Crossed Andreev Reflection in Superconducting Devices

• CAR is a nonlocal superconducting process where an electron in one lead converts into a hole in a distant lead, directly probing Cooper-pair correlations.
• CAR competes with elastic cotunneling and local Andreev reflection, with its effectiveness enhanced by spin filtering, band alignment, and interference techniques.
• CAR facilitates Cooper-pair splitting and quantum entanglement, offering a pathway to study subgap states and nonlocal superconducting phenomena in various hybrid systems.

Searching arXiv for recent and foundational papers on crossed Andreev reflection. Crossed Andreev reflection (CAR) is a nonlocal superconducting scattering process in which an electron incident from one metallic lead is converted into a hole in a different, spatially separated lead, while the superconductor absorbs the missing charge as part of a Cooper pair. In contrast to local Andreev reflection, where the reflected hole remains in the injecting lead, CAR directly couples distant terminals and therefore probes nonlocal superconducting correlations. Across contemporary mesoscopic, magnetic, topological, and quantum Hall platforms, CAR is studied both as a transport channel competing with elastic cotunneling and ordinary reflection, and as a mechanism for Cooper-pair splitting, entanglement generation, and nonlocal spectroscopy (Soori, 2022, Das et al., 2024, Feng et al., 2024).

1. Definition and elementary scattering structure

In the standard two-lead superconducting geometry, an electron incident on a superconducting region can undergo four elementary processes. Electron reflection (ER) returns the electron to the injecting lead as an electron. Local Andreev reflection (AR) returns it as a hole in the same lead. Electron tunneling (ET), also called elastic cotunneling in several works, transfers it to the opposite lead as an electron. Crossed Andreev reflection (CAR) transfers it to the opposite lead as a hole (Soori, 2022, Das et al., 2024, Jakobsen et al., 2021).

This distinction is operationally important because CAR is the only one of these processes that is simultaneously nonlocal and Andreev-like. In the formulation used for a ferromagnet/superconductor/normal-metal/superconductor/ferromagnet heterostructure, current conservation for an electron incident from the left reduces to

$$|r_e|^2 + |t_h|^2 \frac{\sin k_h a}{\sin k_e a} = 1,$$

when the device geometry removes the transmitted electron channel from the relevant bias window (Soori, 2022). In that regime, ER and CAR exhaust the outgoing probability.

The nonlocal conductance signature follows directly from the hole nature of the transmitted quasiparticle. In the same formulation, the differential transconductance is

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$

so a negative nonlocal response tracks CAR probability directly (Soori, 2022). Closely related sign conventions recur across distinct platforms: in antiferromagnet–superconductor–antiferromagnet junctions, topological-insulator nanowires, and diffusive NSN structures, CAR contributes with the sign opposite to ET or elastic cotunneling in nonlocal conductance or current (Jakobsen et al., 2021, Feng et al., 2024, Tjernshaugen et al., 2024).

CAR is also the microscopic inverse of Cooper-pair splitting in the transport sense emphasized for altermagnet-based and quantum Hall-based devices. In one driving convention, current from a lead into a superconductor produces CAR; in the reverse interpretation, a Cooper pair in the superconductor is split between two separated metallic contacts (Das et al., 2024, Zhang et al., 2019).

2. Competition with ET, AR, and ER

A persistent theme in the CAR literature is that generic devices suppress CAR because the injected current partitions among several allowed channels. In ordinary two-normal-lead/superconductor systems, ET is often stronger than CAR, while AR can dominate locally at the injecting interface, and ER further reduces the residual nonlocal probability (Soori, 2022). This competition is the central reason that robust CAR signatures are difficult to obtain without additional selection rules or interference control.

Several works formalize this competition through the sign of nonlocal conductance. In antiferromagnetic junctions, the zero-temperature nonlocal differential conductance contains a positive contribution from cotunneling and a negative one from CAR, so negative nonlocal conductance indicates CAR dominance (Jakobsen et al., 2021). The same logic appears in altermagnet/superconductor/altermagnet structures, where negative nonlocal conductivity signals that CAR outweighs ET (Das et al., 2024), and in conventional NSN structures treated within self-consistent Keldysh-Usadel theory, where

$I_{nl} = (G_E - G_C)V$

for grounded nonlocal measurement, implying $I_{nl}<0$ when CAR dominates (Tjernshaugen et al., 2024).

