Spin-Resolved Josephson Diode Effect

• Spin-resolved Josephson diode effect is a nonreciprocal superconducting phenomenon where asymmetric current-phase relations emerge from broken inversion and time-reversal symmetries.
• It is achieved through mechanisms such as noncoplanar spin textures, spin-orbit coupling, and finite-momentum pairing that generate distinct spin and charge supercurrents.
• Experimental platforms using tailored ferromagnet/superconductor junctions report tunable diode efficiencies, with charge rectification up to 33-41% and spin rectification reaching 100%.

The spin-resolved Josephson diode effect denotes a class of nonreciprocal Josephson phenomena in which the critical supercurrent depends on direction and the nonreciprocity can be resolved into spin channels, equal-spin triplet sectors, or a spin supercurrent. In a conventional Josephson junction the current-phase relation is reciprocal, $I = I_c \sin\phi$, so that $I_{c+} = I_{c-}$. Spin-resolved diode behavior appears when the current-phase relation no longer has the antisymmetry or phase-inversion structure that enforces reciprocity, typically because inversion symmetry and time-reversal symmetry are broken by noncoplanar spin textures, spin-orbit coupling, exchange fields, finite-momentum pairing, or current-induced spin accumulation. Within this broader class, the literature includes quantum-geometric diode effects in strongly spin-polarized ferromagnets, conical and helimagnetic junctions, Rashba-based and field-effect devices, field-free multilayers, topological and Majorana-enhanced junctions, heavy-metal nanopillars exploiting the supercurrent spin Hall effect, and altermagnetic proximity structures (Schulz et al., 21 Jan 2025, Nikolić et al., 26 Dec 2025, Nikolić et al., 7 Aug 2025, Fu et al., 2022, Nayak et al., 16 May 2026, Esin et al., 9 Feb 2026).

1. Formal definition and current-phase structure

In the spin-resolved formulation, the Josephson transport problem is decomposed into equal-spin channels. A representative description writes

$$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$

$$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$

with

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$

so that the superconducting phase bias $\Delta\chi$ and the spin-geometric phase $\Delta\varphi$ enter on comparable footing. In this convention the charge and spin supercurrents are

$$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$

and the diode efficiencies are

$$\eta_x = \frac{\left| I_x^+ \right| - \left| I_x^- \right|}{\left| I_x^+ \right| + \left| I_x^- \right|}, \quad x=ch,sp.$$

This framework makes the spin-resolved content explicit: nonreciprocity can exist in charge current, spin current, or both, depending on which harmonics and spin-transfer processes are present (Schulz et al., 20 Jan 2025).

A central structural criterion is the absence of a phase-inversion center in the current-phase relation. In the phenomenological formulation,

$$I(\Delta\chi) = \sum_{n\geq 0}\left[I_n \sin (n\Delta\chi) + J_n \cos (n\Delta\chi)\right],$$

a junction may break inversion and time-reversal symmetry yet still remain reciprocal if a shifted antisymmetry survives, $I_{c+} = I_{c-}$0. The spin-resolved Josephson diode effect therefore requires more than a $I_{c+} = I_{c-}$1-shifted sinusoid: the current-phase relation must genuinely lack any phase-inversion center (Nikolić et al., 26 Dec 2025).

This distinction is not merely formal. A purely shifted first harmonic describes an anomalous Josephson effect, whereas directional critical-current asymmetry requires higher-order structure or inequivalent spin-channel contributions. This is why the modern literature emphasizes harmonic analysis, multichannel equal-spin transfer, and symmetry constraints rather than only the existence of an anomalous phase shift (Shin et al., 2024, Nikolić et al., 26 Dec 2025).

2. Quantum-geometric origin in strongly spin-polarized hybrids

A particularly systematic route to the spin-resolved Josephson diode effect arises in junctions containing a strongly spin-polarized ferromagnet between two singlet superconductors, coupled through spin-active interfaces. The decisive ingredient is a noncoplanar spin texture: when the magnetizations of the ferromagnetic region and the two interfaces are mutually noncoplanar, they generate a quantum-geometric phase that enters the Josephson relation analogously to the superconducting phase difference. In the spin-resolved channel description, the $I_{c+} = I_{c-}$2 and $I_{c+} = I_{c-}$3 pairs acquire opposite phase shifts, which immediately creates the possibility of charge-current and spin-current rectification (Schulz et al., 21 Jan 2025).

