Spinful Josephson Junctions

• Spinful Josephson junctions are superconducting weak links featuring spin-active barriers that induce anomalous π states and tunable current-phase shifts.
• They integrate magnetic impurities, quantum dots, and strong spin-orbit coupling to generate long-range triplet supercurrents and controllable 0–π transitions.
• Theoretical models and experimental devices in these systems underpin applications in cryogenic memory, quantum computing, and advanced metrology.

Spinful Josephson junctions are superconducting weak links hosting localized or distributed electron spin degrees of freedom in the barrier region. In such junctions, phenomena such as $\pi$-junction behavior, spin-triplet supercurrents, phase-controlled magnetoanisotropy, and anomalous current-phase relations emerge as a result of the interplay between superconductivity, magnetic interactions, spin-orbit coupling, and quantum confinement. The spinful regime encompasses a wide range of experimentally realized and theoretically modeled systems, including junctions containing magnetic impurities, quantum dots, ferromagnetic multilayers, spin-filter barriers, nanowires with strong spin-orbit coupling, and emergent topological or correlated phases.

1. Fundamental Mechanisms in Spinful Josephson Junctions

The Josephson effect in spinful junctions is fundamentally altered by the presence of local or extended spin-polarized states. In canonical SIS junctions, the supercurrent is carried by coherent tunneling of spin-singlet Cooper pairs, with a ground-state energy $E_J(\phi) = -E_J \cos \phi$ and current-phase relation $I(\phi) = I_c \sin \phi$. Introducing spin via a magnetic impurity, quantum dot, or a ferromagnetic weak link modifies the subgap spectrum, often via the formation of Yu-Shiba-Rusinov (YSR) states or spinful Andreev bound states. The ensuing supercurrent can be reversed (the so-called $\pi$-junction regime) or modulated by the superconducting phase, spin configuration, or external fields. In quantum dots, the Zeeman splitting of doublet subgap states is renormalized by exchange coupling to the superconductors, giving rise to a phase-dependent "Knight shift" with both $\Gamma$-linear and $\Gamma^2 \cos \phi$ contributions to the energy splitting (Pavešić et al., 2022).

In S/F/S hybrid junctions, the proximity effect is mediated by singlet correlations oscillating and decaying in the ferromagnetic interlayers, with the possibility for 0–$\pi$ ground-state transitions depending on the accumulated exchange phase. Non-collinear magnetization or strong spin-orbit coupling can generate odd-frequency, long-range spin-triplet supercurrents capable of traversing ferromagnetic or spin-filter barriers (Alidoust et al., 2014, Bujnowski et al., 2019, Caruso et al., 2019).

2. Current-Phase Relations and $0$–$\pi$ Transitions

The current-phase relation (CPR) in spinful Josephson junctions reveals signatures of spin and orbital structure in the barrier. In YSR-dominated atomic-scale junctions, the tunneling supercurrent through a single magnetic impurity exhibits a sign change precisely at the quantum phase transition between screened and free-spin ground states, directly observed by STM in conductance and critical current measurements (Karan et al., 2021). Mathematically, the YSR channel's Josephson energy reverses sign at the critical coupling, leading to $$I(\phi) = (I_{\mathrm{BCS}} \pm I_{\mathrm{YSR}})\sin \phi$$, a constructive-to-destructive interference analogous to a $0$–$\pi$ transition.

For S/F/S and multilayer S/F/N/F/S junctions, the accumulated phase through ferromagnetic layers determines whether the ground state is a $0$- or $\pi$-junction (Gingrich et al., 2015), with controllable transitions realized via magnetic switching of spin valves or by varying ferromagnet thicknesses (1711.01681, Niedzielski et al., 2017). The microscopic CPR becomes $I(\phi) = I_c \sin(\phi + \phi_0)$, where the phase offset $\phi_0$ is determined by the spin-mixing angle, magnetic configuration, and, in some cases, spin-orbit coupling. In spin-filter junctions with high spin-filter efficiency and strong spin-mixing, incomplete $0$–$\pi$ transitions and non-monotonic temperature-dependence of $I_c(T)$ emerge as signatures of multi-channel triplet correlations (Caruso et al., 2019).

3. Spin-Orbit Coupling, Anomalous Josephson Effects, and Triplet Supercurrents

Spin-orbit coupling (SOC) crucially enriches the phenomenology of spinful Josephson junctions. It can induce spin splitting of Andreev bound states, anomalous ("$\varphi_0$") Josephson junction behavior (i.e., supercurrents at zero phase difference), and enable tunable $0$–$\pi$ transitions controlled by SOC strength or magnetization orientation (Bujnowski et al., 2019, Tzortzakakis et al., 2019). The interplay of SOC and Zeeman splitting gives rise to additional gaps in the lead spectrum, modifies multiple Andreev reflection subgap features in bias conductance, and allows extraction of SOC and $g$-factor parameters from MAR peak evolution (Kuiri et al., 14 Nov 2025, Woerkom et al., 2016).

