002

2002
[19]

Quasiclassical equations of the theory of superconductivity for contiguous metals and the properties of constricted microcontacts,

A. Zaitsev, “Quasiclassical equations of the theory of superconductivity for contiguous metals and the properties of constricted microcontacts,”Soviet Physics - JETP, vol. 59, no. 5, pp. 1015–1024,
[20]

Influence of boundary transparency on the critical current of dirty SS’S structures,

M. Kurpianov and V. Lukichev, “Influence of boundary transparency on the critical current of dirty SS’S structures,”Soviet Physics - JETP (English Translation), vol. 67, no. 6, pp. 1163–1168,
[21]

Quasiclassical boundary conditions for spin-orbit coupled interfaces with spin-charge conversion,

J. Linder and M. Amundsen, “Quasiclassical boundary conditions for spin-orbit coupled interfaces with spin-charge conversion,”Physical Review B, vol. 105, p. 064506, Feb. 2022

2022
[22]

Spin-orbit coupling as a source of long-range triplet proximity effect in superconductor-ferromagnet hybrid structures,

F. S. Bergeret and I. V. Tokatly, “Spin-orbit coupling as a source of long-range triplet proximity effect in superconductor-ferromagnet hybrid structures,”Physical Review B, vol. 89, p. 134517, Apr. 2014

2014
[23]

A universal phenomenology of charge-spin intercon- version and dynamics in diffusive systems with spin-orbit coupling,

T. Kokkeler, F. S. Bergeret, and I. Tokatly, “A universal phenomenology of charge-spin intercon- version and dynamics in diffusive systems with spin-orbit coupling,” Jan. 2025. arXiv:2405.06334

2025 arXiv
[24]

D. G. Duffy,Green ’s Functions with Applications. Boca Raton, Fla. London: Chapman and Hall/CRC, 2001

2001
[25]

A. D. Polyanin and A. V. Manzhirov,Handbook of Integral Equations. Boca Raton, Fla.: CRC Press, 1998. Appendix A: F unctionsλ± n (ε) In this section we derive functionsλ ± n (ε) in (8). First we substitute the solution (4) of the Usadel equation in the boundary conditions (3) ...

1998
[1984]

INIS Reference Number: 16069709. 9
[1988]

INIS Reference Number: 22024754

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Boundary Effects in Superconductors,

P. G. De Gennes, “Boundary Effects in Superconductors,”Reviews of Modern Physics, vol. 36, pp. 225–237, Jan. 1964

work page 1964

[2] [2]

Proximity effects in superconductor-ferromagnet heterostructures,

A. I. Buzdin, “Proximity effects in superconductor-ferromagnet heterostructures,”Reviews of Modern Physics, vol. 77, pp. 935–976, Sept. 2005

work page 2005

[3] [3]

Spin-polarized supercurrents for spintronics: a review of current progress,

M. Eschrig, “Spin-polarized supercurrents for spintronics: a review of current progress,”Reports on Progress in Physics, vol. 78, p. 104501, Oct. 2015

work page 2015

[4] [4]

Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures,

F. S. Bergeret, A. F. Volkov, and K. B. Efetov, “Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures,”Reviews of Modern Physics, vol. 77, pp. 1321–1373, Nov. 2005

work page 2005

[5] [5]

General boundary conditions for quasiclassical theory of superconductivity in the diffusive limit: application to strongly spin-polarized systems,

M. Eschrig, A. Cottet, W. Belzig, and J. Linder, “General boundary conditions for quasiclassical theory of superconductivity in the diffusive limit: application to strongly spin-polarized systems,” New Journal of Physics, vol. 17, p. 083037, Aug. 2015

work page 2015

[6] [6]

Introduction to topological superconductivity and majorana fermions,

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and majorana fermions,”Semiconductor Science and Technology, vol. 27, p. 124003, nov 2012. 8

work page 2012

[7] [7]

Majorana Qubits,

F. Hassler, “Majorana Qubits,” 2014

work page 2014

[8] [8]

Density of States in the Heterostructure Ferromagnetic Insulator-Superconductor-Ferromagnetic Insulator,

D. V. Seleznev, S. S. Seidov, N. G. Pugach, D. G. Bezymiannykh, S. I. Mukhin, and B. G. L’vov, “Density of States in the Heterostructure Ferromagnetic Insulator-Superconductor-Ferromagnetic Insulator,”Journal of Superconductivity and Novel Magnetism, vol. 38, p. 9, Feb. 2025

work page 2025