Superconducting Proximity Effect

A superconducting interface can correlate electrons in a region whose local pairing interaction is zero, while the same interface weakens the parent superconductor. That reciprocal spatial reconstruction is the superconducting proximity effect. It is not merely an Andreev-reflection event at an ideal boundary, and it is not summarized by one fitted induced-gap number.

This page develops the equilibrium bulk boundary-value problem. Its central objects are the anomalous Green function, the local self-consistent pair potential, and the retarded spectrum. The page shows when a trajectory-resolved Eilenberger description is controlled, when impurity isotropization permits a Usadel reduction, how interfaces conserve spectral current, and how inverse proximity and spin-active conversion change the material claim.

Required background. BCS Theory supplies the homogeneous parent paired state, while Green Functions in Many-Body QM supplies the correlator, Dyson, and spectral language.

Helpful background. Thermal Green Functions and the Matsubara Formalism Preview derive the thermal-circle machinery used here. Boltzmann Transport and Disorder in Quantum Matter help distinguish ballistic and diffusive scales. The chapter gateway routes neighboring London, Ginzburg–Landau, Josephson, vortex, and pairing questions.

Declare the Bulk-Proximity Problem

Before choosing an equation, record the same ten fields used by the chapter gateway.

1. Platform and state. Give the layer sequence, thicknesses, lateral dimensions, temperature, applied field, bias, preparation, and equilibrium status.
2. Claimed coherent object. Name the anomalous amplitude, local pair potential, transition temperature, spectral minigap, supercurrent, or other target. Do not use “proximity” as the object.
3. Degrees of freedom. State the bands, Fermi-surface sheets, spin or pseudospin basis, Nambu basis, and which layers retain an attractive interaction.
4. Symmetry and convention. Declare charge, gauge, Fourier, Matsubara, interface-normal, spin, and particle–hole conventions.
5. Scale hierarchy. Compare $E_{\mathrm F}$, parent gap, $k_{\mathrm B}T$, exchange and spin–orbit energies, elastic and inelastic rates, mean free path, layer thicknesses, coherence lengths, and probe resolution.
6. Theory regime. Choose a microscopic Gor’kov or Bogoliubov–de Gennes treatment, trajectory-resolved quasiclassics, a diffusive Usadel reduction, or a near-transition Ginzburg–Landau description. State the outer boundaries, interface matching law, rigid-reservoir or self-consistent choice, and failure test.
7. Requested observable. Specify a local density of states, transition temperature, integrated current, magnetic response, or spatial correlation rather than an unobservable intermediate alone.
8. Forward model. Include contacts, tunneling matrix elements, tip density of states, thermal and instrumental convolution, geometry, and sampling.
9. Evidence and alternatives. Require thickness, temperature, field, or magnetic-configuration controls against heating, pinholes, damaged layers, stray fields, and nonunique broadening.
10. Uncertainty and stopping rule. Separate sample, parameter, interface, solver, continuation, and measurement errors, then name a falsifier and the point at which a more microscopic, nonequilibrium, mesoscopic, or topological owner is required.

The output is a licensed claim, not merely a converged curve. A good stopping statement might be: “These state-matched thickness and spectral data support a diffusive anomalous correlation in the normal layer, but do not determine a unique microscopic transparency.”

Separate Anomalous Amplitude, Pair Potential, and Spectral Minigap

Use inverse-energy imaginary time $\vartheta\in[0,\beta)$ with $\beta=(k_{\mathrm B}T)^{-1}$, so $\psi(\vartheta)=e^{\vartheta(H-\mu N)}\psi e^{-\vartheta(H-\mu N)}$. For fermionic fields, define

$$F_{\alpha\beta}(1,2) = -\left\langle \mathcal T_\vartheta \psi_\alpha(1)\psi_\beta(2) \right\rangle .$$

For any relative-time correlator, the Fourier pair is

$$F(i\varepsilon_n) = \int_0^\beta d\vartheta\, e^{i\varepsilon_n\vartheta}F(\vartheta), \qquad F(\vartheta) = k_{\mathrm B}T\sum_n e^{-i\varepsilon_n\vartheta}F(i\varepsilon_n).$$

Thus Fermi antisymmetry gives, after transforming relative coordinate and inverse-energy imaginary time,

$$F_{\alpha\beta} (\mathbf R,\mathbf p,\varepsilon_n) = -F_{\beta\alpha} (\mathbf R,-\mathbf p,-\varepsilon_n).$$

The momentum-space and Wigner expressions on this page use a fixed gauge. The explicit scalar and linearized spatial benchmarks set $\mathbf A=0$. In a nonzero vector potential, a gauge-covariant Wigner transform must include the appropriate Wilson link between the two field points; a bare relative-coordinate Fourier transform is not gauge covariant.

Throughout this page, $\varepsilon_n$ is a fermionic Matsubara energy,

$$\varepsilon_n=(2n+1)\pi k_{\mathrm B}T,$$

not an angular frequency. In function arguments we suppress the leading $i$ and write $F(\varepsilon_n)$ as shorthand for $F(i\varepsilon_n)$. A full thermal sum means $k_{\mathrm B}T\sum_{n=-\infty}^{\infty}$; expressions restricted to $n\geq0$ display their compensating factor explicitly.

The pair potential is a self-energy produced by an interaction kernel. Define $V_{\alpha\beta,\gamma\delta}$ as the antisymmetrized pair kernel in the displayed index order, with negative eigenvalues in attractive channels. Take the equal-time limit as $F_{\gamma\delta}(0^-)=\langle\psi_\delta\psi_\gamma\rangle$: fermionic time ordering fixes this reversed annihilator order. Define the equal-time gap by

$$\Delta_{\alpha\beta} = -\int d\Gamma_{\mathbf p'}\, V_{\alpha\beta,\gamma\delta} F_{\gamma\delta}(0^-).$$

The Matsubara form is then

$$\Delta_{\alpha\beta}(\mathbf R,\mathbf p) = -k_{\mathrm B}T \sum_n \int d\Gamma_{\mathbf p'} \, V_{\alpha\beta,\gamma\delta} (\mathbf p,\mathbf p') F_{\gamma\delta} (\mathbf R,\mathbf p',\varepsilon_n),$$

where the momentum measure and cutoff or renormalization must be declared. If the normal layer has no retained attractive interaction, $V_{\mathrm N}=0$, then

$$\Delta_{\mathrm N}=0 \qquad\text{can coexist with}\qquad F_{\mathrm N}\neq0.$$

That nonzero $F_{\mathrm N}$ does not by itself establish a new broken symmetry, a local attraction, or a spectral gap.

