# Andreev reflection: interface transport and model limits

Understand electron–hole conversion, barrier-dependent conductance and thermal smearing. Keep contact assumptions separate from claims about bulk symmetry or topology.

## Scope

Andreev reflection converts an electron-like excitation in a normal conductor into a reflected hole-like excitation at a superconducting interface. In the conventional subgap process, charge of magnitude $2e$ enters the condensate. Andreev's original calculation treats a slowly varying excitation near the Fermi surface; the selected scan shows subgap evanescence and reversal of velocity. Retroreflection is an approximation with a specified band and energy regime, not a universal trajectory rule. [R1](#ref-source-18bd294c2b4c06a7)

This is an interface probe. Conductance depends on the contact as well as the superconducting state, so a fitted gap scale or an enhanced subgap signal is not automatically a measurement of bulk pairing symmetry. Keep interface scattering, normal-state background and sample conditions attached to an interpretation.

## Mathematical assumptions

For one incident channel, let $A(E)$ and $B(E)$ be Andreev and normal reflection probabilities with consistent flux normalization. The charge-conductance weight is $1+A-B$. Under purely subgap reflection, probability conservation gives $A+B=1$, hence the weight is $2A$. A factor of two in this ideal charge conversion does not imply that every measured junction has twice its normal-state conductance.

The original BTK abstract describes a tunable interface barrier. To check an explicit limit independently, take a spin-degenerate, single-channel, time-reversal-symmetric normal barrier of transmission $\mathcal T$, separated from an ideal conventional s-wave Andreev reflector. At zero energy the round-trip Andreev phase is $-1$, giving

$$A(0)=\left(\frac{\mathcal T}{2-\mathcal T}\right)^2,\qquad \frac{G_{NS}}{G_N}=\frac{2\mathcal T}{(2-\mathcal T)^2}.$$

For a one-dimensional delta barrier with matched velocities, define $Z$ so that $\mathcal T=1/(1+Z^2)$. Summing coherent electron–hole round trips gives $A(0)=1/(1+2Z^2)^2$. The transparent limit has conductance ratio two; the tunnel limit is suppressed. Independent complex-amplitude sums check twelve barrier/energy cases. This is an ideal scattering check, not a reanalysis of BTK's full finite-bias treatment. [R2](#ref-source-bcd31ef90f21ff0a)

Beenakker's later lectures give, at zero temperature, bias and magnetic field, $G_{NS}=(4e^2/h)\sum_n\mathcal T_n^2/(2-\mathcal T_n)^2$ and $G_N=(2e^2/h)\sum_n\mathcal T_n$. These are two-terminal formulas for the stated coherent NS model, with normal scattering separated from the ideal superconducting boundary. The original 1992 paper is retained separately at abstract scope. [R5](#ref-source-02f3aac52d07be56) [R3](#ref-source-8165cabd8e37fbe8)

Equal normal conductance does not determine the superconducting conductance. For example, transmission sets $(1,0)$ and $(1/2,1/2)$ have the same sum but different sums of $\mathcal T_n^2/(2-\mathcal T_n)^2$. This independent counterexample explains why a contact resistance alone cannot specify a multichannel junction.

As a separate local derivation, take an elastic conductance spectrum $g_0(E)$ that is independent of temperature and bias, with reservoirs described by equilibrium Fermi functions at the same $T>0$. Let $e>0$ be the elementary charge and $eV$ the chemical-potential shift. Thermal smearing then gives:

$$G(V,T)=\int dE\,g_0(E)\frac{\operatorname{sech}^2[(E-eV)/(2k_BT)]}{4k_BT}.$$

The kernel integrates to one and preserves a constant conductance. Three numerical checks compare the convolution to the derivative of its corresponding current integral. The check uses a synthetic smooth spectrum, not a BTK fit. A temperature-dependent gap, bias-dependent barrier, nonequilibrium distribution or heating requires additional modeling; arbitrary lifetime broadening is not equivalent to thermal convolution.

## Experimental criteria

Blonder and Tinkham's 1983 experimental abstract reports Cu-Nb point contacts spanning metallic to tunneling behavior and agreement with the model over a range of contact resistances. This entry checks that qualitative observation only. Contact dimensions, individual spectra and parameter uncertainties remain unverified here; no transition temperature is extracted. [R6](#ref-source-851c482722643e96)

Report contact preparation, normal-state resistance, wiring, crystal direction, current or excitation amplitude, field and temperature. Compare contact dimensions with mean free path and coherence length before choosing a ballistic or diffusive description. Repeat contacts and test heating and critical-current artifacts. Preserve the background normalization and its uncertainty; these are editorial requirements for assessing a claim, not one universally valid apparatus recipe.

