Josephson Diode Effect Overview

• Josephson diode effect is a nonreciprocal superconducting phenomenon defined by unequal forward and reverse critical currents due to broken inversion and time-reversal symmetry.
• It spans diverse platforms—such as Rashba nanowires, topological insulators, and Andreev molecules—highlighting the impact of spectral asymmetry and multichannel interference.
• Experimental and dynamical studies demonstrate that tuning parameters like magnetic fields, gating, and junction geometry can optimize the diode efficiency.

The Josephson diode effect (JDE) is the nonreciprocal Josephson response in which the maximal dissipationless supercurrent depends on current direction, so that the forward and backward critical currents are unequal. In microscopic terms, the effect corresponds to a current–phase relation (CPR) that is no longer an odd function of the superconducting phase difference, and in phenomenological terms it is a superconducting rectification effect. Research has established JDE across a broad set of platforms, including spin–orbit-coupled semiconductor junctions, topological Josephson junctions, topological-insulator surfaces, multi-terminal interferometers, Andreev molecules, magnetic-texture hybrids, twisted nodal superconductors, band-asymmetric metals, helimagnetic junctions, and Kitaev-ladder geometries (Liu et al., 2023).

1. Definition and measures of nonreciprocity

The minimal definition of JDE is the inequality of directional critical currents, usually written as $I_c^+ \neq |I_c^-|$. Several normalized figures of merit are used in the literature. A common choice is

$\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$

used for topological Josephson junctions and related systems (Liu et al., 2023). Other works define a diode quality factor

$Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$

or a diode coefficient

$$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$

reflecting different sign conventions and whether the extrema are expressed as critical currents or as maxima and minima of the CPR (Lu et al., 2022, Soori, 2023).

The supercurrent is commonly obtained from the phase derivative of the junction free energy or the phase derivative of the many-body spectrum. In the BdG formulation used for topological junctions,

$$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$

with

$E = -\frac{1}{2}\sum_{E_n\ge 0} E_n(\phi),$

where the sum includes both Andreev bound states (ABS) and continuum contributions (Liu et al., 2023). This formulation is central because many of the strongest diode responses originate not only from discrete ABS asymmetry but also from continuum states.

A later phenomenological formulation distinguishes an ideal diode effect, defined by unequal critical currents, from a pseudo diode effect, defined by equal critical currents but unequal retrapping currents, $|I_{r+}| \ne |I_{r-}|$ (Wang et al., 29 Jun 2025). This distinction is important in underdamped junctions and in systems where phase dynamics, rather than the static CPR alone, dominates the observed nonreciprocity.

2. Symmetry structure and current-phase relations

A recurrent organizing principle is that ideal JDE requires simultaneous breaking of inversion and time-reversal symmetries (Davydova et al., 2022, Wang et al., 29 Jun 2025). In specific microscopic models this statement becomes more refined. For the topological junctions studied in a Rashba nanowire and in a magnetic topological-insulator junction, JDE requires breaking of time-reversal $(\mathcal{T})$, mirror $(\mathcal{M}_x)$, and inversion $(\mathcal{P})$ symmetries, and both Zeeman terms $\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$0 and $\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$1 are necessary to break sufficient symmetries to allow JDE (Liu et al., 2023).

The CPR framework clarifies how symmetry breaking is encoded. In mixed singlet–triplet quantum-wire junctions, the current is expanded as

$\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$2

so that cosine harmonics signal an anomalous Josephson component and multi-harmonic structure can generate critical-current asymmetry (Soori, 2024). In a topological diode model combining conventional and Majorana channels, the CPR takes the form

$\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$3

where the interplay of $\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$4- and $\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$5-periodic terms plus a phase shift $\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$6 produces direction-dependent critical currents (Zhong et al., 13 Apr 2025).

A distinct route to JDE is purely interferometric. In supercurrent interferometers with mesoscopic junctions, the total CPR

$\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$7

becomes nonreciprocal when magnetic flux breaks time-reversal symmetry and the constituent junctions have non-sinusoidal CPRs with higher harmonics; neither Zeeman splitting nor spin–orbit coupling is required in that construction (Souto et al., 2022). A three-terminal Josephson device implements the same idea geometrically: the effective CPR contains first and second harmonics with a flux-tunable phase offset, for example $\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$8, and the multi-terminal geometry itself supplies the inversion-symmetry breaking needed for a diode response (Gupta et al., 2022).

Twisted nodal superconductors add another symmetry-based distinction. Near a twist angle of $\eta = \frac{|I_c^+ - I_c^-|}{I_c^+ + I_c^-},$9, the first harmonic is suppressed while the second harmonic remains, producing a double-well Josephson free energy and spontaneous $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$0-breaking. In that regime an underdamped junction can show a dynamical JDE, whereas explicit $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$1-breaking produces a thermodynamic JDE that survives even in the overdamped limit (Volkov et al., 2023).