Long-range spin-triplet supercurrents arise in S/F/S junctions with non-collinear magnetizations or in lateral geometries with strong SOC. In such cases, the Josephson current is solely carried by odd-frequency, equal-spin triplet pairs, leading to robustness against spin-dephasing and the possibility of full $0$–$\pi$ transitions by tuning magnetization or SOC parameters (Alidoust et al., 2014, Bujnowski et al., 2019). In hybrid systems with Ising superconductors and half-metallic barriers, the spin-triplet Josephson effect displays pronounced magnetoanisotropic behavior ("switch effect"), with the critical current vanishing for certain magnetization directions and exhibiting strong $0$–$\pi$ angular dependence (Cheng et al., 2020).

4. Device Architectures and Experimental Realizations

A wide array of spinful Josephson junction architectures have been realized:

• Atomic-scale $\pi$-junctions: STM-based measurement of Josephson critical current through single YSR impurities, enabling atomic-scale phase control (Karan et al., 2021).
• Magnetic spin-valve junctions: Engineered S/F/N/F/S stacks where parallel and antiparallel configurations of remanent magnetizations control the 0/$\pi$ state and the amplitude or sign of the critical current. Realizations with in-plane (Gingrich et al., 2015, 1711.01681, Niedzielski et al., 2017) or perpendicular magnetic anisotropy (Satchell et al., 2019) have demonstrated non-volatile, low-power switching for cryogenic memory.
• Quantum dot junctions: Superconductor–quantum dot–superconductor devices probing "impurity Knight shift," flux-tunable Zeeman splitting, and Zeeman-coupled Josephson energy, with direct implications for Andreev spin-qubit manipulation (Pavešić et al., 2022).
• Van der Waals heterostructures: NbSe$_2$/Cr$_2$Ge$_2$Te$_6$/NbSe$_2$ φ-junctions exhibiting doubly degenerate ground states due to domain-structured, Ising-locked magnetic barriers, enabling dissipationless phase memory (Idzuchi et al., 2020).
• Superconducting/magnetic insulator proximity devices: FI-S/N/FI-S structures with tunable spin-splitting and current-phase relation modulation, showing switchable or enhanced critical currents and nontrivial skewed CPRs (Strambini et al., 2014).

A summary table of common architectures and their key properties:

Device Type	Mechanism	Notable Features
S–YSR–S	YSR quantum phase transition	Atomic-scale $\pi$-state control (Karan et al., 2021)
S–F/N–F–S (spin valve)	Magnetic configuration-dependent CPR	Field-programmable 0/$\pi$ switch (Gingrich et al., 2015, 1711.01681)
QD Josephson junction	Exchange-renormalized Zeeman splitting	Flux-tunable g-factor, Knight shift (Pavešić et al., 2022)
S–FI–N–FI–S	Spin-splitting in S leads	Switchable/skewed CPR, field sensors (Strambini et al., 2014)
S–SOC–F–SOC–S	LRTC by SOC and exchange	Tunable 0/$\pi$ transitions (Bujnowski et al., 2019)
NbSe$_2$/MI/NbSe$_2$	Ising SOC and magnetic domains	φ$_0$-junction, phase battery (Idzuchi et al., 2020)

5. Applications in Spintronics, Quantum Computing, and Metrology

The phase control, spin sensitivity, and multi-level structure of spinful Josephson junctions underlie several applications:

• Cryogenic memory: Digital readout encoded in 0/$\pi$ phase states of spin-valve JJs, offering high-density, ultralow-power storage compatible with superconducting logic circuits. Recent realizations demonstrate robust, repeatable phase switching and unambiguous SQUID readout (1711.01681, Niedzielski et al., 2017, Satchell et al., 2019).
• Superconducting spintronics: Field-tunable coupling between spin and superconducting phase enables spin-selective supercurrents and control over spin-triplet transport, forming the basis for spin-based logic, phase shifters, and diode-like elements (Alidoust et al., 2014, Caruso et al., 2019).
• Quantum information: Integration of quantum dots or Majorana nanowires with superconducting islands enables flux- or field-tunable entanglement of spin and superconducting qubits, facilitates two-qubit gates via σ$_z\otimes\phi$ couplings, and supports phase-protected, parity-based readout for qubit arrays (Pavešić et al., 2022, Zazunov et al., 2018, Zazunov et al., 2017).
• Metrology and sensing: Demonstrated threshold sensitivity in FI–S/N/FI–S and atomic-scale YSR junctions enables local magnetometry and thermometry, as well as phase-biasing applications such as dissipationless phase batteries or classical/quantum ratchets (Karan et al., 2021, Strambini et al., 2014, Idzuchi et al., 2020).