A constant series resistance $R_s$ gives $G_{\mathrm{meas}}=G_{\mathrm{int}}/(1+R_sG_{\mathrm{int}})$ when the internal response is differentiated consistently. Ignoring it can change an apparent enhancement or fitted transparency. This circuit identity assumes the series element is bias independent; it does not describe thermal feedback or nonlinear leads.

## Limits and disputes

Asano, Tanaka and Kashiwaya's theoretical abstract associates a sign-changing pair potential with zero-energy surface states and a zero-bias conductance peak. This is a model result about a surface, not a new experimental proof of bulk symmetry. The broad Kashiwaya–Tanaka review is retained as a bibliographic pointer; its detailed calculations have not been checked here. [R7](#ref-source-cbbdb7f1a746a8f4) [R4](#ref-source-eb20b5a729cf98da)

A peak alone does not establish a Majorana mode, spin-triplet pairing or a unique mechanism. Model comparison should address ordinary bound states and junction effects, and test the predicted orientation, field, temperature and spatial dependence. Surface roughness and mixed trajectories can change the inference. A phase-sensitive experiment and a local conductance spectrum answer different questions.

## Discovery implications

Use interface spectroscopy to constrain candidate gap and scattering models, then seek independent bulk and phase-sensitive evidence. Report competing parameter sets and failed alternatives alongside a best fit. A good fit can support consistency with a model while leaving parameter uniqueness and microscopic causation unresolved. Preserve that distinction in both prose and machine-readable records.

## Related topics

Compare [gap probes](/scwiki/superconducting-gap), [zero resistance](/scwiki/zero-resistance), [BCS theory](/scwiki/bcs-weak-coupling), [multiband pairing](/scwiki/multiband-pairing), and [s± pairing](/scwiki/s-plus-minus).

## Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.

| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |
|---|---|---|---|---|---|---|
| Cu / Nb; Blonder–Tinkham Cu-Nb point contacts; abstract scope | No transition temperature extracted at this review scope | unassigned; point-contact-conductance | Point-contact differential conductance across metallic/tunneling regimes; full geometry and uncertainty not checked | unknown; unknown GPa | unknown | experiment; [source](#ref-source-851c482722643e96) |

## Sources

### Ref source-18bd294c2b4c06a7

[R1] The Thermal Conductivity of the Intermediate State in Superconductors (1964). [Source publication](https://www.jetp.ras.ru/cgi-bin/dn/e_019_05_1228.pdf). Publication: journal-article. Check: primary-scan-scoped-check. Checked: 2026-10-18. Andreev, Sov. Phys. JETP 19, 1228–1231. Selected pp. 1228–1229; p. 1229 visually checked for Eqs. (6)–(10), subgap evanescence and reversed velocity. Thermal-resistance derivation not fully reproduced. No DOI established.
### Ref source-bcd31ef90f21ff0a

[R2] Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion (1982). [DOI 10.1103/PhysRevB.25.4515](https://doi.org/10.1103/PhysRevB.25.4515). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. BTK publisher abstract only: barrier crossover, Bogoliubov scattering and excess current. Full original derivation and finite-bias curves not checked.
### Ref source-8165cabd8e37fbe8

[R3] Quantum transport in semiconductor-superconductor microjunctions (1992). [DOI 10.1103/PhysRevB.46.12841](https://doi.org/10.1103/PhysRevB.46.12841). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Original 1992 theory abstract only. The later same-title lectures are separately referenced; do not attribute their full-text check to this paper.
### Ref source-eb20b5a729cf98da

[R4] Tunnelling effects on surface bound states in unconventional superconductors (2000). [DOI 10.1088/0034-4885/63/10/202](https://doi.org/10.1088/0034-4885/63/10/202). Publication: journal-article. Check: crossref-metadata-only. Checked: 2026-10-18. Bibliographic pointer. Full review inaccessible at the attempted publisher endpoint; no detailed claim verified from it.
### Ref source-02f3aac52d07be56

