Nonreciprocal Josephson Transport

• Nonreciprocal Josephson Transport is the phenomenon where superconducting junctions exhibit direction-dependent critical currents due to broken inversion and time-reversal symmetry.
• The mechanism relies on engineering asymmetric current-phase relations using Rashba spin-orbit coupling, magnetic fields, and noncentrosymmetric barrier structures.
• Recent experiments report diode efficiencies up to 90%, with implications for superconducting rectifiers, microwave routing, and quantum device applications.

Searching arXiv for the cited papers to ground the article in current literature. arXiv search: (Nayak et al., 16 May 2026) "Supercurrent spin Hall effect enabled nanopillar Josephson diodes" Nonreciprocal Josephson transport denotes superconducting transport regimes in which reversing current bias, exchanging active ports, or reversing propagation direction does not produce a response related by simple odd symmetry. In the two-terminal case its canonical manifestation is the Josephson diode effect, for which the forward and backward critical currents are unequal, $I_c^+ \neq |I_c^-|$, commonly quantified by $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$. Closely related phenomena include anomalous Josephson currents at zero phase bias, transverse Hall-like supercurrents, direction-dependent depinning of vortices, and nonreciprocal microwave transmission in extended junctions. Across recent work, the field has evolved from Rashba- and field-driven short junctions to intrinsically noncentrosymmetric barriers, multi-terminal phase networks, textured magnets, topological and helical channels, and explicitly nonequilibrium dynamical schemes (Baumgartner et al., 2021).

1. Symmetry conditions and operational definitions

The standard symmetry logic is that a Josephson diode requires the simultaneous breaking of inversion symmetry and time-reversal symmetry. In highly transparent InAs-based Josephson junctions, inversion breaking is supplied by Rashba spin-orbit coupling and time-reversal breaking by an in-plane magnetic field, which yields an asymmetric current-phase relation and unequal critical currents (Baumgartner et al., 2021). The same symmetry principle is formulated in vertical van der Waals junctions with Td-WTe$_2$, where the non-centrosymmetric crystal structure breaks inversion symmetry and an in-plane magnetic field breaks time reversal, producing an anomalous phase shift and supercurrent asymmetry (Kim et al., 2023). In heavy-metal nanopillars, inversion symmetry is argued to be broken naturally at the Nb/Pt interfaces, while an in-plane field breaks time reversal and converts a current-induced spin moment into a diode response (Nayak et al., 16 May 2026).

This two-symmetry rule is not exhausted by the Rashba-plus-field paradigm. Several platforms realize time-reversal breaking without an external magnetic field. In a graphene Josephson triode, a control supercurrent through one branch biases the common superconducting island and thereby breaks the time-reversal symmetry of the transport response, enabling field-free nonreciprocity with efficiencies above $90\%$ (Chiles et al., 2022). In a conical magnet, the helical spin rotation generates Rashba-like inversion breaking while the out-of-plane canting acts as a Zeeman-like time-reversal-breaking component, so a single magnetic material supplies both ingredients (Kamra et al., 2024). In planar $s$-wave/$d$-wave/$s$-wave junctions, asymmetric interface coupling can induce a spontaneous $\pm \pi/2$ phase difference and thus spontaneous time-reversal-symmetry breaking without a magnetic field, although the resulting $\phi_0$-junction state alone does not guarantee a diode effect (Guo et al., 2 Dec 2025).

Operational definitions also vary with geometry. In ordinary two-terminal junctions, nonreciprocity is usually identified from $I_c^+ \neq |I_c^-|$. In multi-terminal graphene devices this criterion is insufficient: a reciprocal device can appear asymmetric in a single-port measurement if an additional bias current is present. The appropriate quantity is then a two-port reciprocity relation defined through a critical-current tensor $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$0, with nonreciprocity meaning $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$1 rather than merely $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$2 (Zhang et al., 2023). In transverse Josephson transport, the asymmetry is instead between oppositely directed transverse critical currents, $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$3, quantified by a transverse diode quality factor $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$4 (Zeng, 2024).