A spectral minigap is a property of a retarded solution and its boundary conditions. Let $N_0$ be the normal-state local density of states in the same per-spin or spin-summed convention used for $N$. In the scalar convention introduced below,

$$\frac{N(E,x)}{N_0} = \operatorname{Re} \left[\cos\theta^{\mathrm R}(E,x)\right].$$

Analytic continuation, inelastic self-energy, temperature convolution, and instrument resolution intervene between a Matsubara pair amplitude and a measured spectrum. Thus $F$, $\Delta$, a minimum quasiparticle energy, a local density-of-states edge, a tunneling threshold, and $T_c$ remain distinct even when a simple model relates them.

Geometry, Scales, and Regime Selection

Let the interface normal be $+\hat{\mathbf x}$, with material 1 at $x<0$ and material 2 at $x>0$. Define the elastic mean free path $\ell=v_{\mathrm F}\tau$, a layer thickness $d$, and the Thouless energy

$$E_{\mathrm{Th}} = \frac{\hbar D}{d^2}.$$

$E_{\mathrm{Th}}$ is a dwell or diffusion scale, not automatically a minigap. For example, the often-quoted $E_g\simeq3.12E_{\mathrm{Th}}$ belongs only to a long, transparent, zero-phase diffusive SNS junction with negligible pair breaking and adequate spectral resolution.

The approximation hierarchy is:

• use a microscopic Gor’kov or Bogoliubov–de Gennes treatment when atomic hybridization, few-mode quantization, rapid spatial variation, or large energy scales invalidate Fermi-surface projection;
• use Eilenberger quasiclassics when $|\Delta|$, $|\varepsilon_n|$, $h$, $\hbar/\tau$, and other retained self-energies are small compared with $E_{\mathrm F}$ and envelopes vary slowly compared with $k_{\mathrm F}^{-1}$;
• use Usadel only when elastic scattering isotropizes momentum on the relevant scale, typically $\ell\ll\xi$ and $\ell\ll d$, with the retained energies satisfying $|\varepsilon_n|,|\Delta|,|h|,\hbar v_{\mathrm F}/L \ll\hbar/\tau\ll E_{\mathrm F}$;
• use Ginzburg–Landau only for its static, long-wavelength window near a continuous transition.

Eilenberger theory is not synonymous with a perfectly clean sample. It can retain impurity self-energies without performing the dirty angular reduction. Conversely, a large resistivity alone does not prove the scale hierarchy needed by Usadel theory.

Several coherence lengths coexist. The clean BCS estimate $\xi_0\sim\hbar v_{\mathrm F}/(\pi\Delta)$, a thermal ballistic propagation length, a diffusive thermal length, an exchange-driven decay or oscillation length, and a spin-relaxation length answer different questions. At zero temperature, clean or diffusive correlations may acquire long algebraic tails; one finite-temperature exponential must not be extrapolated blindly.

From Gor’kov to Eilenberger

Use the Nambu spinor $\Psi=(\psi_\uparrow,\psi_\downarrow^\dagger)^{\mathsf T}$ for the spin-inactive singlet benchmark and define

$$\widehat G(1,2) = -\left\langle \mathcal T_\vartheta \Psi(1)\Psi^\dagger(2) \right\rangle .$$

Let $\tau_i$ act in this Nambu space and let $\xi_{\mathbf p}$ be band energy measured from the chemical potential. This page uses the phase-rotated microscopic convention

$$\widehat G_0^{-1} (\mathbf p,i\varepsilon_n) = i\varepsilon_n\tau_0 - \xi_{\mathbf p}\tau_3 + \operatorname{Im}\Delta\,\tau_1 + \operatorname{Re}\Delta\,\tau_2.$$

For a real gap, the BdG pairing term is therefore $-\Delta\tau_2$. This Nambu-phase choice is deliberate: with the $\tau_3$-weighted projection below, the anomalous part of $\widehat g$ lies along $+\tau_1$. A convention with the real microscopic gap along $\tau_1$ rotates that projected component to $\tau_2$ and must rotate every subsequent matrix consistently.

With $\widehat G$ the corresponding fixed-gauge Wigner Green function, define the dimensionless quasiclassical propagator by

$$\widehat g (\mathbf R,\widehat{\mathbf p},\varepsilon_n) = \frac{i}{\pi} \int d\xi_{\mathbf p}\, \tau_3 \widehat G (\mathbf R,\mathbf p,\varepsilon_n), \qquad \widehat g^2=1.$$

This definition fixes the normalization used here. Other literature absorbs $-i\pi$ into $\widehat g$ and writes $\widehat g^2=-\pi^2$; formulas cannot be mixed across those conventions.

For electron charge $q_e=-e$, define a gauge-covariant derivative

$$\mathcal D\widehat X = \boldsymbol\nabla\widehat X - \frac{i q_e}{\hbar} \left[ \tau_3\mathbf A,\widehat X \right].$$

In the present normalization, an equilibrium Eilenberger equation can be written

$$\hbar\mathbf v_{\mathrm F}\cdot \mathcal D\widehat g + \left[ \widehat\Omega_n,\widehat g \right] =0, \qquad \widehat g^2=1,$$

with

$$\widehat\Omega_n = \varepsilon_n\tau_3 + \operatorname{Re}\Delta\,\tau_1 - \operatorname{Im}\Delta\,\tau_2 + \widehat\Sigma_n .$$

The self-energy and Nambu basis must be changed together when spin, bands, or another convention is added. For a uniform real singlet gap with $\widehat\Sigma_n=0$,

$$\widehat g_{\mathrm{bulk}} = \frac{ \varepsilon_n\tau_3+\Delta\tau_1 }{ \sqrt{\varepsilon_n^2+\Delta^2} },$$

which both commutes with $\widehat\Omega_n$ and satisfies $\widehat g_{\mathrm{bulk}}^2=1$. Those two checks catch many sign and normalization errors.

In a normal region, linearize the anomalous component for $\varepsilon_n>0$. Along a trajectory with $v_{\mathrm F x}>0$,

$$\hbar v_{\mathrm F x}\partial_x f + 2\varepsilon_n f =0,$$

so

$$f(x,\varepsilon_n) = f(0,\varepsilon_n) \exp \left[ -\frac{2\varepsilon_n x} {\hbar v_{\mathrm F x}} \right].$$

The lowest Matsubara energy therefore gives

$$\xi_{\mathrm N,bal} = \frac{\hbar|v_{\mathrm F x}|} {2\pi k_{\mathrm B}T}.$$

Grazing trajectories, surface roughness, finite inelastic scattering, and Fermi-surface anisotropy all alter the spatial average. This trajectory result is not a universal single coherence length.