[R5] Quantum transport in semiconductor-superconductor microjunctions (Les Houches lectures; arXiv v2) (1994). [DOI 10.48550/arXiv.cond-mat/9406083](https://doi.org/10.48550/arXiv.cond-mat/9406083). Publication: preprint. Check: author-review-scoped-check. Checked: 2026-10-18. Selected pp. 2–4, Eqs. (1.1)–(1.2): two-terminal, zero-temperature/voltage/field conductance and spatially separated normal/Andreev scattering. Lectures dated June 1994; archive v2 September 2005; linked book chapter 1995. Full review not audited.
### Ref source-851c482722643e96

[R6] Metallic to tunneling transition in Cu-Nb point contacts (1983). [DOI 10.1103/PhysRevB.27.112](https://doi.org/10.1103/PhysRevB.27.112). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Experimental publisher abstract: Cu-Nb contacts across the crossover. Numerical curves, contact size, heating and fitting uncertainties not independently checked.
### Ref source-cbbdb7f1a746a8f4

[R7] A phenomenological theory of zero-energy Andreev resonant states (2004). [DOI 10.1103/PhysRevB.69.134501](https://doi.org/10.1103/PhysRevB.69.134501). Publication: journal-article. Check: author-abstract-scoped-check. Checked: 2026-10-18. Author abstract and journal identity: theoretical sign-changing surface states and zero-bias peaks. Detailed surface/field calculations not reviewed.

## Claim evidence and conditions

### scwiki-a9-claim-scope-1

Andreev reflection converts an electron-like excitation in a normal conductor into a reflected hole-like excitation at a superconducting interface. In the conventional subgap process, charge of magnitude $2e$ enters the condensate. Andreev's original calculation treats a slowly varying excitation near the Fermi surface; the selected scan shows subgap evanescence and reversal of velocity. Retroreflection is an approximation with a specified band and energy regime, not a universal trajectory rule.

Evidence: theory. Sources: source-18bd294c2b4c06a7.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-scope-2

This is an interface probe. Conductance depends on the contact as well as the superconducting state, so a fitted gap scale or an enhanced subgap signal is not automatically a measurement of bulk pairing symmetry. Keep interface scattering, normal-state background and sample conditions attached to an interpretation.

Evidence: hypothesis. Sources: source-02f3aac52d07be56.

Conditions: Editorial evidence criteria; contact, regime and competing models required. No unique mechanism or topology from a fitted peak.

Verification scope: Pinned A9 chapter edited with limited source checks.

### scwiki-a9-claim-mathematical-assumptions-1

For one incident channel, let $A(E)$ and $B(E)$ be Andreev and normal reflection probabilities with consistent flux normalization. The charge-conductance weight is $1+A-B$. Under purely subgap reflection, probability conservation gives $A+B=1$, hence the weight is $2A$. A factor of two in this ideal charge conversion does not imply that every measured junction has twice its normal-state conductance.

Evidence: theory. Sources: source-02f3aac52d07be56.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-mathematical-assumptions-2

The original BTK abstract describes a tunable interface barrier. To check an explicit limit independently, take a spin-degenerate, single-channel, time-reversal-symmetric normal barrier of transmission $\mathcal T$, separated from an ideal conventional s-wave Andreev reflector. At zero energy the round-trip Andreev phase is $-1$, giving

Evidence: theory. Sources: source-02f3aac52d07be56.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-mathematical-assumptions-3

$$A(0)=\left(\frac{\mathcal T}{2-\mathcal T}\right)^2,\qquad \frac{G_{NS}}{G_N}=\frac{2\mathcal T}{(2-\mathcal T)^2}.$$

Evidence: theory. Sources: source-02f3aac52d07be56.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-mathematical-assumptions-4

For a one-dimensional delta barrier with matched velocities, define $Z$ so that $\mathcal T=1/(1+Z^2)$. Summing coherent electron–hole round trips gives $A(0)=1/(1+2Z^2)^2$. The transparent limit has conductance ratio two; the tunnel limit is suppressed. Independent complex-amplitude sums check twelve barrier/energy cases. This is an ideal scattering check, not a reanalysis of BTK's full finite-bias treatment.