This competition can be removed or greatly reduced by band filtering, spin selection, or geometry. Fully spin-polarized antiparallel ferromagnetic leads suppress AR and ET in the FM/SC/NM/SC/FM proposal, leaving essentially only ER and CAR (Soori, 2022). Oppositely doped antiferromagnetic leads can suppress CT and local AR over a finite bias window and yield robust, nearly perfect CAR (Jakobsen et al., 2021). In bipolar magnetic semiconductors, opposite-sign lead chemical potentials make AR and ET vanish entirely, leaving transport dominated by CAR and ER (Naveen et al., 15 May 2026). In Dirac-semimetal quantum Hall junctions operated in the zeroth-Landau-level regime, channel mismatch blocks local Andreev reflection and electron cotunneling, permitting perfect CAR with probability unity in suitable parameter regimes (Zhang et al., 2019).

A related but conceptually distinct route is to enhance CAR not by direct suppression of ET through filtering, but by engineering propagating subgap states that convert the central superconductor from an evanescent barrier into a resonant nonlocal converter. The superconducting ladder proposal does precisely this: a phase-biased transverse Josephson junction produces subgap Andreev states extended along the transport direction, and coherent interference can drive the device smoothly between ET-dominant and CAR-dominant regimes (Soori et al., 2016).

3. Interference, resonance, and tunability

A large class of CAR platforms exploit interference. In the FM/SC/NM/SC/FM heterostructure, a gate-tunable normal-metal segment functions as a Fabry–Pérot interferometer for electron and hole modes. The relevant wave numbers are

$$k_{e0}a=\cos^{-1}\!\left[-\frac{\mu_0+E}{2t}\right], \qquad k_{h0}a=\cos^{-1}\!\left[-\frac{\mu_0-E}{2t}\right],$$

and at zero bias they become equal. Resonant maxima occur when successive values of $\mu_0$ satisfy

$$(k_{e0,i+1}-k_{e0,i})(N_0+1)a=(k_{h0,i+1}-k_{h0,i})(N_0+1)a=\pi.$$

This produces oscillatory conversion between ER-dominant and CAR-dominant transport, with conductance tunable from $0$ to $-e^2/h$ (Soori, 2022).

Fabry–Pérot-type oscillations also appear when the superconducting region itself is the interferometric cavity. In bipolar magnetic semiconductors, resonances arise from multiple reflections inside the superconductor with phase condition

$2\,\mathrm{Re}(q_x)L=2\pi n,$

and peak spacing

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$0

(Naveen et al., 15 May 2026). In altermagnet junctions, the conductivities oscillate with superconducting length $$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$1 with spacing

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$2

again interpreted as Fabry–Perot interference (Das et al., 2024). In the superconducting ladder, the subgap Andreev modes act as a resonant cavity; recurrent CAR enhancement with chemical potential obeys approximately

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$3

in the large-$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$4 limit (Soori et al., 2016).

Interference need not be purely electrostatic. In focused CAR, weak perpendicular magnetic fields steer quasiparticles semiclassically in a ballistic 2DEG. The cyclotron diameter

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$5

matches the device geometry at

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$6

or equivalently

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$7

At integer multiples of $$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$8, trajectories are directed through the superconducting contact and CAR is enhanced; at half-integer multiples, ET is enhanced instead (Haugen et al., 2010). This replaces phase tuning in a confined cavity with orbital focusing as the control knob.

A further class of tunable proposals exploits Berry-phase or flux control in topological nanowires. In a topological-insulator nanowire T-junction, a flux near

$$G_{RL} = -|t_h|^2 \frac{e^2}{h}\frac{\sin k_h a}{\sin k_e a},$$9

through the superconducting arm converts the interface into a regime supporting perfect or near-perfect CAR in the single-mode quantum Hall limit (Fuchs et al., 2020). In proximitized topological-insulator nanowires, CAR appears only at certain gate settings, consistent with the requirement that local spectra on both sides support participating states (Feng et al., 2024).