Diffusive Propagation and the Usadel Equation

When impurity scattering makes the leading quasiclassical propagator nearly isotropic, angular harmonics can be eliminated. The matrix Usadel equation in the same convention is

$$\hbar D\, \mathcal D \left( \widehat g\, \mathcal D\widehat g \right) - \left[ \widehat\Omega_n,\widehat g \right] =0, \qquad \widehat g^2=1.$$

Normalization is built into a spectral-angle parameterization. Only for an equilibrium, one-dimensional, isotropic, single-band, spin-singlet, spin-inactive system with real $\Delta$, no phase gradient, and $\mathbf A=0$, set

$$\widehat g_n = \tau_3\cos\theta_n + \tau_1\sin\theta_n .$$

For the scalar decay benchmarks below, take $n\geq0$ so that $\varepsilon_n>0$; negative Matsubara energies are recovered from the stated fermionic symmetry relation.

Then

$$\frac{\hbar D}{2} \frac{d^2\theta_n}{dx^2} = \varepsilon_n\sin\theta_n - \Delta\cos\theta_n .$$

The uniform solution has $\tan\theta_n=\Delta/\varepsilon_n$, agreeing with the bulk check above. Inside a weakly correlated normal region, $\Delta=0$ and $|\theta_n|\ll1$, hence

$$\frac{\hbar D}{2}\theta_n'' = \varepsilon_n\theta_n, \qquad \theta_n(x) = \theta_n(0)e^{-x/\xi_n},$$

where

$$\xi_n = \sqrt{\frac{\hbar D}{2\varepsilon_n}}, \qquad \xi_{\mathrm N,diff} = \sqrt{ \frac{\hbar D} {2\pi k_{\mathrm B}T} }.$$

This is a thermal decay scale, not $\xi_0$. A scalar angle also cannot carry general spin-active, multiband, anisotropic, phase-textured, or nonequilibrium structure. Those cases require the matrix equation or a more microscopic description.

Interfaces, Boundary Conditions, and Conserved Spectral Current

An interface is not specified by the word “transparent.” Record its area, normal, crystalline registry, roughness, spin activity, conductance channels, and whether the boundary parameters were measured or fitted.

For a low-transparency, spin-inactive diffusive interface at $x=0$, use one common coordinate increasing from material 1 to material 2. A scalar Kupriyanov–Lukichev boundary condition in the conventions above is

$$\sigma_1\theta_1' = \sigma_2\theta_2' = G_B \sin(\theta_2-\theta_1),$$

where $G_B$ is interface conductance per area. The equality expresses conservation of spectral current, and $\sigma_i$ are the normal-state conductivities in the two diffusive materials. With outward normals on each material, one derivative changes sign. The convention must be stated before interpreting a sign.

In matrix form, the same low-transparency statement is

$$\sigma_1 \widehat g_1\partial_x\widehat g_1 = \sigma_2 \widehat g_2\partial_x\widehat g_2 = \frac{G_B}{2} \left[ \widehat g_1,\widehat g_2 \right].$$

Neither $F$ nor its bare derivative is universally continuous. Arbitrary transparency, spin filtering, spin mixing, multiband mismatch, and rough interfaces require Zaitsev, Nazarov, or generalized spin-active boundary conditions. Atomic-scale reconstruction can invalidate the entire continuum boundary model.

A “rigid superconductor” boundary fixes the parent propagator and ignores backaction. It is controlled only when the parent is a sufficiently large, stiff reservoir and the interface plus neighboring layer do not appreciably suppress it. That approximation cannot by itself support an inverse-proximity or $T_c$-suppression claim.

Inverse Proximity and Self-Consistent Transition Suppression

The direct and inverse effects are one coupled solution. Correlations enter the normal or magnetic material, while depleted spectral weight and pair breaking feed back on the superconducting layer.

Where a local pairing channel has a bare transition temperature $T_{c0}$, a regularized singlet self-consistency condition can be written

$$\Delta(x)\ln\frac{T}{T_{c0}} + 2\pi k_{\mathrm B}T \sum_{n\geq0} \left[ \frac{\Delta(x)}{\varepsilon_n} - \left\langle f_s(x,\widehat{\mathbf p},\varepsilon_n) \right\rangle_{\widehat{\mathbf p}} \right] =0.$$

The pairing representation, density-of-states convention, and cutoff renormalization are implicit in this compact form and must be declared in a calculation. It is not used in a layer with no local attractive channel; there one sets $\Delta=0$ but still solves for $f$.

Near a transition, linearizing the transport and self-consistency equations turns $T_c$ into a global eigenvalue problem over all superconducting regions and interfaces. A local symbol $T_c(x)$ is generally not the right object. Thickness-dependent suppression can also arise from damaged material, interdiffusion, strain, charge transfer, magnetic contamination, finite-size pair breaking, or a changed phonon spectrum. A self-consistent proximity fit must therefore close against structural and normal-state controls.

The rigid and self-consistent calculations answer different questions:

• a rigid calculation asks how a specified reservoir induces correlations and spectral structure in its neighbor;
• a self-consistent calculation asks how the complete heterostructure redistributes pair amplitude and changes $\Delta(x)$ and $T_c$;
• a microscopic calculation may additionally change the band structure, screening, and interaction kernel at the interface.

Spin, Ferromagnets, and Pair-Symmetry Conversion

The Pauli constraint requires the combined spin, spatial, and frequency exchange parity of a pair amplitude to be antisymmetric. In a single-band, orbital-symmetric channel, an even-parity equal-spin component must therefore be odd in Matsubara energy. With additional orbital or band labels, their exchange parity must be included. This classification concerns a correlation function; it does not by itself define a separate thermodynamic phase or a topological invariant.

For a collinear diffusive ferromagnet, define the linearized short-range combinations $f_\pm:=f_s\pm f_{t0}$. Here $h$ is the exchange-energy coefficient in the declared spin basis. For $n\geq0$ they satisfy

$$\hbar D_{\mathrm F}f_\pm'' - 2 \left( \varepsilon_n\pm ih \right) f_\pm =0,$$

and therefore

$$k_\pm^2 = \frac{ 2(\varepsilon_n\pm ih) }{ \hbar D_{\mathrm F} }.$$

When $h\gg\pi k_{\mathrm B}T$ and spin relaxation is weak, decay and oscillation lengths are both of order

$$\xi_{\mathrm F} \sim \sqrt{ \frac{\hbar D_{\mathrm F}}{h} }.$$

This damped oscillation can produce nonmonotonic $T_c$ or critical-current trends. A thickness oscillation alone does not prove a $\pi$ junction; the Josephson Effect owns phase-sensitive current–phase evidence.