Evidence: theory. Sources: source-bcd31ef90f21ff0a.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-mathematical-assumptions-5

Beenakker's later lectures give, at zero temperature, bias and magnetic field, $G_{NS}=(4e^2/h)\sum_n\mathcal T_n^2/(2-\mathcal T_n)^2$ and $G_N=(2e^2/h)\sum_n\mathcal T_n$. These are two-terminal formulas for the stated coherent NS model, with normal scattering separated from the ideal superconducting boundary. The original 1992 paper is retained separately at abstract scope.

Evidence: theory. Sources: source-02f3aac52d07be56, source-8165cabd8e37fbe8.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-mathematical-assumptions-6

Equal normal conductance does not determine the superconducting conductance. For example, transmission sets $(1,0)$ and $(1/2,1/2)$ have the same sum but different sums of $\mathcal T_n^2/(2-\mathcal T_n)^2$. This independent counterexample explains why a contact resistance alone cannot specify a multichannel junction.

Evidence: theory. Sources: source-02f3aac52d07be56.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-mathematical-assumptions-7

As a separate local derivation, take an elastic conductance spectrum $g_0(E)$ that is independent of temperature and bias, with reservoirs described by equilibrium Fermi functions at the same $T>0$. Let $e>0$ be the elementary charge and $eV$ the chemical-potential shift. Thermal smearing then gives:

Evidence: theory. Sources: source-02f3aac52d07be56.

Source relationship: context-for-local-derivation. Beenakker v2 pp. 3-4 supplies zero-temperature coherent-interface context. This finite-temperature convolution is independently derived from the equilibrium Fermi occupation difference for a fixed spectrum; it is not attributed to those zero-temperature formulas.

Conditions: Fixed elastic spectrum independent of temperature and bias; equal positive reservoir temperature; chemical-potential displacement eV with e>0. No heating, nonequilibrium distributions, self-consistent gap change or nonlinear barrier.

Verification scope: Local Fermi-function differentiation, normalization and synthetic quadrature; cited review provides zero-temperature context only, not direct finite-temperature derivation verification.

### scwiki-a9-claim-mathematical-assumptions-8

$$G(V,T)=\int dE\,g_0(E)\frac{\operatorname{sech}^2[(E-eV)/(2k_BT)]}{4k_BT}.$$

Evidence: theory. Sources: source-02f3aac52d07be56.

Source relationship: context-for-local-derivation. Beenakker v2 pp. 3-4 supplies zero-temperature coherent-interface context. This finite-temperature convolution is independently derived from the equilibrium Fermi occupation difference for a fixed spectrum; it is not attributed to those zero-temperature formulas.

Conditions: Fixed elastic spectrum independent of temperature and bias; equal positive reservoir temperature; chemical-potential displacement eV with e>0. No heating, nonequilibrium distributions, self-consistent gap change or nonlinear barrier.

Verification scope: Local Fermi-function differentiation, normalization and synthetic quadrature; cited review provides zero-temperature context only, not direct finite-temperature derivation verification.

### scwiki-a9-claim-mathematical-assumptions-9

The kernel integrates to one and preserves a constant conductance. Three numerical checks compare the convolution to the derivative of its corresponding current integral. The check uses a synthetic smooth spectrum, not a BTK fit. A temperature-dependent gap, bias-dependent barrier, nonequilibrium distribution or heating requires additional modeling; arbitrary lifetime broadening is not equivalent to thermal convolution.

Evidence: theory. Sources: source-02f3aac52d07be56.

Source relationship: context-for-local-derivation. Beenakker v2 pp. 3-4 supplies zero-temperature coherent-interface context. This finite-temperature convolution is independently derived from the equilibrium Fermi occupation difference for a fixed spectrum; it is not attributed to those zero-temperature formulas.

Conditions: Fixed elastic spectrum independent of temperature and bias; equal positive reservoir temperature; chemical-potential displacement eV with e>0. No heating, nonequilibrium distributions, self-consistent gap change or nonlinear barrier.

Verification scope: Local Fermi-function differentiation, normalization and synthetic quadrature; cited review provides zero-temperature context only, not direct finite-temperature derivation verification.

### scwiki-a9-claim-experimental-criteria-1

Blonder and Tinkham's 1983 experimental abstract reports Cu-Nb point contacts spanning metallic to tunneling behavior and agreement with the model over a range of contact resistances. This entry checks that qualitative observation only. Contact dimensions, individual spectra and parameter uncertainties remain unverified here; no transition temperature is extracted.

Evidence: experiment. Sources: source-851c482722643e96.