The necessary conditions in this framework have been stated explicitly. First, noncoplanarity is required to produce a finite geometric phase and to break spatial inversion symmetry. Second, both spin bands must contribute to transport, so the effect is absent in half-metallic junctions. Third, strong spin polarization is required, in practice meaning different band-specific densities of states for the two spin bands. Fourth, higher harmonics in the current-phase relation are necessary, so the effect is absent in the tunneling limit. Finally, even when all of these hold, the geometric phase must avoid the special values $I_{c+} = I_{c-}$4, $I_{c+} = I_{c-}$5, because those values restore a phase-inversion center (Nikolić et al., 26 Dec 2025).

These conditions are compactly represented by a minimal Josephson energy,

$I_{c+} = I_{c-}$6

Here $I_{c+} = I_{c-}$7 and $I_{c+} = I_{c-}$8 describe transport of a single majority-spin or minority-spin equal-spin pair, while $I_{c+} = I_{c-}$9 is a crossed-pair process. The model makes transparent why neither a half-metal nor a weakly polarized ferromagnet suffices, and why crossed-pair or higher-harmonic processes are indispensable for genuine diode behavior (Nikolić et al., 26 Dec 2025).

Within the diffusive quasiclassical theory of these sFM hybrids, the charge Josephson diode efficiency exceeds $$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$0 and the spin-diode efficiency reaches $$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$1. The harmonic analysis identifies crossed-pair processes as especially important for the charge diode effect, while a distinct $$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$2 contribution enables ideal spin rectification. The same framework also yields a SQUID geometry in which reversing the magnetic flux switches between nearly pure spin-up and spin-down equal-spin supercurrents across the ferromagnet (Schulz et al., 21 Jan 2025, Schulz et al., 20 Jan 2025).

3. Noncoplanar magnets, helimagnets, and frustrated spin textures

Conical and helimagnetic magnets provide an intrinsic realization of the same general principle: noncoplanar spin textures imprint a spin-dependent geometric phase on equal-spin triplet pairs. In strongly spin-polarized conical magnets, the adiabatic spin gauge phase is

$$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$3

for the case $$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$4 and $$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$5. The spin-resolved current-phase relation then takes the form

$$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$6

with opposite signs for the two spin channels. In this setting, strong spin polarization and a helical pitch comparable to the superconducting coherence length are essential, and the reported diode efficiency exceeds $$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$7, up to approximately $$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$8, in optimal parameter regimes (Nikolić et al., 7 Aug 2025).

Related work on a single conical magnetic material attributes the diode effect to the combination of helical spin rotation and spin canting. The helical rotation generates a Rashba-like band splitting inversely proportional to the rotation period,

$$I_{\uparrow\uparrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \mu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$9

while out-of-plane canting contributes a Zeeman-like splitting $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$0. Efficient nonreciprocity is found near the $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$1–$$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$2 transition, where the second harmonic in

$$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$3

becomes significant (Kamra et al., 2024).

In one-dimensional superconductor/helimagnet/superconductor junctions, the Josephson diode effect can arise without spin-orbit coupling. For spin-singlet $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$4-wave superconductors, the necessary and sufficient conditions are a conical magnetic configuration and nonzero chemical potential. For spin-triplet $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$5-wave superconductors, the conical configuration remains necessary and sufficient, but the chemical potential is no longer necessary because equal-spin Cooper pair-mediated transport can sustain the effect even at $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$6. The efficiency is strongly dependent on chirality, tilt angle, exchange coupling, and chemical potential, and values close to $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$7 occur for specific parameter choices; the sign can be switched by reversing chirality or tuning the other magnetic parameters (Cheng et al., 28 May 2026).

A further extension replaces spin-orbit-driven anisotropy by frustrated magnetic structure. In triplet superconductors coupled through frustrated spin textures, the Josephson free energy acquires Heisenberg-like, Dzyaloshinskii-Moriya-like, and $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$8-type couplings between $$I_{\downarrow\downarrow} = \frac{1}{2}\sum_{\mu,\nu=-\infty}^{\infty} \nu (-1)^{\mu+\nu} I_{\mu,\nu} \sin\psi_{\mu,\nu},$$9-vectors,

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$0

and the diode effect originates either from nonvanishing spin chirality in the barrier or from antisymmetric Josephson coupling between noncollinear $$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$1-vectors, both of which break inversion and time-reversal symmetries (Frazier et al., 29 Oct 2025).