2. Current-phase relations, anomalous phases, and harmonic content

The current-phase relation is the central organizing object of nonreciprocal Josephson transport. A purely sinusoidal CPR, $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$5, is reciprocal. A simple phase shift $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$6 is also insufficient by itself, because it can be absorbed into a redefinition of the phase origin and does not force unequal critical currents. What repeatedly produces diode behavior in the literature is the coexistence of a phase-shifted first harmonic with higher harmonics or other nontrivial anharmonic structure (Zhong et al., 13 Apr 2025).

In short, high-transparency junctions this structure commonly emerges from Andreev bound states. In InAs quantum-well junctions, cosine components enter the CPR when both inversion and time-reversal symmetries are broken, and in the highly transparent regime these cosine terms cannot simply be removed by a $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$7 shift, leading to a genuinely asymmetric CPR. Josephson inductance measurements were used there to connect the diode effect directly to CPR asymmetry and to extract the supercurrent magnetochiral anisotropy coefficient $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$8 (Baumgartner et al., 2021). In Ge-based SQUIDs embedding JoFETs, the single-junction CPR contains up to three harmonics with gate-tunable amplitude; the reported ratios reach $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$9 and $_2$0, and flux and gate tuning drive the device between a Josephson-diode regime with $_2$1 asymmetry and an effectively $_2$2-periodic $_2$3 regime associated with charge-$_2$4 transport (Leblanc et al., 2023).

Higher harmonics are also central in magnetically textured junctions. For a conical magnet, the numerically extracted CPR is written as

$_2$5

and a finite diode effect requires $_2$6. Near the $_2$7-$_2$8 transition, $_2$9 becomes small, $90\%$0 becomes comparatively important, and the diode efficiency becomes large (Kamra et al., 2024). An analogous crossover physics is invoked in Nb/Pt/Nb nanopillars, where the thickness dependence of the normalized transition temperature is fit by $90\%$1, and in a $90\%$2 nm Pt device a sudden change in the slope of $90\%$3 near $90\%$4 K is accompanied by a Fraunhofer-pattern bump interpreted as a second-harmonic Josephson contribution (Nayak et al., 16 May 2026).

Several papers sharpen the distinction between anomalous phase and true diode behavior. In planar $90\%$5-$90\%$6-$90\%$7 junctions, spontaneous time-reversal breaking can generate a $90\%$8-junction-like state, but diode transport appears only when single-Cooper-pair tunneling is symmetry-allowed and coexists with even-harmonic terms, giving a CPR of the form

$90\%$9

for which $s$0 and $s$1 are reported (Guo et al., 2 Dec 2025). In a topological diode based on a conventional $s$2-periodic channel in parallel with a $s$3-periodic Majorana channel,

$s$4

and it is precisely the coexistence of the $s$5 and $s$6 contributions that makes a Zeeman-induced phase offset unabsorbable and the critical current nonreciprocal (Zhong et al., 13 Apr 2025).

3. Microscopic routes in two-terminal junctions

One major route is the Rashba-plus-Zeeman mechanism. In Al/InAs junction arrays, strong Rashba SOC, a sizeable in-plane $s$7-factor, and high transparency produce short, ballistic SNS junctions whose diode response is strongest for transverse in-plane field and vanishes for field parallel to the current. The same work measured a finite-bias interval satisfying $s$8, directly identifying a superconducting diode regime (Baumgartner et al., 2021).

A distinct route replaces interfacial Rashba engineering with intrinsic SOC in the barrier. In Nb/Pt/Nb vertical nanopillars, first-principles calculations show that broken inversion symmetry at the Nb/Pt interfaces and strong atomic SOC in Pt split interfacial bands near the Fermi level by about $s$9–$d$0 meV and generate spin textures. The resulting singlet-triplet mixing is argued to produce a nonequilibrium spin accumulation in the Pt barrier via a supercurrent spin-Hall effect. Under an in-plane field, the field-induced phase $d$1 and the spin-moment phase $d$2 combine as

$d$3

with opposite signs for opposite current directions. Experimentally, the reported diode efficiency reaches about $d$4, strongest for $d$5–15 nm, with the best device operating around $d$6–$d$7 K and the asymmetry absent at zero field (Nayak et al., 16 May 2026). The paper explicitly distinguishes this induced spin moment from a normal quasiparticle spin current.

