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Andreev reflection
Andreev reflection
Andreev reflection
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An electron (red) meeting the interface between a normal conductor (N) and a superconductor (S) produces a Cooper pair in the superconductor and a retroreflected hole (green) in the normal conductor. Vertical arrows indicate the spin band occupied by each particle.

Andreev reflection, named after the Russian physicist Alexander F. Andreev, is a type of particle scattering which occurs at interfaces between a superconductor (S) and a normal state material (N). It is a charge-transfer process by which normal current in N is converted to supercurrent in S. Each Andreev reflection transfers a charge 2e across the interface, avoiding the forbidden single-particle transmission within the superconducting energy gap.

This effect is generally called Andreev reflection but it is also be referred to as Andreev–Saint-James reflection, as it was predicted independently by Saint-James and de Gennes and by Andreev in the early sixties.[1]

Overview

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The process involves an electron incident on the interface from the normal state material at energies less than the superconducting energy gap. The incident electron forms a Cooper pair in the superconductor with the retroreflection of a hole of opposite spin and velocity but equal momentum to the incident electron, as seen in the figure. The barrier transparency is assumed to be high, with no oxide or tunnel layer which reduces instances of normal electron-electron or hole-hole scattering at the interface. Since the pair consists of an up and down spin electron, a second electron of opposite spin to the incident electron from the normal state forms the pair in the superconductor, and hence the retroreflected hole. Through time-reversal symmetry, the process with an incident electron will also work with an incident hole (and retroreflected electron).

The process is highly spin-dependent – if only one spin band is occupied by the conduction electrons in the normal-state material (i.e. it is fully spin-polarized), Andreev reflection will be inhibited due to inability to form a pair in the superconductor and impossibility of single-particle transmission. In a ferromagnet or material where spin-polarization exists or may be induced by a magnetic field, the strength of the Andreev reflection (and hence conductance of the junction) is a function of the spin-polarization in the normal state.

The spin-dependence of Andreev reflection gives rise to the point contact Andreev reflection technique, whereby a narrow superconducting tip (often niobium, antimony or lead) is placed into contact with a normal material at temperatures below the critical temperature of the tip. By applying a voltage to the tip, and measuring differential conductance between it and the sample, the spin polarization of the normal metal at that point (and magnetic field) may be determined. This is of use in such tasks as measurement of spin-polarized currents or characterizing spin polarization of material layers or bulk samples, and the effects of magnetic fields on such properties.

In an Andreev process, the phase difference between the electron and hole is −π/2 plus the phase of the superconducting order parameter.

Crossed Andreev reflection

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Crossed Andreev reflection, also known as non-local Andreev reflection, occurs when two spatially separated normal state material electrodes form two separate junctions with a superconductor, with the junction separation of the order of the BCS superconducting coherence length of the material in question. In such a device, retroreflection of the hole from an Andreev reflection process, resulting from an incident electron at energies less than the superconducting gap at one lead, occurs in the second spatially separated normal lead with the same charge transfer as in a normal Andreev reflection process to a Cooper pair in the superconductor.[2] For crossed Andreev reflection to occur, electrons of opposite spin must exist at each normal electrode (so as to form the pair in the superconductor). If the normal material is a ferromagnet this may be guaranteed by creating opposite spin polarization via the application of a magnetic field to normal electrodes of differing coercivity.

Crossed Andreev reflection occurs in competition with elastic cotunneling, the quantum mechanical tunneling of electrons between the normal leads via an intermediate state in the superconductor. This process conserves electron spin. As such, a detectable potential at one electrode on the application of current to the other may be masked by the competing elastic cotunneling process, making clear detection difficult. In addition, normal Andreev reflection may occur at either interface, in conjunction with other normal electron scattering processes from the normal/superconductor interface.

The process is of interest in the formation of solid-state quantum entanglement, via the formation of a spatially separated entangled electron-hole (Andreev) pair, with applications in spintronics and quantum computing.

References

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Further reading

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Andreev reflection

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Andreev reflection is a charge-transfer process at the interface between a normal metal (N) and a superconductor (S), in which an incident from side is retro-reflected as a hole of opposite spin and momentum, while the missing charge (equivalent to two electrons) is transmitted into the superconductor as a Cooper pair, enabling the conversion of normal current to supercurrent without dissipation within the superconducting energy gap. This phenomenon was first theoretically described by Alexander F. Andreev in 1964 while investigating thermal conductivity in the intermediate state of type-I superconductors, where he derived the electron-to-hole reflection mechanism at NS boundaries. The process is most efficient at energies below the superconducting gap Δ (typically on the order of meV), where single-particle transmission is forbidden, and it relies on the proximity effect, which induces superconducting correlations in the normal metal near the interface. Andreev reflection plays a central role in mesoscopic , underpinning the formation of Andreev bound states—localized subgap excitations that mediate Josephson currents in superconducting weak links and influence transport in hybrid nanostructures such as superconductor-ferromagnet junctions. It has been experimentally observed through , where the doubled charge (2e) manifests as enhanced subgap conductance, and serves as a probe for unconventional , including d-wave pairing in high-temperature cuprates. In ballistic NS junctions, multiple Andreev reflections can lead to multiple charge transfer with effective charges larger than 2e and are key to applications in superconducting qubits and nanoscale electronics.