4. Spin-orbit, finite-momentum, and topological routes

A separate but closely related family of mechanisms is built on spin-momentum locking, finite center-of-mass momentum of Cooper pairs, and anomalous higher-harmonic current-phase relations. In asymmetric spin-momentum-locking states, the effective normal-state Hamiltonian

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$2

yields a finite pairing momentum

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$3

a Doppler shift

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$4

and direction-dependent gaps

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$5

Because the decisive combination is $$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$6, gate voltage can play the same role as a Zeeman field in generating finite-momentum pairing, enabling a field-effect Josephson diode based on purely electrical control. The topological implementation in time-reversal-broken quantum spin Hall systems is predicted to exhibit strongly enhanced efficiency, including a two-edge effect up to $$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$7, while semiconductor implementations are reduced by backscattering (Fu et al., 2022).

In planar Al-InAs Josephson junctions with in-plane magnetic field, the nonreciprocal current-phase relation is written as

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$8

with diode efficiency

$$\psi_{\mu,\nu} = (\mu+\nu)\Delta\chi - (\mu-\nu)\Delta\varphi,$$9

Here finite Cooper-pair momentum $\Delta\chi$0 originates from the orbital effect of the in-plane field, while the interplay of Rashba and Dresselhaus spin-orbit couplings controls the harmonic content. Polarity reversal occurs when $\Delta\chi$1 crosses $\Delta\chi$2, and the reversal field as well as the magnitude of the effect are tuned by the local electric field through Rashba coupling (Shin et al., 2024).

Quantum-dot junctions provide a more localized version of the same symmetry logic. In a Rashba-coupled S/QD/S junction, Rashba interaction appears as a spin-dependent phase $\Delta\chi$3 in the tunneling amplitudes, the Zeeman field lifts spin degeneracy, and the combination of the two produces asymmetric critical currents. The rectification coefficient

$\Delta\chi$4

can be tuned by gate voltage to as high as $\Delta\chi$5 for realistic Rashba strength in the presence of magnetic field and chirality (Debnath et al., 2024).

Topological superconductivity introduces an additional enhancement channel. In Rashba nanowire Josephson junctions, a Zeeman component parallel to the spin-orbit axis produces an Andreev spectrum asymmetric with respect to $\Delta\chi$6. This asymmetry exists already in the trivial phase but becomes strongly enhanced in the topological phase because of the spatial nonlocality of Majorana bound states, leading to a larger difference between the positive and negative critical currents (Cayao et al., 2023).

Field-free diode behavior can also be produced by combining static exchange fields with Rashba spin-orbit interaction in multilayer S/F/RM/F/S structures. In that regime the current is exclusively carried by spin-triplet Cooper pairs because spin-singlet components are strongly suppressed by the ferromagnet, and the efficiency can be enhanced by tuning the thickness of the normal metal and the strength of the Rashba spin-orbit interaction (Hikino, 28 Jul 2025).

5. Experimental platforms and spin-origin diagnostics

Experimental work has concentrated on charge-current nonreciprocity with explicit diagnostics of spin origin. In Nb-Pt-Nb vertical nanopillars, nonreciprocity is generated by the supercurrent spin Hall effect in the Pt barrier. A bias supercurrent induces a non-equilibrium spin segregation, interpreted as partial conversion of singlet Cooper pairs into odd-parity spin-triplet pairs with opposite $\Delta\chi$7 projections that accumulate at opposite sides of the Pt layer. The induced spin moment $\Delta\chi$8 acts as an internal, bias-reversible pseudo-magnetic field, and in an external in-plane field the total phase is written as

$\Delta\chi$9

leading to

$\Delta\varphi$0

These devices exhibited field-tunable Josephson diode efficiencies as high as $\Delta\varphi$1, measured above liquid Helium temperature, with a non-monotonic Pt-thickness dependence, $\Delta\varphi$2–$\Delta\varphi$3-like crossover behavior, second-harmonic current near the crossover, and a spin-valve-like effect in Nb-Ni-Pt-Nb structures that changes resistance when the current-induced $\Delta\varphi$4 is reversed relative to Ni magnetization (Nayak et al., 16 May 2026).

In InAs nanosheet-based Josephson junctions with Al contacts, the diode effect is controlled by the orientation of the in-plane magnetic field and by electrostatic gating. The strength of the effect reaches its maximum when $\Delta\varphi$5 and drops to nearly zero when $\Delta\varphi$6, consistent with a Rashba spin-orbit mechanism in which finite-momentum pairing is generated when the Zeeman field aligns with the effective spin-orbit field. Gate voltage suppresses the effect completely at certain values while the supercurrent remains sizable, which was taken as evidence that the diode effect follows the tuning of Rashba spin-orbit interaction rather than trivial interface degradation (Yan et al., 26 Jan 2025).