3. Microscopic mechanisms

Although symmetry arguments determine whether JDE is allowed, the magnitude and sign of the effect are set by specific spectral mechanisms. A widely cited microscopic mechanism is finite Cooper-pair momentum in a short junction. In that case the diode response arises from a Doppler shift of ABS energies together with a phase-independent asymmetric current from the continuum, with

$Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$2

and diode efficiency up to $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$3 reported for the short-junction theory (Davydova et al., 2022). This mechanism was presented as universal because the finite-momentum pairing need not rely on spin–orbit coupling.

In topological Josephson junctions, the key microscopic driver is asymmetry in the Andreev spectrum. In the Rashba nanowire studied in (Liu et al., 2023), JDE arises from an imbalance between contributions of “inner” and “outer” modes near the Fermi points, and suppression of certain modes near the topological boundary enhances the asymmetry. The same work emphasizes that the presence of Majorana bound states is not a sufficient condition for a large diode effect.

A complementary analytic theory for singlet-superconductor/Rashba/singlet-superconductor junctions attributes the asymmetry to the combined action of Rashba spin splitting, an in-plane magnetic field, and a bias-current-induced Fermi-momentum shift

$Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$4

In that model the asymmetry scales as $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$5, while the magnitude and even the sign oscillate with the electrode separation $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$6; stronger Rashba splitting can suppress the diode effect rather than enhance it (Mori et al., 10 Jan 2025).

On the surface of a topological insulator, the relevant mechanism is channel-resolved phase misalignment. Each transverse mode acquires an angle-dependent Doppler shift $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$7, so the total current is a superposition of many transverse CPRs with different extrema. In long junctions this produces strong tunability and even sign reversal of the diode quality factor with magnetic field, gate voltage, or junction length (Lu et al., 2022).

Another microscopically distinct route is band asymmetry in the normal layer. In SC–BAM–SC junctions the right- and left-moving modes acquire unequal phases, and the current asymmetry can be traced to unequal Andreev-state weights in forward and reverse processes. In that setting band asymmetry is necessary but not sufficient: for some parameters the phases become equal in magnitude and the diode effect disappears despite the asymmetric band structure (Soori, 2023).

Mixed-parity superconducting leads introduce still another mechanism. In a one-dimensional quantum wire connected to superconductors with mixed singlet–triplet pairing, the study of (Soori, 2024) found that SOC and a Zeeman field in the wire are not sufficient when the leads are purely singlet, whereas finite triplet pairing enables JDE by allowing same-spin Andreev processes with unequal dynamical phases. In that model JDE is always accompanied by the anomalous Josephson effect, and the diode coefficient oscillates with the wire chemical potential because of Fabry–Pérot interference.

4. Topology, Majorana physics, and Andreev molecularity

The relation between JDE and topological superconductivity is substantial but nontrivial. In topological Josephson junctions based on a 1D Rashba nanowire and a 2D magnetic topological insulator, higher diode efficiency can occur in topological phases, but the behavior is explicitly not universal. The diode efficiency changes substantially only in specific regions along the topological phase-transition boundary, and significant JDE often coincides with the topological phase without being unique to it (Liu et al., 2023). In the nanowire model the sharp rise occurs as parameters approach $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$8.

A central misconception addressed by that work is that Majorana bound states automatically imply a large diode response. The stated conclusion is the opposite: MBS are not sufficient for a large JDE, some topological regions have small or negligible $Q = \frac{J_c^{+} - J_c^{-}}{J_c^{+} + J_c^{-}},$9, and regions just short of the topological transition can already show large $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$0 because the decisive ingredient is strong asymmetry in the Andreev spectrum rather than topology alone (Liu et al., 2023).

A more detailed ABS–MBS analysis in superconductor–semiconductor hybrids reached a related but sharper conclusion. When a Zeeman field has a component parallel to the spin–orbit axis, the low-energy spectrum becomes asymmetric in the phase difference in both trivial and topological phases. The diode effect is particularly promoted when zero-energy ABS and MBS are both present, and the field evolution of the diode efficiency can map the topological phase transition through oscillatory behavior that becomes more visible in long superconductors. In the tunneling regime, however, only the topological-phase diode effect remains because the Majorana contribution stays finite while the ABS contribution is suppressed (Mondal et al., 11 Mar 2025).

Andreev molecules realize JDE through nonlocal hybridization rather than through a single junction alone. Two closely spaced Josephson junctions with separation $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$1 hybridize their ABS into molecular states, and a phase $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$2 applied across one junction nonlocally breaks time-reversal symmetry for the other. In the microscopic treatment of (Pillet et al., 2023), the diode efficiency can reach $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$3, and the crucial contribution comes largely from the continuum, specifically from “leaky” Andreev states formed when ABS merge into the continuum.