Introduction and Fundamentals

Definition and Physical Process

Andreev reflection is a charge-transfer process that occurs at the interface between a normal metal (N) and a superconductor (S), converting an incident normal current in the N region into a supercurrent in the S region by transferring a charge of 2e across the interface. In this process, an from the normal metal with below the superconducting gap impinges on the interface and pairs with another of opposite spin to form a that enters the superconductor, while a is retroreflected back into the normal metal. The retroreflected possesses the opposite spin to the incident and has equal magnitude but reversed , effectively retracing the electron's path due to the time-reversal symmetry inherent in the superconducting state. This phenomenon is dominant for incident electron energies |E| satisfying |E| < Δ, where Δ is the superconducting energy gap, as the gap prohibits single-particle excitations within the superconductor; above the gap (|E| > Δ), the process is suppressed in favor of direct transmission or normal reflection. The spin dependence arises from the requirement that Cooper pairs consist of electrons with opposite spins, imposing spin selectivity on the reflection: the incident electron's spin determines the reflected hole's spin, enabling applications in spin-polarized . Additionally, the retroreflected hole acquires a phase shift of −π/2 relative to the incident electron, plus the phase of the superconducting order parameter, which plays a crucial role in phase-coherent effects at the interface. This qualitative process was later formalized in the Blonder-Tinkham-Klapwijk model.

Historical Development

The groundwork for understanding reflection processes at superconductor-normal metal (N-S) interfaces was laid in 1963 by D. Saint-James and P. G. de Gennes, who predicted the existence of bound states at normal metal-superconductor interfaces, providing early insights into subgap excitations that would later connect to proximity-induced phenomena. In 1964, Alexander F. Andreev extended this framework by predicting a charge-transfer process at clean N-S interfaces within the context of proximity effects in superconductors, arising from solutions to the Bogoliubov-de Gennes equations that describe behavior across the boundary. Andreev's work specifically addressed transport in the intermediate state of type I superconductors, where electrons incident from the normal side undergo retroreflection as holes, effectively inducing Cooper pairs in the superconductor. This prediction built directly on the bound-state concepts from Saint-James and de Gennes, establishing the theoretical basis for interface . Early experimental hints emerged in the through studies of proximity effects, such as enhanced conductance observed in N-S contacts by J. I. Pankove in , though these were not immediately attributed to the predicted reflection mechanism due to limitations in resolution and interpretation. Full confirmation remained elusive until the 1980s, when advances in tunneling and point-contact provided clearer evidence of subgap conductance features consistent with the process, enabling quantitative validation in high-quality junctions. The phenomenon is named after Andreev for his seminal prediction of the reflection process, while the combined electron-hole reflection at interfaces is often referred to as Andreev-Saint-James reflections to honor the earlier contributions. A key review in 2005 by G. Deutscher synthesized these developments, emphasizing their application to and underscoring the historical linkage between the 1960s predictions and modern .

Theoretical Framework

Basic Model and Derivation

The Bogoliubov-de Gennes (BdG) equations provide the foundational theoretical framework for describing quasiparticle excitations in superconductors, treating them as coherent superpositions of electron-like and hole-like components. The wavefunction is expressed as a spinor ψ=(uv)\psi = \begin{pmatrix} u \\ v \end{pmatrix}, where uu represents the electron amplitude and vv the hole amplitude. These equations arise from linearizing the BCS Hamiltonian around the Fermi surface and are given by (H0ΔΔ∗−H0)(uv)=E(uv),\begin{pmatrix} H_0 & \Delta \\ \Delta^* & -H_0 \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} = E \begin{pmatrix} u \\ v \end{pmatrix}, with H0=−ℏ22m∇2−μH_0 = -\frac{\hbar^2}{2m} \nabla^2 - \mu the single-particle Hamiltonian and Δ\Delta the superconducting order parameter. In the basic model for Andreev reflection, a one-dimensional normal metal-superconductor (N-S) interface is considered, with the normal region occupying x<0x < 0 (characterized by Fermi wavevector kFk_F) and the superconductor region x>0x > 0 (with pairing gap Δ\Delta). An with subgap ∣E∣<∣Δ∣|E| < |\Delta| is incident from the normal metal. In the normal region, the wavefunction consists of an incident plane wave and reflected components: an electron-like reflection and a hole-like Andreev reflection. In the superconductor, since propagation is forbidden within the gap, the wavefunction decays evanescently as a combination of - and hole-like quasiparticles. The Andreev approximation assumes excitation much smaller than the Fermi energy (E,Δ≪EFE, \Delta \ll E_F), allowing neglect of interband mixing and approximate conservation of momentum parallel to the interface, which results in retroreflection of the hole with opposite group velocity but nearly the same wavevector. At the N-S interface, the boundary conditions enforce continuity of the wavefunction ψ\psi and its derivative dψdx\frac{d\psi}{dx} to ensure current conservation. For a perfectly transparent interface (no potential barrier), matching these conditions in the Andreev approximation yields the reflection coefficients. The probability of normal reflection B(E)≈0B(E) \approx 0, while the Andreev reflection probability approaches unity A(E)≈1A(E) \approx 1 for subgap incidence, as the electron pairs with another from the normal metal to form a Cooper pair in the superconductor, retroreflecting a hole. The energy dependence of the Andreev reflection probability for a transparent interface is A(E)=Δ2E2+(Δ2−E2),A(E) = \frac{\Delta^2}{E^2 + (\Delta^2 - E^2)}, which simplifies to 1 for ∣E∣<Δ|E| < \Delta. The Andreev reflection amplitude, describing the coherent electron-to-hole conversion, is derived from the boundary matching and given by rhe=ΔE+iΔ2−E2 eiϕ,r_{he} = \frac{\Delta}{E + i \sqrt{\Delta^2 - E^2}} \, e^{i \phi},
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