Conditions: Cu-Nb point-contact crossover, publisher abstract only; no raw curves, contact size or uncertainty reanalysis.

Verification scope: Blonder–Tinkham 1983 publisher abstract.

### scwiki-a9-claim-experimental-criteria-2

Report contact preparation, normal-state resistance, wiring, crystal direction, current or excitation amplitude, field and temperature. Compare contact dimensions with mean free path and coherence length before choosing a ballistic or diffusive description. Repeat contacts and test heating and critical-current artifacts. Preserve the background normalization and its uncertainty; these are editorial requirements for assessing a claim, not one universally valid apparatus recipe.

Evidence: hypothesis. Sources: source-02f3aac52d07be56.

Conditions: Editorial evidence criteria; contact, regime and competing models required. No unique mechanism or topology from a fitted peak.

Verification scope: Pinned A9 chapter edited with limited source checks.

### scwiki-a9-claim-experimental-criteria-3

A constant series resistance $R_s$ gives $G_{\mathrm{meas}}=G_{\mathrm{int}}/(1+R_sG_{\mathrm{int}})$ when the internal response is differentiated consistently. Ignoring it can change an apparent enhancement or fitted transparency. This circuit identity assumes the series element is bias independent; it does not describe thermal feedback or nonlinear leads.

Evidence: theory. Sources: source-02f3aac52d07be56.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-limits-and-disputes-1

Asano, Tanaka and Kashiwaya's theoretical abstract associates a sign-changing pair potential with zero-energy surface states and a zero-bias conductance peak. This is a model result about a surface, not a new experimental proof of bulk symmetry. The broad Kashiwaya–Tanaka review is retained as a bibliographic pointer; its detailed calculations have not been checked here.

Evidence: theory. Sources: source-cbbdb7f1a746a8f4, source-eb20b5a729cf98da.

Conditions: Conventional NS interface under explicitly stated band, flux, symmetry, temperature and response assumptions; independent ideal models do not certify real contacts.

Verification scope: Selected original scan/review passages or author abstract as cited; conditional complex-amplitude and thermal-kernel checks, not full paper reproduction.

### scwiki-a9-claim-limits-and-disputes-2

A peak alone does not establish a Majorana mode, spin-triplet pairing or a unique mechanism. Model comparison should address ordinary bound states and junction effects, and test the predicted orientation, field, temperature and spatial dependence. Surface roughness and mixed trajectories can change the inference. A phase-sensitive experiment and a local conductance spectrum answer different questions.

Evidence: hypothesis. Sources: source-02f3aac52d07be56.

Conditions: Editorial evidence criteria; contact, regime and competing models required. No unique mechanism or topology from a fitted peak.

Verification scope: Pinned A9 chapter edited with limited source checks.

### scwiki-a9-claim-discovery-implications-1

Use interface spectroscopy to constrain candidate gap and scattering models, then seek independent bulk and phase-sensitive evidence. Report competing parameter sets and failed alternatives alongside a best fit. A good fit can support consistency with a model while leaving parameter uniqueness and microscopic causation unresolved. Preserve that distinction in both prose and machine-readable records.

Evidence: hypothesis. Sources: source-02f3aac52d07be56.

Conditions: Editorial evidence criteria; contact, regime and competing models required. No unique mechanism or topology from a fitted peak.

Verification scope: Pinned A9 chapter edited with limited source checks.

## Revision and review

Article ID: scwiki-a9. Revision: 1a23723887a9a1495c50. Updated: 2026-10-19.

Review scope: Selected Andreev original scan, Beenakker lecture passages and experimental/theoretical abstracts; independent ideal scattering, channel-distribution and thermal-kernel calculations. Thermal smearing is an explicitly separate local fixed-spectrum derivation with positive-temperature and equilibrium conditions.

- Scoped checks are not full scientific acceptance or independent reproduction of contact spectra.
- BTK original and Beenakker 1992 paper remain abstract-scoped; later same-title lectures have a separate source/version.
- Kashiwaya–Tanaka review is metadata-only; Asano surface-state claim is abstract-scoped.
- No contact-specific transition temperature, unique pairing mechanism or topological state is inferred.

Source: [A_phenomenology/A9_andreev_reflection.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/A_phenomenology/A9_andreev_reflection.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 ee5d88be874334d860bd656000d56f866c565409d1e4c7abef85f8400cd9f378.