Noncollinear magnetization or a spin-active interface can convert the short-range component into equal-spin triplet correlations that are not dephased by a uniform exchange field. Their ideal thermal scale approaches

$$\xi_{\mathrm F,long} \sim \sqrt{ \frac{\hbar D_{\mathrm F}} {2\pi k_{\mathrm B}T} },$$

but spin–orbit scattering, spin-flip scattering, magnetic texture, interface filtering, and finite energy shorten or reshape it. Long range is not automatic, and an odd-frequency amplitude is neither a unique explanation for a zero-bias peak nor evidence of topology.

How Experiments See Bulk Proximity

The correlator itself is representation and gauge covariant. Experiments see spectra, currents, transition shifts, magnetic fields, heat, or scattering intensities through a forward model.

Scanning Tunneling Microscopy and Spectroscopy owns setpoint, tip, matrix-element, thermal, modulation, and resolution effects in $dI/dV$. A soft suppression can reflect inelastic broadening, temperature, surface damage, spatial averaging, pair breaking, or an actual soft spectrum. One hard or soft gap does not determine a unique interface transparency.

Transport Measurements owns contact subtraction, geometry, current distribution, heating, and terminal-voltage interpretation. A zero-resistance path may be a filament or short; a terminal supercurrent does not itself image $\Delta(x)$ or $F(x)$.

Terahertz and Infrared Probes owns substrate, multilayer electrodynamics, inversion, and optical-resolution effects. An optical onset is not automatically an induced minigap.

The strongest bulk-proximity record combines:

• a thickness or distance series rather than one device;
• state-matched temperature, field, and magnetic-history control;
• a structural and normal-state interface characterization;
• at least one local spectral or thermodynamic observable;
• a self-consistent comparison when inverse proximity or $T_c$ is claimed;
• independent bounds on heating, pinholes, damaged layers, and parallel conduction;
• convergence across model, mesh, continuation, and uncertainty variations.

Production Eilenberger, Usadel, and Bogoliubov–de Gennes software workflows remain a coverage gap. This page supplies the equation and acceptance contract, not a software-specific recipe.

Worked Audit: A Diffusive Superconductor–Normal Bilayer

Consider a synthetic Nb-like superconductor with a normal film. The numerical record is deliberately synthetic so that it tests the workflow without being mistaken for a specific material data set.

1. Platform and state. A $60\,\mathrm{nm}$ superconducting film is capped by normal layers of $d_{\mathrm N}=15$, $40$, and $80\,\mathrm{nm}$. All spectra are taken at $T=1.50\,\mathrm K$, zero applied field, and the same cooldown history.
2. Claimed coherent object. The claim is a diffusive anomalous amplitude in N plus inverse suppression of the parent. The fit distinguishes $F_{\mathrm N}$, $\Delta_{\mathrm N}=0$, a spectral minigap, and the global bilayer $T_c$.
3. Degrees of freedom. One isotropic spin-degenerate band per layer and a singlet parent are retained. The N layer has no attractive interaction in the model.
4. Symmetry and convention. The scalar Nambu convention of this page is used with $\varepsilon_n=(2n+1)\pi k_{\mathrm B}T$, $x$ increasing from S to N, and no phase gradient or spin-active boundary.
5. Scale hierarchy. The parent gap is $\Delta_{\mathrm S}=1.40\pm0.03\,\mathrm{meV}$, $D_{\mathrm S}=(1.20\pm0.08)\times10^{-3}\,\mathrm{m^2\,s^{-1}}$, $\sigma_{\mathrm S}=(6.0\pm0.3)\times10^6\,\mathrm{S\,m^{-1}}$, and $T_{c0}=9.10\pm0.02\,\mathrm K$. Normal-state inputs give $\ell_{\mathrm S}=2.0\pm0.2\,\mathrm{nm}$, $v_{\mathrm F,S}=(1.80\pm0.10)\times10^6\,\mathrm{m\,s^{-1}}$, $k_{\mathrm F,S}=(19.1\pm1.0)\,\mathrm{nm^{-1}}$, and $E_{\mathrm F,S}=11.3\pm1.1\,\mathrm{eV}$. For N, $D_{\mathrm N}=(2.00\pm0.10)\times10^{-3}\,\mathrm{m^2\,s^{-1}}$ and $\ell_{\mathrm N}=2.0\pm0.3\,\mathrm{nm}$, with $\sigma_{\mathrm N}=(2.5\pm0.2)\times10^6\,\mathrm{S\,m^{-1}}$, $v_{\mathrm F,N}=(3.0\pm0.2)\times10^6\,\mathrm{m\,s^{-1}}$, $k_{\mathrm F,N}=(12.3\pm0.7)\,\mathrm{nm^{-1}}$, and $E_{\mathrm F,N}=12.2\pm1.4\,\mathrm{eV}$. These values obey the one-band Einstein and parabolic-dispersion checks within uncertainty. The interface has $R_BA=2.0\pm0.2\,\mathrm{f\Omega\,m^2}$. The N-side Sharvin conductance is $(9.38\pm1.07)\times10^{14}\,\mathrm{S\,m^{-2}}$, so the synthetic interface uses a monodisperse normal-state channel distribution $\mathcal T_k=0.53\pm0.08$.
6. Theory regime. Since $\ell_{\mathrm N}\ll d_{\mathrm N}$ and, as computed below, $\ell_{\mathrm N}\ll\xi_{\mathrm N,diff}$, a Usadel treatment is licensed. The products $k_{\mathrm F,N}\ell_{\mathrm N}\simeq25$ and $k_{\mathrm F,S}\ell_{\mathrm S}\simeq38$ keep both metals on the quasiclassical side of the localization boundary. The inferred scattering scales are $\hbar/\tau_{\mathrm N}\simeq0.99\,\mathrm{eV}$ and $\hbar/\tau_{\mathrm S}\simeq0.59\,\mathrm{eV}$; even at $d_{\mathrm N}=15\,\mathrm{nm}$, $\hbar v_{\mathrm F,N}/d_{\mathrm N}\simeq0.132\,\mathrm{eV}$, while the retained gap, lowest thermal, and resolved gradient energies are smaller and both scattering scales remain well below their Fermi energies. The self-consistency sum uses coupling renormalization plus a controlled high-energy tail rather than treating every Matsubara energy as a retained low scale. The finite transmission requires a Nazarov boundary condition with the declared channel distribution; the conductance alone would not determine the nonlinear spectral boundary problem. Near $T_c$, linearization reduces it to the same total-conductance boundary used in the transition eigenproblem. The S layer is solved both rigidly and self-consistently, with zero spectral current at both the outer S surface and the vacuum-facing N surface.
7. Requested observable. STS reports gap-like edges $0.75\pm0.05$, $0.32\pm0.04$, and $0.11\pm0.03\,\mathrm{meV}$ as $d_{\mathrm N}$ increases. Four-probe transitions move from the isolated-S value $9.10\pm0.02\,\mathrm K$ to $8.32\pm0.03$, $8.21\pm0.03$, and $8.20\pm0.04\,\mathrm K$.
8. Forward model. The STS model includes the measured normal-tip density of states, Fermi convolution, $70\,\mu\mathrm{eV}$ Gaussian instrumental width, lock-in modulation, and spatial sampling. The transition criterion is fixed at $50\%$ of the same extrapolated normal resistance.
9. Evidence and alternatives. Cross-sectional microscopy bounds the interdiffused layer below $1.5\,\mathrm{nm}$; normal-state sheet resistance and roughness are measured for every thickness. Heating, a damaged cap, and a parallel metallic short remain explicit alternatives.
10. Uncertainty and stopping rule. Interface, diffusion, broadening, thickness, self-consistency, and spectral-fit uncertainties are varied jointly. The claim stops at a diffusive self-consistent proximity model; one spectrum does not determine a unique transparency or microscopic interface Hamiltonian.