Vertical van der Waals junctions provide an intrinsic crystal-symmetry route. In NbSe$d$8/Td-WTe$d$9/NbSe$s$0 junctions, the non-centrosymmetric Td-WTe$s$1 crystal furnishes a polar axis and strong spin-orbit interaction. The diode efficiency obeys magneto-linear and magneto-chiral relations,

$s$2

and increases with barrier thickness up to a critical regime before deviating from linearity, interpreted as a ballistic-to-diffusive transition (Kim et al., 2023). Twist-angle engineering in a double-barrier structure further modifies the effective polar axis and hence the magneto-chirality.

Magnetic-texture mechanisms can encode both required symmetry breakings within a single barrier. In a conical magnet, the helical rotation produces quasi-1D Rashba-like splitting inversely proportional to the helix period $s$3, while the canting term $s$4 breaks time reversal and shifts the superconducting gap differently for opposite directions. The diode efficiency is largest near $s$5-$s$6 transitions and changes sign when either $s$7 or the helix rotation direction is reversed (Kamra et al., 2024). In antiferromagnetic planar junctions, by contrast, nonreciprocity arises from triplet Cooper diffusion modes that are odd under sublattice exchange; if tunneling into the two antiferromagnetic sublattices is different, sublattice-odd propagators survive and generate $s$8 terms so that $s$9 (Mal'shukov, 2024).

The microscopic origin need not lie in the equilibrium supercurrent branch. Atomic-scale Pb–Pb STM junctions are reciprocal when stabilized by a single Pb atom, but insertion of a single Cr or Mn atom produces a bias-direction-dependent retrapping current. The asymmetry is traced not to an intrinsically asymmetric CPR, but to asymmetric quasiparticle dissipation through Yu–Shiba–Rusinov states inside the gap. The dominant signature is retrapping-current asymmetry, with opposite signs for Cr and Mn, reproduced by an extended RCSJ model with asymmetric damping (Trahms et al., 2022). This establishes that nonreciprocal Josephson transport can be fundamentally dynamical and dissipative even in atomically small junctions.

4. Multiterminal, transverse, and network-based nonreciprocity

Multi-terminal geometries generalize nonreciprocity beyond the two-terminal critical-current asymmetry. In the graphene Josephson triode, three graphene weak-link junctions share a common superconducting island, so the phase drops are linked by current conservation and flux-free constraints. A control current through one branch reshapes the phase landscape seen by another branch, generating nonreciprocal supercurrent transport without external magnetic field. The control branch can remain in a dissipationless regime, and square-wave rectification is demonstrated for amplitudes down to $\pm \pi/2$0 nA, with efficiencies above $\pm \pi/2$1 (Chiles et al., 2022).

A related but conceptually distinct case is the three-terminal graphene multi-terminal Josephson junction treated as a two-port network. There, nonreciprocity is governed by broken spatial mirror symmetry between the active ports, not by the standard two-terminal diode criterion. In the asymmetric configuration, the measured critical currents differ substantially, with $\pm \pi/2$2 nA and $\pm \pi/2$3 nA, corresponding to $\pm \pi/2$4 at $\pm \pi/2$5 (Zhang et al., 2023). This is polarity-independent port nonreciprocity rather than the usual $\pm \pi/2$6 versus $\pm \pi/2$7 asymmetry.