4. Material and device platforms

CAR has been studied in a wide range of superconducting hybrids, with the dominant design principle being selective removal of ET or AR, or selective enhancement of nonlocal hole conversion.

Platform	Main control mechanism	Characteristic CAR result
FM/SC/NM/SC/FM	Gate-controlled Fabry–Pérot interference with antiparallel fully spin-polarized leads	CAR and ER probabilities tunable between $I_{nl} = (G_E - G_C)V$0 and $I_{nl} = (G_E - G_C)V$1 (Soori, 2022)
BMS–SC–BMS	Independent chemical potentials in oppositely spin-polarized band edges	AR and ET vanish entirely for opposite-sign lead chemical potentials (Naveen et al., 15 May 2026)
AM–SC–AM	$I_{nl} = (G_E - G_C)V$2 rotation of altermagnets	Strong phase supports CAR-dominant nonlocal transport (Das et al., 2024)
AF–S–AF	Opposite doping in antiferromagnetic leads	Robust signature of perfect CAR in nonlocal differential conductance (Jakobsen et al., 2021)
QH–S–QH Dirac semimetal	Zeroth-Landau-level single-channel transport	Perfect CAR probability can reach unity without applying bias voltage (Zhang et al., 2019)
TI nanowire T-junction	Perpendicular and axial magnetic fields	Perfect CAR in a restricted range, robust to disorder (Fuchs et al., 2020)
Proximitized TI nanowire	Gate tuning of local states; disorder-assisted extended ABSs	Negative nonlocal conductance and CAR over $I_{nl} = (G_E - G_C)V$3m (Feng et al., 2024)
SC ladder between NMs	Phase difference and inter-leg coupling	Negative transconductance from CAR-dominant transport (Soori et al., 2016)

Magnetic systems furnish several distinct CAR mechanisms. Ferromagnetic spin valves suppress AR and ET through half-metal filtering (Soori, 2022). Altermagnets provide momentum-space spin splitting with zero net polarization, enabling field-free CAR filtering through crystallographic orientation rather than net magnetization (Das et al., 2024). Antiferromagnets preserve spin degeneracy while exploiting exchange-induced band gaps and opposite doping to suppress competing channels (Jakobsen et al., 2021). Bipolar magnetic semiconductors replace ferromagnets entirely with leads whose conduction and valence edges are fully spin-polarized in opposite directions (Naveen et al., 15 May 2026).

Topological and Dirac platforms exploit chiral or helical transport. In QH–S–QH junctions formed from a time-reversal symmetric Dirac semimetal, only zeroth Landau levels participate in the quantum limit,

$I_{nl} = (G_E - G_C)V$4

so one side supports only electron channels and the other only hole channels within a given BdG block, eliminating local Andreev reflection and EC (Zhang et al., 2019). In QSHI–SC–QSHI junctions, edge helicity spatially separates ET and CAR onto opposite edges of the outgoing lead, enabling all-electrical discrimination between the two (Reinthaler et al., 2012).

Quantum Hall systems provide a separate experimental regime. In graphene-based van der Waals heterostructures with narrow NbN electrodes, negative edge resistance evidences CAR across the superconductor separating two fractional quantum Hall edges, with particle-like fractional fillings showing markedly higher CAR probabilities than integer and hole-conjugate fractional fillings (Gül et al., 2020).

5. Conductance, noise, and geometric signatures

The most common observable is nonlocal differential conductance, whose sign usually distinguishes CAR from ET. In the FM/SC/NM/SC/FM proposal, the nonlocal differential transconductance is tunable across

$I_{nl} = (G_E - G_C)V$5

indicating a crossover from suppressed to nearly perfect nonlocal hole transmission (Soori, 2022). In Dirac-semimetal QH–S–QH junctions, ideal perfect CAR gives

$I_{nl} = (G_E - G_C)V$6

with the local and nonlocal responses equal in magnitude and opposite in sign (Zhang et al., 2019). By contrast, in TI nanowire T-junctions the convention is

$I_{nl} = (G_E - G_C)V$7

so positive nonlocal conductance signals CAR dominance (Fuchs et al., 2020). This sign difference reflects geometry and current convention rather than any physical ambiguity.