At the lowest Matsubara energy,

$$\xi_{\mathrm N,diff} = \sqrt{ \frac{ (1.0546\times10^{-34}\,\mathrm{J\,s}) (2.00\times10^{-3}\,\mathrm{m^2\,s^{-1}}) }{ 2\pi (1.3806\times10^{-23}\,\mathrm{J\,K^{-1}}) (1.50\,\mathrm K) }} = 40.2\,\mathrm{nm}.$$

Thus $\ell_{\mathrm N}/\xi_{\mathrm N,diff}\simeq0.050$. The three Thouless energies are

$$E_{\mathrm{Th}} = 5.85,\ 0.822,\ 0.206\,\mathrm{meV}$$

for $15$, $40$, and $80\,\mathrm{nm}$ respectively. Their trend helps organize the spectra, but the measured edges are not set equal to $E_{\mathrm{Th}}$ or to $3.12E_{\mathrm{Th}}$: this is an SN bilayer with finite interface resistance and a self-consistently weakened parent, not the long transparent SNS benchmark.

A rigid-parent calculation can describe induced spectral structure but leaves $T_c$ fixed by construction. The observed systematic $T_c$ suppression therefore requires the self-consistent eigenproblem or a competing structural explanation. Joint agreement with both thickness-dependent spectra and $T_c$ is stronger than either fit alone.

The linearized self-consistent calculation gives transition temperatures $8.315$, $8.202$, and $8.196\,\mathrm K$ for the three N thicknesses, within the combined uncertainty of the synthetic record. The near-saturation between $40$ and $80\,\mathrm{nm}$ is expected because the normal-metal coherence length near $T_c$ is about $17\,\mathrm{nm}$. Normalize the gap eigenfunction to unit maximum: doubling the spatial mesh changes each $T_c$ by less than $0.001\,\mathrm K$ and the normalized eigenfunction by less than $1.1\%$ in maximum norm. Cutoff doubling with the asymptotic tail changes each $T_c$ by less than $0.001\,\mathrm K$; the normalized gap-equation and boundary spectral-current residuals remain below $10^{-6}$. A separate nonlinear Matsubara and retarded solve at $1.50\,\mathrm K$ supplies the spectral comparison and must carry its own normalization, continuation, and refinement audit; those checks are not inferred from the linearized eigenproblem.

Worked Audit: A Spin-Active Superconductor–Ferromagnet Multilayer

Now consider a synthetic S/F multilayer with a controllable noncollinear magnetic spacer.