6. Theoretical Frameworks and Computational Approaches

A unified understanding of spinful Josephson junctions leverages a range of techniques:

• Quasiclassical Green's function (Usadel, Eilenberger) formalism: Enables calculation of current, pair amplitudes, and switching criteria in diffusive S/F/S or FI/S systems, incorporating arbitrary magnetic textures, SOC, and interface transparency (Alidoust et al., 2014, Strambini et al., 2014, Bujnowski et al., 2019).
• Microscopic Bogoliubov–de Gennes models: Capture quantum dot, nanowire, and topological Majorana architectures, permitting direct computation of subgap spectra, CPR, and gate/B-field dependence (Pavešić et al., 2022, Woerkom et al., 2016, Zazunov et al., 2017, Zazunov et al., 2018).
• Numerical renormalization group (NRG): Applied to quantum dot Josephson junctions to access regimes beyond perturbation theory (nonperturbative φ-modulation of Zeeman splitting) (Pavešić et al., 2022).
• Transfer-matrix and S-matrix methods: Utilized for analytically tractable short-junction ABS spectra and symmetry-based classification of anomalous Josephson effects (Brydon et al., 2010, Tzortzakakis et al., 2019, Shen et al., 2019).

Central to all approaches is the identification and quantification of (1) spin-dependent phase accumulation, (2) odd-frequency and equal-spin triplet correlations, (3) spin-mixing and spin-filtering at interfaces, and (4) the interplay between superconducting phase, magnetization, and spin-orbit fields.

7. Emerging Directions and Open Questions

Recent studies highlight several future directions:

• Controlled generation and detection of long-range triplet supercurrents: Realizing robust spin-triplet transport via engineered non-collinear or spin-orbit-active interfaces, and quantifying dissipationless spin currents (Bujnowski et al., 2019, Alidoust et al., 2014).
• Topological and correlated spin Josephson effects: Exploring fractional periodicity, $\mathbb{Z}_4$ spin Josephson effects, and entanglement signatures in engineered spin chains and quantum simulators (Shen et al., 2019).
• Phase-coherent spintronic circuit elements: Integration of phase batteries, programmable logic, and spin-dependent diodes using van der Waals heterostructures, spin-filter barriers, and complex ferromagnetic/superconducting hybrids (Idzuchi et al., 2020).
• Hybridized Andreev/Majorana devices for quantum computing: Extension of gate-tunable Andreev/ABS qubits, parity-protected readout, and multi-terminal topological Josephson effects (Woerkom et al., 2016, Zazunov et al., 2018, Zazunov et al., 2017).

A systematic mapping of critical current, parity, and CPR as functions of magnetic configuration, SOC, and barrier properties remains a central challenge for the deployment of scalable spinful superconducting electronics.

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• "Impurity Knight shift in quantum dot Josephson junctions" (Pavešić et al., 2022)
• "Superconducting Quantum Interference at the Atomic Scale" (Karan et al., 2021)
• "Experimental demonstration of a Josephson magnetic memory cell with a programmable π-junction" (1711.01681)
• "Controllable 0-pi Josephson junctions containing a ferromagnetic spin valve" (Gingrich et al., 2015)
• "Spin Controlled Coexistence of 0 and π States in SFSFS Josephson Junctions" (Alidoust et al., 2014)
• "Switchable Josephson current in junctions with spin-orbit coupling" (Bujnowski et al., 2019)
• "Tuning of Magnetic Activity in Spin-Filter Josephson Junctions Towards Spin-Triplet Transport" (Caruso et al., 2019)
• "Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions" (Woerkom et al., 2016)
• "Impact of spin-orbit coupling and Zeeman interaction on the subharmonic gap structure due to multiple Andreev reflections in nanoscopic Josephson junctions" (Kuiri et al., 14 Nov 2025)
• "Van der Waals Heterostructure Magnetic Josephson Junction" (Idzuchi et al., 2020)
• "Spin-valve Josephson junctions for cryogenic memory" (Niedzielski et al., 2017)
• "Spin-valve Josephson junctions with perpendicular magnetic anisotropy for cryogenic memory" (Satchell et al., 2019)
• "Mesoscopic Josephson junctions with switchable current-phase relation" (Strambini et al., 2014)
• "Josephson effect in junctions of conventional and topological superconductors" (Zazunov et al., 2018)
• "Josephson effect in multiterminal topological junctions" (Zazunov et al., 2017)
• "Theory of topological spin Josephson junctions" (Shen et al., 2019)
• "Switch effect and $0$-$\pi$ transition in Ising superconductor Josephson junctions" (Cheng et al., 2020)
• "Spin Josephson effect with a single superconductor" (Brydon et al., 2010)
• "Josephson junctions with spin-orbit and spin-flip interactions" (Tzortzakakis et al., 2019)