In In-CrSb and In-CrSb-In proximity devices on the altermagnet CrSb, two complementary signatures were reported. First, the $\Delta\varphi$7 curves for opposite magnetic-field sweep directions are mirrored with respect to zero field, which is characteristic behavior of a Josephson spin valve. Second, direct critical-current measurements show $\Delta\varphi$8, establishing a Josephson diode effect in external field. The interpretation is a joint effect of spin-polarized topological surface states and altermagnetic spin splitting in CrSb. In single In-CrSb interfaces, the superconducting gap oscillates with magnetic field for both field orientations, resembling a transition into the Fulde-Ferrell-Larkin-Ovchinnikov state and thereby linking the diode effect to finite-momentum Cooper pairing in a spin-split environment (Esin et al., 9 Feb 2026).

Platform	Spin-resolved ingredient	Reported result
Nb-Pt-Nb nanopillar	supercurrent spin Hall effect and induced $\Delta\varphi$9 in Pt	diode efficiency as high as $$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$0 above liquid Helium temperature
InAs nanosheet-Al junction	Rashba SOI with in-plane field orientation control	maximum for $$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$1, nearly zero for $$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$2
In-CrSb-In junction	spin-polarized topological surface states plus altermagnetic spin splitting	mirrored $$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$3 and direct critical-current asymmetry

These experiments do not all measure spin current directly, but they do isolate spin-related mechanisms through field-angle selection rules, spin-valve response, current-induced spin segregation, and material-specific spin-split band structure. This distinction is important for interpreting present evidence: spin-resolved theory is broader than presently realized experimental observables, yet the experimental signatures are designed to identify the spin origin of charge-current nonreciprocity (Nayak et al., 16 May 2026, Yan et al., 26 Jan 2025, Esin et al., 9 Feb 2026).

6. Necessary conditions, conceptual boundaries, and open directions

Several conceptual boundaries recur across the literature. Coplanar spin textures do not generate the quantum-geometric phase required in the noncoplanar sFM frameworks, so the diode effect vanishes there. Half-metallic transport also fails to produce a charge diode effect in those frameworks because the current-phase relation reduces to a shifted sinusoid with a phase-inversion center. Likewise, in the tunneling limit, where only the first harmonic survives, one obtains at most a $$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$4- or $$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$5-junction rather than directional critical-current asymmetry (Schulz et al., 20 Jan 2025, Nikolić et al., 26 Dec 2025).

A frequent misconception is that any anomalous phase shift implies diode behavior. The cited theories explicitly distinguish these phenomena: a first-harmonic phase shift can break $$I_{ch} = I_{\uparrow\uparrow} + I_{\downarrow\downarrow}, \qquad I_{sp} = I_{\uparrow\uparrow} - I_{\downarrow\downarrow},$$6 about the origin while preserving an equivalent shifted antisymmetry, and therefore still produce reciprocal critical currents. Genuine Josephson-diode behavior requires either higher harmonics, inequivalent spin-channel amplitudes, or both (Shin et al., 2024, Nikolić et al., 26 Dec 2025).

Another boundary concerns the role of spin-orbit coupling. Several influential proposals rely on Rashba or Dresselhaus coupling, yet conical magnets, helimagnets, and frustrated spin textures demonstrate that spin-orbit coupling is not universally required. Conversely, spin-orbit coupling can be sufficient when combined with Zeeman or exchange fields, finite-momentum pairing, or asymmetric spin-momentum locking. The field-free multilayer proposals underscore this point by showing that a static exchange field plus Rashba spin-orbit interaction can produce a diode current carried exclusively by spin-triplet Cooper pairs even in the absence of an external magnetic field (Cheng et al., 28 May 2026, Hikino, 28 Jul 2025).

The current status of the field is therefore stratified. The strongest claims of perfect spin-diode behavior and flux-controlled switching between nearly pure spin-up and spin-down equal-spin supercurrents remain theoretical results in strongly spin-polarized, noncoplanar hybrid structures (Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025). Experimental works in the cited set instead establish charge-current nonreciprocity with increasingly specific spin-origin diagnostics, including supercurrent-induced spin segregation, Rashba-vector selectivity, Josephson spin-valve behavior, and FFLO-like gap oscillations (Nayak et al., 16 May 2026, Yan et al., 26 Jan 2025, Esin et al., 9 Feb 2026). A plausible implication is that progress toward direct spin-supercurrent rectification will depend on combining the explicit spin-channel control of the quantum-geometric proposals with the higher-temperature and materials-scalable platforms demonstrated in heavy-metal, semiconductor, and altermagnetic junctions.