Shot noise and current cross correlations are especially useful when average conductance is insufficiently selective. In the four-terminal superconducting proximity junction proposed as a mesoscopic beam splitter, a single injected electron gives perfect anticorrelation between distinct output leads,

$I_{nl} = (G_E - G_C)V$8

while for two injected electrons the cross-correlation coefficient becomes

$I_{nl} = (G_E - G_C)V$9

The negative sign is the shot-noise signature of fermionic partition noise in CAR-mediated splitting (Marga et al., 29 Oct 2025).

Noise is also central in proposals using CAR to diagnose nonlocal states in topological superconductors. In multiband one-dimensional topological superconductors, zero-temperature shot noise is summarized by Fano factors $I_{nl}<0$0. CAR-dominated transport gives $I_{nl}<0$1 and $I_{nl}<0$2, whereas local Andreev reflection yields $I_{nl}<0$3 and $I_{nl}<0$4 (Wu et al., 2015). This use of noise as a nonlocality probe is distinct from conductance-based identification.

CAR can also have geometric observables. In NSN junctions with strong spin-orbit-coupled terminals, the outgoing CAR hole beam can exhibit a transverse displacement relative to the incident electron beam. For identical Weyl-semimetal terminals, the CAR transverse shift is

$I_{nl}<0$5

while the corresponding EC shift vanishes,

$I_{nl}<0$6

This suggests that CAR can be probed not only through charge transport but also through beam geometry and transverse accumulation (Liu et al., 2018).

6. CAR, bound states, and topological or entanglement contexts

CAR is frequently linked to Andreev bound states, Majorana physics, and entanglement generation, but these connections are not interchangeable.

A short superconducting segment can mediate both CAR and elastic cotunneling between quantum dots through Andreev bound states. In semiconductor-superconductor heterostructures, ABSs possess electron and hole coherence factors $I_{nl}<0$7 and $I_{nl}<0$8, leading to constructive interference for CAR and destructive interference for ECT: $I_{nl}<0$9 Near the charge-neutral ABS point $$k_{e0}a=\cos^{-1}\!\left[-\frac{\mu_0+E}{2t}\right], \qquad k_{h0}a=\cos^{-1}\!\left[-\frac{\mu_0-E}{2t}\right],$$0, CAR peaks while ECT is suppressed (Bordin et al., 2022). In that context, the tunable CAR/ECT ratio is relevant to “poor man’s Majorana bound states” and minimal Kitaev-chain engineering rather than direct topological proof.

The relation between CAR and Majorana modes is more subtle in multiband one-dimensional topological superconductors. Approximate chiral symmetry can render both multiple Majorana modes and low-energy Andreev bound states nonlocal, allowing both to drive nearly zero-bias CAR. The paper on symmetry-protected and topologically protected CAR emphasizes that this makes CAR alone insufficient to identify Majorana modes in clean or non-magnetically disordered systems (Wu et al., 2015). Magnetic disorder breaks chiral symmetry, localizes low-energy ABSs, and can leave only topologically protected CAR from Majorana modes. This is an explicit caution against treating any zero-bias nonlocal CAR signal as uniquely Majorana-derived.

Experimental work on proximitized topological-insulator nanowires adds a related complication. Local conductance shows a hard gap and Andreev bound states that can reach zero bias, while CAR appears as occasional negative nonlocal conductance upon sweeping chemical potential. The reported long-range CAR over $$k_{e0}a=\cos^{-1}\!\left[-\frac{\mu_0+E}{2t}\right], \qquad k_{h0}a=\cos^{-1}\!\left[-\frac{\mu_0-E}{2t}\right],$$1m is interpreted not as a simple coherence-length effect but as possibly due to disorder-induced overlapping ABSs hybridizing into an “Andreev band” (Feng et al., 2024). This suggests that disorder may enable rather than merely suppress long-range nonlocality in some proximitized nanowires.