Four-terminal geometries allow simultaneous longitudinal and transverse Josephson responses. In a square-lattice junction with a spin-orbit-coupled central region, four superconducting leads, and an in-plane Zeeman field $\pm \pi/2$8, breaking $\pm \pi/2$9 generates longitudinal anomalous Josephson effect and diode effect, while breaking $\phi_0$0 generates transverse current, transverse AJE, and transverse JDE. For $\phi_0$1, $\phi_0$2, $\phi_0$3, $\phi_0$4, and $\phi_0$5, the transverse CPR is reported to be unidirectional over the range studied, and for $\phi_0$6 at $\phi_0$7 both transverse critical currents keep the same sign (Sahoo et al., 25 Mar 2025).

The transverse Josephson diode effect can also arise in tilted Dirac materials with valley-dependent gaps. In that proposal, a finite transverse Josephson Hall current requires both a finite tilt velocity $\phi_0$8 and broken time-reversal symmetry quantified by $\phi_0$9. The longitudinal current remains reciprocal, $I_c^+ \neq |I_c^-|$0, whereas the transverse current is nonsinusoidal, can remain finite at $I_c^+ \neq |I_c^-|$1, and admits a quality factor

$I_c^+ \neq |I_c^-|$2

By tuning $I_c^+ \neq |I_c^-|$3, one transverse critical current can be driven to zero, yielding $I_c^+ \neq |I_c^-|$4, i.e. a fully polarized $I_c^+ \neq |I_c^-|$5 transverse diode effect with decoupled input and output paths (Zeng, 2024).

A broader implication is that nonreciprocal Josephson transport is not confined to collinear source-drain geometries. It can be encoded in port assignment, transverse conversion, or network topology, and in those settings reciprocity must be defined at the level of the full multi-port response rather than by a single pair of switching currents.

5. Nonequilibrium, voltage-biased, and collective dynamical regimes

Nonreciprocal Josephson transport extends beyond equilibrium CPR asymmetry. In single-channel junctions between helical superconductors with finite Cooper-pair momentum $I_c^+ \neq |I_c^-|$6, the equilibrium current-biased diode efficiency reaches a model maximum $I_c^+ \neq |I_c^-|$7, but under voltage bias the situation changes qualitatively. Multiple Andreev reflections acquire Doppler-shifted thresholds at $I_c^+ \neq |I_c^-|$8, where $I_c^+ \neq |I_c^-|$9, and in the ballistic low-voltage limit the rectification efficiency can reach the ideal value $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$00 (Zazunov et al., 2023). This sharp contrast shows that nonequilibrium transport can be more strongly rectifying than the equilibrium CPR.

The inverse viewpoint is developed in the topological Josephson diode with coexisting $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$01- and $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$02-periodic channels. There a pure ac current drive $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$03 generates a finite time-averaged dc voltage,

$\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$04

whenever the ac amplitude lies between the two directional critical currents. The dc voltage is thus a rectified output enabled by the nonreciprocal thresholds themselves, rather than an independent effect (Zhong et al., 13 Apr 2025).

Extended junctions furnish a wave-transport form of nonreciprocity. In long Josephson junctions biased into the flux-flow regime, a moving chain of fluxons acts as a traveling nonlinear medium for microwaves. Analytical and numerical work based on the perturbed sine-Gordon equation shows that the largest attenuation occurs when the flux-flow direction is opposite to the microwave propagation direction, and that the effect is enhanced by increasing junction length and by impedance matching at the junction ends (Pankratov et al., 2014). An annular version converts this principle into circulation: in an extended Josephson junction ring, a DC-driven moving fluxon train splits counter-propagating microwave resonances, enabling near-ideal three-port circulation around $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$05 GHz with a reported bandwidth of about $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$06 MHz at the optimal bias point (Le et al., 1 Dec 2025).

Collective vortex dynamics in arrays provides yet another regime. In two-dimensional arrays of multiterminal $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$07-junctions, next-nearest-neighbor couplings make anomalous phase shifts ungaugeable and generate spontaneous supercurrent loops. These loops distort the vortex pinning landscape into a ratchet-like potential, producing nonreciprocal vortex depinning currents with $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$08 reaching values up to $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$09 in the dilute-vortex regime. Commensurate frustrations such as $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$10, $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$11, $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$12, and $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$13 enhance the effect, and a sign reversal near $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$14 is reported as an open collective-physics feature (Reinhardt et al., 2024).