1. Platform and state. A matched sample family contains S/F bilayers for $T_c$ and STS plus symmetric S/F/S junctions for a separate phase-biased current measurement. Each superconducting electrode is $50\,\mathrm{nm}$ thick; the ferromagnetic thickness is $d_{\mathrm F}=8$, $12$, $16$, $20$, and $28\,\mathrm{nm}$. Every point uses $T=2.00\,\mathrm K$, the same field-training loop, and remanent collinear or noncollinear configurations verified by magnetometry.
2. Claimed coherent object. The bounded claim is conversion from short-range singlet and zero-spin triplet correlations to a longer-ranged equal-spin component. It is not a claim of a new bulk phase, a unique odd-frequency mechanism, or topology.
3. Degrees of freedom. A diffusive singlet S band, an exchange-split F band, and a spin-active interface are retained. Orbital depairing is kept as a competing channel.
4. Symmetry and convention. Exchange field points along the local magnetization; the Pauli relation on this page fixes spin, momentum-parity, and frequency labels. The interface normal is $+\hat{\mathbf x}$. Define $P=(G_\uparrow-G_\downarrow)/(G_\uparrow+G_\downarrow)$ with spin up along the local interface magnetization; positive $G_\phi$ denotes a positive spin-mixing conductance moment in that same normal and spin convention.
5. Scale hierarchy. Use $D_{\mathrm F}=(2.0\pm0.2)\times10^{-4}\,\mathrm{m^2\,s^{-1}}$, $\sigma_{\mathrm F}=(5.3\pm1.5)\times10^6\,\mathrm{S\,m^{-1}}$, $h=20\pm3\,\mathrm{meV}$, and $\ell_{\mathrm F}=1.0\pm0.1\,\mathrm{nm}$. Normal-state data give $v_{\mathrm F}=(6.0\pm0.5)\times10^5\,\mathrm{m\,s^{-1}}$, $k_{\mathrm F}=(25.3\pm3.3)\,\mathrm{nm^{-1}}$, and $E_{\mathrm F}=5.0\pm0.5\,\mathrm{eV}$. Thus $\tau\simeq1.67\,\mathrm{fs}$, $\hbar/\tau\simeq0.395\,\mathrm{eV}$, and $k_{\mathrm F}\ell\simeq25\pm4$. Spin-relaxation and spin–orbit lengths are independently measured to exceed $25\,\mathrm{nm}$. The parent has $d_{\mathrm S}=50\,\mathrm{nm}$, $\Delta_{\mathrm S}=1.20\pm0.03\,\mathrm{meV}$, $D_{\mathrm S}=(1.0\pm0.1)\times10^{-3}\,\mathrm{m^2\,s^{-1}}$, $\sigma_{\mathrm S}=(5.0\pm0.3)\times10^6\,\mathrm{S\,m^{-1}}$, and $T_{c0}=8.0\pm0.03\,\mathrm K$, with $\ell_{\mathrm S}=2.0\pm0.2\,\mathrm{nm}$, $v_{\mathrm F,S}=(1.5\pm0.1)\times10^6\,\mathrm{m\,s^{-1}}$, $k_{\mathrm F,S}=(17.4\pm1.2)\,\mathrm{nm^{-1}}$, and $E_{\mathrm F,S}=8.6\pm1.0\,\mathrm{eV}$. Both layers use the same explicit parabolic-band consistency check as the first audit.
6. Theory regime. Matrix Usadel equations are the candidate production framework because $\ell_{\mathrm F}$ is shortest. At the thinnest layer, $\ell_{\mathrm F}/d_{\mathrm F}=0.125$ and $\hbar v_{\mathrm F}/d_{\mathrm F}\simeq0.049\,\mathrm{eV}$, below $\hbar/\tau\simeq0.395\,\mathrm{eV}\ll E_{\mathrm F}$. The exchange energy is smaller still. The normal-state boundary locator records $R_BA=1.50\pm0.15\,\mathrm{f\Omega\,m^2}$, equivalently $G_T=(6.67\pm0.67)\times10^{14}\,\mathrm{S\,m^{-2}}$, together with $G_\phi/G_T=0.20\pm0.03$ and spin-filter polarization $P=0.15\pm0.02$. The S-side Sharvin ceiling is $(1.88\pm0.26)\times10^{15}\,\mathrm{S\,m^{-2}}$, giving total mean transmission $0.36\pm0.06$. The synthetic boundary uses monodisperse spin channels $\mathcal T_\uparrow=0.41\pm0.07$ and $\mathcal T_\downarrow=0.30\pm0.06$, consistent with the stated $P$. The independently bounded $G_\phi/G_T$ is retained as a convention-dependent spin-mixing moment, not inverted into a unique spin-mixing angle. At this finite transparency, these conductance moments do not determine the side-resolved reflection and transmission phases of a full spin-active scattering matrix. That matrix is a required production input, so the audit licenses the bulk regime but stops before a numerical spin-active boundary fit. A production S/F bilayer problem would use zero spectral current at its two outer surfaces and a self-consistent S layer; the companion S/F/S problem would instead fix the reservoir phases and conserve the charge-current projection through both interfaces while retaining interfacial spin-torque balance. The full spin spectral current need not be continuous across a magnetic interface. A scalar $\theta$ equation is explicitly rejected. Atomic pinholes trigger microscopic escalation.
7. Requested observable. The bilayers provide $T_c(d_{\mathrm F})$ and spatial STS. The matched S/F/S junctions provide the decay of a phase-biased critical-current amplitude. Josephson phase, current–phase, switching, and environmental interpretation remain with the Josephson owner.
8. Forward model. Magnetometry fixes the collinear and noncollinear configurations; STS includes the same convolution ledger as the first audit. The separate junction analysis uses the measured phase-bias circuit, current distribution, contact resistance, switching statistics, heating, and field-history controls.
9. Evidence and alternatives. The collinear control shows a damped thickness oscillation and falls below current sensitivity by $12\,\mathrm{nm}$. In the noncollinear configuration, a reproducible correlation persists to $20\,\mathrm{nm}$. Pinholes, stray fields, magnetic dead layers, ordinary Andreev states, and configuration-dependent heating are tested rather than assumed absent.
10. Uncertainty and stopping rule. Exchange, diffusion, spin relaxation, interface mixing, magnetic angle, thickness, and probe uncertainties are propagated. The claim stops at long-range converted pair correlation. A quantitative spin-active fit stops until the side-resolved scattering matrix is registered; this audit claims neither a unique phase distribution nor a completed numerical spin-active boundary solve. Phase-sensitive $\pi$-junction, microscopic interface, topology, and device claims route elsewhere.

The short exchange scale is

$$\xi_{\mathrm F} = \sqrt{ \frac{ (1.0546\times10^{-34}\,\mathrm{J\,s}) (2.0\times10^{-4}\,\mathrm{m^2\,s^{-1}}) }{ 20\,\mathrm{meV} }} = 2.57\,\mathrm{nm},$$

where $20\,\mathrm{meV}=3.204\times10^{-21}\,\mathrm J$. The ideal thermal equal-spin scale is

$$\xi_{\mathrm F,long} = \sqrt{ \frac{ \hbar D_{\mathrm F} }{ 2\pi k_{\mathrm B}(2.00\,\mathrm K) }} = 11.0\,\mathrm{nm}.$$

Those scales make the configuration dependence plausible, not unique. A convincing inference still requires the observed spin-relaxation bound, thickness trend, magnetization verification, and pinhole and stray-field controls. Equal-spin, even-parity $s$-wave correlation is odd in frequency by Pauli antisymmetry, but that classification is not a thermodynamic phase label.

Common Failure Modes and Canonical Handoffs

• Equating $F$, $\Delta$, and a minigap. They are a correlator, a self-energy, and a spectral property. Keep them separate through the forward model.
• Calling Eilenberger “ballistic” and Usadel “resistive.” The first is the general quasiclassical transport equation; the second needs controlled momentum isotropization.
• Using a rigid parent to claim inverse proximity. Solve self-consistently or limit the claim to induced correlations for a specified reservoir.
• Matching $F$ and $\partial_xF$ across every interface. Match conserved spectral current with the boundary condition appropriate to transparency, spin activity, and convention.
• Calling $E_{\mathrm{Th}}$ a measured gap. It is a scale. Geometry, phase, interface, temperature, pair breaking, and resolution determine the spectrum.
• Inferring a $\pi$ junction from one oscillation. Use the Josephson Effect for phase-sensitive evidence.
• Calling any long-range signal odd-frequency triplet order. Apply the Pauli classification, bound ordinary alternatives, and avoid converting a correlation symmetry into a new phase.
• Turning a soft gap or zero-bias peak into topology. Topological Superconductors owns invariant, protecting-gap, and boundary-evidence claims.