The entanglement interpretation is more direct in CAR beam-splitter proposals and in quantum Hall-superconductor hybrids. CAR naturally splits a Cooper pair into spatially separated outputs, and shot-noise or Bell-type logic is therefore central to its significance (Marga et al., 29 Oct 2025). In antiferromagnet-based proposals, the preservation of spin degeneracy is highlighted as advantageous for spin-entangled pair generation relative to ferromagnetic schemes that suppress one spin species (Jakobsen et al., 2021).

7. Open issues, design principles, and recurring misconceptions

A recurring misconception is that CAR is generically short-ranged on the scale of the superconducting coherence length. Focused CAR in ballistic 2DEGs explicitly argues that, with semiclassical focusing, the effective separation scale becomes the mean free path rather than $$k_{e0}a=\cos^{-1}\!\left[-\frac{\mu_0+E}{2t}\right], \qquad k_{h0}a=\cos^{-1}\!\left[-\frac{\mu_0-E}{2t}\right],$$2 (Haugen et al., 2010). Experimental work on proximitized topological-insulator nanowires reports CAR over $$k_{e0}a=\cos^{-1}\!\left[-\frac{\mu_0+E}{2t}\right], \qquad k_{h0}a=\cos^{-1}\!\left[-\frac{\mu_0-E}{2t}\right],$$3m, far beyond the estimated dirty-limit coherence length, with disorder-induced extended subgap states proposed as the mechanism (Feng et al., 2024). These results do not invalidate the usual coherence-length intuition, but they show that device geometry and subgap-state structure can supersede it.

A second misconception is that magnetic polarization always helps CAR. Several papers support a more conditional statement. Fully spin-polarized ferromagnets can suppress AR and ET and thereby help CAR (Soori, 2022), but self-consistent diffusive NSN theory finds that spin splitting in the superconductor always favors EC over CAR because it increases the subvoltage density of states (Tjernshaugen et al., 2024). Likewise, the ferromagnet/superconductor/ferromagnet review emphasizes that fully transparent interfaces suppress CAR even in spin-resolved structures (Kalenkov et al., 2011). Thus spin selectivity is beneficial only when it blocks competing channels without simultaneously increasing quasiparticle leakage.

A third misconception is that perfect CAR requires bias fine tuning. In the Dirac-semimetal QH–S–QH proposal, perfect CAR can reach unity without applying bias voltage by exploiting zeroth-Landau-level channel structure (Zhang et al., 2019). By contrast, in gate-tuned antiferromagnetic or magnetic-semiconductor proposals, wide but finite bias windows are important, and the robustness of CAR depends on alignment between band edges and superconducting pairing channels (Jakobsen et al., 2021, Naveen et al., 15 May 2026).

Several design principles recur across the literature. First, CAR is favored when the opposite lead supports hole transmission while the corresponding transmitted electron channel is absent or strongly mismatched. This appears in ferromagnets, antiferromagnets, altermagnets, bipolar magnetic semiconductors, and quantum Hall single-channel regimes (Soori, 2022, Das et al., 2024, Jakobsen et al., 2021, Naveen et al., 15 May 2026, Zhang et al., 2019). Second, coherent subgap propagation rather than simple interface transparency is often decisive. Fabry–Pérot interference, bound-state hybridization, and ABS-mediated transport are recurring mechanisms (Soori, 2022, Soori et al., 2016, Bordin et al., 2022, Feng et al., 2024). Third, nonlocal conductance alone is sometimes insufficient; spatial separation, shot noise, and disorder response can be essential discriminants (Reinthaler et al., 2012, Marga et al., 29 Oct 2025, Wu et al., 2015).

Taken together, the literature presents CAR not as a single transport anomaly but as a family of nonlocal electron–hole conversion phenomena whose observability depends on how effectively a device engineers channel selectivity, phase coherence, and the suppression or reinterpretation of competing processes. This suggests that progress in CAR research will continue to come less from any universal material choice than from architectures that make the nonlocal hole channel uniquely available or uniquely diagnosable.