A purely dynamical route dispenses with material asymmetry altogether. In the Kapitza supercurrent diode, parametric modulation of a conventional tunnel-junction supercurrent amplitude generates an effective second harmonic in the averaged CPR,

$\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$15

and thereby a diode response in otherwise reciprocal junctions. Two implementations are proposed, based on gate-controlled interferometers or flux-driven double-loop SQUIDs, with experimentally accessible drive frequencies $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$16–$\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$17 and numerical examples showing diode efficiencies up to about $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$18 (Yerin et al., 27 Feb 2026). This suggests that nonequilibrium driving can serve as a design principle alongside SOC and magnetism.

6. Conceptual distinctions, misconceptions, and emerging directions

Several recurrent misconceptions are explicitly addressed in the literature. First, a $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$19-junction or anomalous Josephson effect is not automatically a diode. A finite current at zero phase, or a mere phase shift of the first harmonic, does not by itself imply unequal forward and backward critical currents; additional anharmonicity or symmetry reduction is required (Guo et al., 2 Dec 2025). Second, in multi-terminal structures a single-port asymmetry can be misleading: the proper notion of reciprocity involves exchange of source and probe ports, not only reversal of current polarity (Zhang et al., 2023). Third, not every observed asymmetry originates in the supercurrent branch. Atomic-scale magnetic-atom junctions show that asymmetric quasiparticle damping through YSR states can dominate the observed nonreciprocity, especially in retrapping rather than switching currents (Trahms et al., 2022).

A related point concerns the microscopic origin of “field-free” diode effects. Field-free operation does not imply a universal mechanism. In graphene triodes, time-reversal breaking is generated by a control supercurrent (Chiles et al., 2022); in conical magnets it is supplied by the noncoplanar magnetic texture itself (Kamra et al., 2024); in MATBG it is attributed to valley polarization together with trigonal warping, which produces finite anomalous current at $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$20 and makes the valley-polarization potential $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$21 a flux-free control parameter (Soufy et al., 26 May 2026); in gyrotropic S/N/FI structures the relevant ingredients are a polar axis, FI-induced time-reversal breaking, and spin Hall conversion, which together generate a $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$22 shift and, in a related trilayer geometry, a diode effect (Kokkeler et al., 2023). These are structurally different realizations of the same broad nonreciprocal phenomenology.

The practical trajectory of the field is toward stronger rectification, higher operating temperature, and broader functionality. Heavy-metal Nb/Pt/Nb nanopillars report about $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$23 field-tunable efficiency above liquid-helium temperature (Nayak et al., 16 May 2026); Ge-based hybrid SQUIDs achieve $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$24 asymmetry while remaining gate- and flux-tunable and also access a $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$25-periodic charge-$\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$26 regime (Leblanc et al., 2023); graphene triodes exceed $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$27 at zero field (Chiles et al., 2022); transverse tilted-Dirac proposals reach a fully polarized $\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)$28 (Zeng, 2024). A plausible implication is that “nonreciprocal Josephson transport” is becoming less a single device concept than a unifying framework for superconducting rectification, anomalous phase control, Hall-like supercurrent conversion, microwave routing, and even caloritronic functionality, as illustrated by the MATBG Josephson-diode thermal-machine proposals (Soufy et al., 26 May 2026).

Viewed at this breadth, nonreciprocal Josephson transport is not one mechanism but a family of symmetry-engineered superconducting responses. The common structure is the production of direction-dependent Andreev transport, phase dynamics, or collective excitation dynamics under conditions where the reciprocal Josephson relation would otherwise forbid it. The differences between platforms—static versus dynamical, equilibrium versus voltage-biased, local versus collective, longitudinal versus transverse, two-terminal versus multi-port—now define the field’s main taxonomic divisions rather than its exceptions.