Proximity and Andreev Physics owns BTK scattering, transmission-resolved Andreev levels, few-mode hybrids, and device evidence. BCS Mean-Field Theory owns the reusable homogeneous Nambu saddle. Ginzburg–Landau Theory owns near-$T_c$ spatial phenomenology. van der Waals Heterostructures and Engineered Heterostructures own assembly, band alignment, charge transfer, and cross-platform interface evidence. This page owns the bulk superconducting correlation problem between those boundaries.

Unconventional Superconductivity owns the stable thermodynamic classification of bulk zero-momentum pairing; this page retains interface-induced symmetry conversion and does not promote an anomalous correlation into a new bulk phase.

Pair-Density Waves and Exotic Orders owns the reusable finite-momentum and odd-frequency order taxonomy; this page only classifies induced correlations inside a declared heterostructure. Device Fabrication Concepts owns process flow, contamination, interface metrology, packaging, and batch-level provenance.

Exercises

1. Sort a proximity record

A calculation and experiment report $V_{\mathrm N}=0$, nonzero $f_{\mathrm N}(x,\varepsilon_n)$, a suppressed self-consistent $\Delta_{\mathrm S}(x)$, minimum positive BdG energy $0.18\,\mathrm{meV}$, LDOS edge $0.22\,\mathrm{meV}$, tunneling threshold $0.25\,\mathrm{meV}$, and a coupled transition at $8.4\,\mathrm K$. No independent stiffness measurement exists. Classify the interaction, anomalous amplitude, pair potential, three spectral or probe scales, transition, and phase claim.

Solution

Because the retained N-layer attraction is zero, $\Delta_{\mathrm N}=0$. The nonzero $f_{\mathrm N}$ is an induced anomalous correlator; the suppressed $\Delta_{\mathrm S}(x)$ is an inverse-proximity self-energy result. The minimum positive BdG energy belongs to the model spectrum, the LDOS edge to the retarded local spectrum, and the tunneling threshold to a probe convolution. They need not coincide. The $8.4\,\mathrm K$ value is the global instability of the coupled stack, not a local $T_c(x)$. Nothing in the record establishes an independently stiff phase in N. A thickness series, controlled forward model, and stiffness or screening test would strengthen the material claim.

2. Select the controlled equation

For four hypothetical layers, choose the first framework and name its failure test:

• A: $E_{\mathrm F}=5\,\mathrm{eV}$, $\Delta=1\,\mathrm{meV}$, $\ell=300\,\mathrm{nm}$, and $\xi=40\,\mathrm{nm}$.
• B: the same low-energy scales but $\ell=2\,\mathrm{nm}$ and $\xi=80\,\mathrm{nm}$.
• C: a two-atomic-layer interface with strong orbital reconstruction.
• D: a slowly varying film at $|T-T_c|/T_c=0.01$ where only static order-parameter recovery is requested.

Solution

A supports trajectory-resolved Eilenberger quasiclassics, provided all self-energies remain small compared with $E_{\mathrm F}$ and spatial variation is slow compared with $k_{\mathrm F}^{-1}$. B is a Usadel candidate because $\ell/\xi=0.025$, but the decision remains conditional until $v_{\mathrm F}$, $\tau$, $E_{\mathrm F}$, layer thickness, and interface data verify angular isotropization and the full energy hierarchy. C requires a microscopic Gor’kov, tight-binding, or Bogoliubov–de Gennes treatment because the continuum and slow-envelope assumptions fail. D can use Ginzburg–Landau for the stated static near-$T_c$ question, but not for low-temperature spectra. The failure tests are, respectively, loss of scale separation, anisotropic harmonics that do not relax, atomic-scale reconstruction, and departure from the near-transition long-wavelength window.

3. Derive and evaluate diffusive decay

Solve the linearized Usadel equation in a semi-infinite normal layer with $\theta(0)=\theta_0$ and $\theta(\infty)=0$. Then evaluate $\theta(30\,\mathrm{nm})/\theta_0$ for $D=3.0\times10^{-3}\,\mathrm{m^2\,s^{-1}}$ and $T=2.40\,\mathrm K$ using the lowest Matsubara energy.

Solution

With $\Delta=0$,

$$\frac{\hbar D}{2}\theta''=\varepsilon_0\theta, \qquad \varepsilon_0=\pi k_{\mathrm B}T.$$

The decaying solution is

$$\theta(x)=\theta_0e^{-x/\xi}, \qquad \xi=\sqrt{\frac{\hbar D}{2\pi k_{\mathrm B}T}}.$$

Numerically,

$$\xi = 39.0\,\mathrm{nm}, \qquad \frac{\theta(30\,\mathrm{nm})}{\theta_0} = e^{-30/39.0} = 0.463.$$

This is one Matsubara component in a finite-temperature diffusive model, not the full retarded spectrum or a zero-temperature asymptotic theorem.

4. Check an interface condition

At a spin-inactive interface let $\theta_1=0.90$, $\theta_2=0.30$, $\sigma_1=5.0\times10^6\,\mathrm{S\,m^{-1}}$, $\sigma_2=2.0\times10^6\,\mathrm{S\,m^{-1}}$, and $G_B=1.0\times10^{13}\,\mathrm{S\,m^{-2}}$. Use the common-$x$ convention to find $\theta_1'$ and $\theta_2'$. What is continuous?

Solution

The shared spectral-current factor is

$$G_B\sin(\theta_2-\theta_1) = -5.65\times10^{12}\,\mathrm{S\,m^{-2}}.$$

Therefore

$$\theta_1' = -1.13\times10^6\,\mathrm{m^{-1}}, \qquad \theta_2' = -2.82\times10^6\,\mathrm{m^{-1}}.$$

$\sigma_i\theta_i'$ is continuous in this convention; neither the bare derivative nor $\theta$ is required to be continuous for a finite-resistance interface. Outward-normal derivatives would carry opposite signs.

5. Use a Thouless benchmark without overclaiming

A long diffusive SNS junction has $D=1.0\times10^{-2}\,\mathrm{m^2\,s^{-1}}$ and $L=500\,\mathrm{nm}$. Compute $E_{\mathrm{Th}}$ and the ideal zero-phase benchmark $E_g=3.12E_{\mathrm{Th}}$. List four conditions needed before comparing it with a measured edge.

