Charge & Spin Josephson Diode Effects

• Charge and spin Josephson diode effects are superconducting phenomena featuring unequal forward and reverse critical currents due to broken inversion and time-reversal symmetry.
• They are realized in various systems such as Ising superconductors, S–F–S junctions with conical magnets, and Rashba nanowires, where anomalous current-phase relations with higher harmonics and geometric phases emerge.
• Quantification shows up to 40% charge efficiency and perfect (100%) spin rectification, offering promising applications in superconducting spintronics and low-power device technologies.

Charge and Spin Josephson Diode Effects

The charge and spin Josephson diode effects denote the occurrence of nonreciprocal superconducting critical currents—where the critical current for forward and reverse bias are unequal—and, in the case of spin, nonreciprocal transport of spin supercurrents in Josephson junctions. These phenomena result from concomitant breaking of inversion and time-reversal symmetries, often realized via spin-orbit coupling, magnetic textures, and/or noncoplanar magnetization configurations. Central to the theoretical description is a generalized current-phase relation (CPR) that incorporates additional geometric or magnetic phases, higher harmonics, and band-selective effects, enabling large diode efficiencies in both charge and spin sectors and robust operation even in diffusive, disordered systems.

1. Microscopic Mechanisms and Symmetry Requirements

Charge and spin Josephson diode effects require the joint breaking of spatial inversion and time-reversal symmetry. This is typically achieved through combinations of spin-orbit coupling, exchange fields, and noncoplanar spin structures. In strongly spin-polarized Josephson junctions, the CPR acquires dependence not only on the conventional superconducting phase difference $\Delta\chi$, but also on a quantum-geometric phase $\Delta\varphi$ associated with noncoplanarity of the local magnetization or interfacial spin axes.

The general harmonic decomposition for spin-resolved charge and spin Josephson currents in a spin-polarized superconductor–ferromagnet–superconductor (S–F–S) system can be written as: $$I_{\uparrow\uparrow}=\tfrac{1}{2}\sum_{\mu,\nu}\mu\,I_{\mu,\nu}\sin[(\mu+\nu)\Delta\chi-(\mu-\nu)\Delta\varphi],$$

$$I_{\downarrow\downarrow}=\tfrac{1}{2}\sum_{\mu,\nu}\nu\,I_{\mu,\nu}\sin[(\mu+\nu)\Delta\chi-(\mu-\nu)\Delta\varphi].$$

The total charge and spin currents are: $$I_{\rm ch}=e\,(I_{\uparrow\uparrow}+I_{\downarrow\downarrow}),\qquad I_{\rm sp}=\tfrac{\hbar}{2}\,(I_{\uparrow\uparrow}-I_{\downarrow\downarrow}).$$ Nonreciprocity arises when this CPR lacks a phase inversion center, i.e., when inversion symmetry is broken uniquely by sufficient noncoplanarity (geometric phase $\Delta\varphi\neq k\pi/2$), two-band transport is active (both spin bands contribute), band densities of states are unequal (strong spin polarization), and higher harmonics are present (Nikolić et al., 26 Dec 2025, Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025).

2. Model Systems: Junction Types and Theoretical Frameworks

A variety of junction architectures and material systems exhibit charge and spin Josephson diode effects:

• Ising superconductor junctions: Monolayer transition metal dichalcogenides with strong Ising spin-orbit coupling, proximitized by ferromagnets, support noncollinear in-plane exchange fields, yielding large spin-split Andreev bound states and non-sinusoidal CPRs, with diode efficiency tunable by field orientation and magnitude (Patil et al., 2024).
• S–F–S with conical or chiral magnets: Strongly spin-polarized ferromagnets with conical or helical magnetization (e.g., Ho, Dy, Cr$_{1/3}$NbS$_2$) facilitate geometric phases through spin precession. S–F–S junctions with spin-active insulating barriers (misaligned magnetizations at the two interfaces) directly realize the full two-phase CPR structure (Nikolić et al., 7 Aug 2025, Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025, Nikolić et al., 26 Dec 2025, Beach et al., 1 Dec 2025).
• Ballistic Rashba nanowire junctions: Single-channel nanowires with strong Rashba SOC and Zeeman fields support charge and spin diode effects via different Fermi velocities and phase shifts in pseudo-spin bands (Meyer et al., 2024).
• Diffusive/Disordered junctions: The diode effect survives well into the diffusive regime, where quasiclassical Usadel theory with spin-mixing boundary conditions and quantum-geometric phases accurately reproduces the phenomenon (Schulz et al., 21 Jan 2025, Schulz et al., 20 Jan 2025).
• Josephson junctions with Rashba regions under bias: Tuning Fermi momentum shift via a bias current in the presence of in-plane magnetic fields and Rashba SOC offers a direct handle on diode efficiency, which can be optimized by junction length and band structure (Mori et al., 10 Jan 2025).

3. Current-Phase Relations and Harmonic Structure

The Josephson diode effects are manifest as nonreciprocal critical currents due to the interplay of higher-order harmonics and phase shifts in the CPR. The generic expression is modeled as: $I(\phi) = I_1 \sin\phi + I_2 \sin 2\phi + ...,$ or equivalently, as an anomalous-phase-shifted form,

$$I(\phi) = I_c \sin(\phi - \phi_0) + I_d \sin 2\phi.$$

In systems with spin-polarized triplet transport, the CPR becomes multivariate: $$I_{\rm ch}(\Delta\chi, \Delta\varphi) = \sum_{n,m} I_{n,m} \sin(n\Delta\chi + m\Delta\varphi),$$ where $n$ and $m$ index the number of transferred Cooper pairs and geometric phase windings, respectively. The presence of harmonics with $|n|>1$ is essential for diode functionality.

In the minimal model including lowest-order transmission processes, the CPR immediately shows that diode efficiency vanishes unless higher harmonics ($I_{1,1}\neq 0$) and quantum-geometric phase ($\Delta\varphi \notin k\pi /2$) are present (Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025, Nikolić et al., 26 Dec 2025).

4. Quantification of Diode Efficiencies

Charge and spin diode efficiencies are defined as: $$\eta_{\rm ch} = \frac{I_{\rm ch}^+ - |I_{\rm ch}^-|}{I_{\rm ch}^+ + |I_{\rm ch}^-|}, \qquad \eta_{\rm sp} = \frac{|I_{\rm sp}^+| - |I_{\rm sp}^-|}{|I_{\rm sp}^+| + |I_{\rm sp}^-|},$$ where $I_{\rm ch}^\pm$ are the maximal/minimal charge critical currents as a function of $\Delta\chi$ (at fixed $\Delta\varphi$), and $I_{\rm sp}^{\pm}$ similarly for spin. Theoretical analysis and numerical simulations demonstrate the possibility of:

• $\eta_{\rm ch}$ up to $33\%$ for S–F–S systems with strong polarization and optimal geometric phase (Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025);
• $\eta_{\rm ch}$ up to $40\%$ in Ising superconductor hybrids and conical magnet junctions (Patil et al., 2024, Nikolić et al., 7 Aug 2025);
• $\eta_{\rm sp}$ reaching $100\%$, i.e., perfect rectification of spin supercurrent (Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025).

The optimal regime involves strong but not full spin polarization (avoiding half-metallicity), high-transparency interfaces, and maximized quantum-geometric phase. Disorder and moderate temperature do not appreciably suppress the effect as long as higher harmonics remain.