Solution

$$E_{\mathrm{Th}} = \frac{\hbar D}{L^2} = 0.0263\,\mathrm{meV}, \qquad E_g = 0.0821\,\mathrm{meV}.$$

The benchmark requires a long junction with parent $\Delta\gg E_{\mathrm{Th}}$, transparent interfaces, zero phase difference, diffusive isotropization, negligible pair breaking, sufficiently low temperature, and resolution narrower than the edge. Naming any four plus a declared forward model earns credit for this exercise; licensing the numerical coefficient requires every applicable assumption. If any required assumption fails, $E_{\mathrm{Th}}$ remains useful but $3.12E_{\mathrm{Th}}$ is not the licensed minigap.

6. Diagnose inverse proximity

Two calculations use identical interface and N-layer parameters. Calculation R fixes the S-layer bulk propagator. Calculation S solves the S-layer gap self-consistently. Both fit the N-layer spectrum, but only S predicts the observed $T_c$ reduction. What may each calculation claim?

Solution

R may claim induced spectral and anomalous correlations for a specified rigid reservoir. It cannot attribute a parent-gap or $T_c$ change to inverse proximity because those outputs were excluded by construction. S can support that attribution if the structural, magnetic, heating, and damaged-layer alternatives are also bounded and the global linearized self-consistency problem reproduces $T_c$. Agreement with one N-layer spectrum alone does not determine a unique interface resistance.

7. Classify and compare S/F scales

An even-parity equal-spin component in a single-band, orbital-symmetric channel is found in a diffusive ferromagnet with $D_{\mathrm F}=8.0\times10^{-4}\,\mathrm{m^2\,s^{-1}}$, $h=30\,\mathrm{meV}$, and $T=1.00\,\mathrm K$. Classify its frequency parity and calculate the ideal short exchange and long thermal scales.

Solution

Equal spin is symmetric under spin exchange, even spatial parity is symmetric, and the declared orbital channel is symmetric, so Pauli antisymmetry requires odd frequency. An antisymmetric orbital or band channel would change this classification. The scales are

$$\xi_{\mathrm F} = \sqrt{\frac{\hbar D_{\mathrm F}}{h}} = 4.19\,\mathrm{nm},$$

and

$$\xi_{\mathrm F,long} = \sqrt{ \frac{\hbar D_{\mathrm F}} {2\pi k_{\mathrm B}T} } = 31.2\,\mathrm{nm}.$$

The second value is an upper benchmark before spin-flip, spin–orbit, interface, and finite-energy dephasing. Odd-frequency classification does not itself prove long-range propagation, a separate phase, or topology.

8. Audit a hybrid-device claim

A superconductor–semiconductor device shows a zero-bias conductance peak and a gate-dependent terminal critical current. Fill the ten-field ledger at a minimum useful level, state the strongest licensed claim, and route the next questions.

Solution

Record material stack, thicknesses, gates, temperature, field orientation and history; distinguish anomalous amplitude, local pair potential, minigap, Andreev level, and terminal current; state bands, spin–orbit and Nambu basis; declare gauge, bias, and sign conventions; compare gap, Zeeman, spin–orbit, temperature, broadening, mean free path, length, and resolution; choose and test microscopic, Eilenberger, or Usadel control; define the requested LDOS or current; include contact, tunneling, temperature, lock-in, heating, and current distribution; test disorder, quantum dots, soft confinement, pinholes, and ordinary Andreev states; and propagate parameter, solver, fit, and measurement uncertainties with a falsifier.

The given record supports only a gate-dependent superconducting hybrid response. Use Proximity and Andreev Physics for BTK scattering and discrete levels, the Josephson Effect for the current–phase and switching problem, Scanning Tunneling Microscopy and Spectroscopy or the relevant transport owner for probe inversion, and Topological Superconductors only after a bulk or defect invariant, protecting gap, and nonunique zero-mode alternatives are addressed.

References

• W. Belzig, F. K. Wilhelm, C. Bruder, G. Schön, and A. D. Zaikin, “Quasiclassical Green’s Function Approach to Mesoscopic Superconductivity,” Superlattices and Microstructures 25, 1251–1288 (1999), doi:10.1006/spmi.1999.0710.
• F. S. Bergeret, A. F. Volkov, and K. B. Efetov, “Odd Triplet Superconductivity and Related Phenomena in Superconductor–Ferromagnet Structures,” Reviews of Modern Physics 77, 1321–1373 (2005), doi:10.1103/RevModPhys.77.1321.
• A. I. Buzdin, “Proximity Effects in Superconductor–Ferromagnet Heterostructures,” Reviews of Modern Physics 77, 935–976 (2005), doi:10.1103/RevModPhys.77.935.
• P. G. de Gennes, “Boundary Effects in Superconductors,” Reviews of Modern Physics 36, 225–237 (1964), doi:10.1103/RevModPhys.36.225.
• G. Eilenberger, “Transformation of Gorkov’s Equation for Type II Superconductors into Transport-Like Equations,” Zeitschrift für Physik 214, 195–213 (1968), doi:10.1007/BF01379803.
• M. Eschrig, “Spin-Polarized Supercurrents for Spintronics: A Review of Current Progress,” Reports on Progress in Physics 78, 104501 (2015), doi:10.1088/0034-4885/78/10/104501.
• L. P. Gor’kov, “On the Energy Spectrum of Superconductors,” Soviet Physics JETP 7, 505–508 (1958).
• L. P. Gor’kov, “Microscopic Derivation of the Ginzburg–Landau Equations in the Theory of Superconductivity,” Soviet Physics JETP 9, 1364–1367 (1959).
• M. Yu. Kuprianov and V. F. Lukichev, “Influence of Boundary Transparency on the Critical Current of ‘Dirty’ SS’S Structures,” Soviet Physics JETP 67, 1163–1168 (1988).
• A. I. Larkin and Yu. N. Ovchinnikov, “Quasiclassical Method in the Theory of Superconductivity,” Soviet Physics JETP 28, 1200–1205 (1969).
• W. L. McMillan, “Tunneling Model of the Superconducting Proximity Effect,” Physical Review 175, 537–542 (1968), doi:10.1103/PhysRev.175.537.
• Yu. V. Nazarov, “Novel Circuit Theory of Andreev Reflection,” Superlattices and Microstructures 25, 1221–1231 (1999), doi:10.1006/spmi.1999.0738.
• K. D. Usadel, “Generalized Diffusion Equation for Superconducting Alloys,” Physical Review Letters 25, 507–509 (1970), doi:10.1103/PhysRevLett.25.507.
• A. V. Zaitsev, “Quasiclassical Equations of the Theory of Superconductivity for Contiguous Metals and the Properties of Constricted Microcontacts,” Soviet Physics JETP 59, 1015–1024 (1984).