5. Experimental Realizations and Materials Platforms

Key experimental platforms exhibiting charge and spin Josephson diode effects include:

• Transition metal dichalcogenide heterostructures with Ising superconductivity and adjacent ferromagnets, allowing large, disorder-robust diode efficiencies tunable via the relative angle of in-plane exchange fields (Patil et al., 2024).
• Chiral helimagnets (e.g., Cr$_{1/3}$NbS$_2$), which due to broken inversion symmetry and intrinsic spin chirality, show the diode effect at zero field, with nonreciprocal efficiencies up to $20\%$ observed. The role of pinned Abrikosov vortices in generating phase offsets is supported by asymmetric Fraunhofer patterns and corroborated by Green's function simulations (Beach et al., 1 Dec 2025).
• Strongly spin-polarized S–F–S junctions employing conical magnets (Ho, Dy), where the diode effect arises from quantum spin-geometric phases accumulated by equal-spin triplet pairs; maximal efficiencies are obtained when the conical angle and helical pitch are tuned relative to the coherence length (Nikolić et al., 7 Aug 2025, Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025).
• Ballistic Rashba nanowires under Zeeman fields, accessible in InSb and InAs heterostructures (Meyer et al., 2024), and 1D Rashba junctions with tunable diode action via bias and interelectrode separation (Mori et al., 10 Jan 2025).

Measurement of spin diode effects requires, in addition to standard transport techniques, either spin-sensitive detection of the Josephson current or flux-controlled SQUID geometries for readout and switching (Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025).

6. Role of Quantum-Geometric Phases and Equal-Spin Triplet Transport

Quantum-geometric phases induced by noncoplanar magnetization configurations or conical magnetic texture act as effective secondary phase variables in the CPR. Equal-spin triplet Cooper pairs are the natural carriers of supercurrent in these strongly spin-polarized environments. Crossed-pair transmission processes—simultaneous transfer of both $\uparrow\uparrow$ and $\downarrow\downarrow$ pairs—are essential for achieving large diode efficiencies, especially in the spin channel. These features afford robust diode action even in disordered (diffusive) junctions and underlie the strongly nonreciprocal current-phase relations (Nikolić et al., 7 Aug 2025, Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025).

7. Applications and Outlook

Realization of charge and spin Josephson diode effects opens new pathways for superconducting electronics with rectification functionality, spin-polarized supercurrent control, and nonvolatile memory elements. The possibility of perfect spin-current rectification and SQUID-based switching between nearly pure spin-up and spin-down supercurrents provide a route towards low-dissipation superconducting spintronics. The robust theoretical basis and applicability to a wide family of material systems, including van der Waals heterostructures, chiral magnets, and engineered Rashba interfaces, suggest broad relevance for future quantum and superconducting device technologies (Schulz et al., 20 Jan 2025, Schulz et al., 21 Jan 2025, Patil et al., 2024, Beach et al., 1 Dec 2025, Nikolić et al., 7 Aug 2025).

---

Summary Table: Representative Systems Exhibiting Charge and Spin Josephson Diode Effects

System Type	Key Symmetry Breaking	Max. $\eta_{\rm ch}$	Max. $\eta_{\rm sp}$	References
Ising SC junction	ISOC + in-plane $J$	$\sim 40\%$	$\sim 40\%$	(Patil et al., 2024)
Conical FM S–F–S	Noncoplanar spins	$> 40\%$	$> 40\%$	(Nikolić et al., 7 Aug 2025)
S–F–S w/ noncoplanar FI	Phase ($\Delta\varphi$)	$33\%$	$100\%$	(Schulz et al., 20 Jan 2025)
Chiral helimagnet JJ	Spin chirality, Berry $\mathcal{B}$	$20\%$	—	(Beach et al., 1 Dec 2025)
Ballistic Rashba nanowire	SOC + Zeeman	$O(10\%)$	$O(10\%)$	(Meyer et al., 2024)

In all cases, nonreciprocity in both charge and spin channels is fundamentally governed by broken inversion and time-reversal symmetries, presence of odd harmonics or anomalous phases in the CPR, and, in magnetic systems, quantum geometric phases arising from noncoplanar magnetization textures. These effects are maximized under conditions of strong spin polarization, optimal phase geometry, and high interface transparency, and remain robust against disorder and moderate temperature variations.