That mechanism has subsequently been observed in nanowire-based Andreev molecules. The experimental work of (Zhu et al., 19 Aug 2025) reported non-local control of the diode effect by phase and gate voltages, including sign reversal of the diode efficiency. The sign reversal was interpreted as regulation of the probabilities of double elastic cotunneling and double crossed Andreev reflection.

Topological geometry alone can also generate JDE. In a Kitaev-ladder Josephson junction, a leg-to-leg phase difference $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$4 breaks time-reversal symmetry, the absence of leg-exchange symmetry removes CPR antisymmetry, and transport decomposes into bonding and antibonding channels. Near the topological transition a $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$5-periodic Majorana channel interferes with $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$6-periodic Andreev channels, producing a dome-like diode efficiency as a function of the interleg hopping $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$7, with the maximum at intermediate coupling (Xie et al., 6 Nov 2025).

5. Nonequilibrium, dynamical, and ac manifestations

Not all diode-like Josephson transport is an equilibrium property of a symmetry-broken Hamiltonian. In a four-terminal Andreev interferometer under voltage bias, a diode asymmetry can emerge solely due to a dissipative current in the normal region of an otherwise symmetric Josephson junction. The resulting nonequilibrium CPR was written as

$$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$8

where the phase-independent term $$\gamma = \frac{2(J_{\max} + J_{\min})}{J_{\max} - J_{\min}},$$9 is the key source of the diode effect. Within certain ranges of voltage, temperature, and geometry, the diode coefficient can exceed its nominal perfect value (Shaffer et al., 30 Apr 2025). This directly contradicts the simplified expectation that explicit inversion-symmetry breaking is always required.

The ac response of Josephson diodes is likewise distinctive. For a topological Josephson diode with nonreciprocal critical currents generated by the interplay of $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$0- and $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$1-periodic CPR components, a pure ac current $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$2 can induce a finite dc voltage. This inverse ac Josephson effect is a rectification phenomenon: the ac drive exceeds the lower critical current in one half-cycle but not the higher critical current in the opposite half-cycle. At low frequency the induced dc voltage obeys

$$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$3

and the effect is optimized for certain amplitudes and low frequencies (Zhong et al., 13 Apr 2025).

Dynamical phase evolution is especially important in underdamped junctions. In twisted nodal superconductors, the double-well Josephson free energy near $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$4 produces a dynamical JDE only in the hysteretic regime; the diode direction can then be trained by sweeping the external current bias. The same system supports a thermodynamic diode effect from explicit $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$5-breaking that persists in the overdamped limit and also vanishes exactly at $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$6, even though time-reversal symmetry is maximally frustrated there (Volkov et al., 2023).

The generalized RCSJ treatment of (Wang et al., 29 Jun 2025) unified these cases into intrinsic, extrinsic, and pseudo diode effects and proposed Shapiro-step diagnostics to distinguish them. Interferometer-based Josephson diodes likewise exhibit characteristic ac signatures, including asymmetric Shapiro-step widths and fractional steps when higher harmonics dominate (Souto et al., 2022).

6. Experimental platforms, control parameters, and emerging directions

Experimental realizations have confirmed that JDE is highly tunable and highly platform-dependent. In high-mobility InSb nanoflag SNS junctions, an in-plane magnetic field produces unequal switching currents for opposite current directions. The asymmetry increases linearly at small field, then saturates, and finally decreases to zero at higher fields; it is maximal when the field is perpendicular to the current, identifying Rashba SOC as the main symmetry-breaking mechanism. In that experiment carrier concentration did not significantly influence the effect in the explored range, whereas increasing temperature strongly suppressed it (Turini et al., 2022).

A different experimental strategy uses a three-terminal Josephson device on an InAs quantum-well platform proximitized by epitaxial aluminum. There the diode efficiency can be tuned by a small out-of-plane magnetic field or by electrostatic gating, and the effect was interpreted as a consequence of an artificially engineered CPR containing higher harmonics. The same multi-terminal geometry enabled nonlinear dc intermodulation and simultaneous two-signal rectification (Gupta et al., 2022).

At the atomic scale, a Pb–Pb Josephson junction in a scanning tunneling microscope showed no nonreciprocity when stabilized by a single Pb atom, but a Josephson diode effect emerged when a single magnetic atom was inserted. For Cr and Mn atoms the preferred direction depended on the atomic species, and the asymmetry was traced to quasiparticle currents flowing through Yu–Shiba–Rusinov states inside the superconducting gap. In this case the dominant nonreciprocity appeared in the retrapping current rather than in the switching current (Trahms et al., 2022).

Several theoretical platforms aim at broader control knobs or reduced hardware constraints. A planar $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$7-wave Josephson junction coupled to a skyrmion crystal underneath yields a field-free, gate-tunable diode effect because the spatially varying exchange field of the skyrmion crystal breaks inversion and time-reversal symmetries intrinsically. In that model the efficiency can be tuned by gate voltage and skyrmion radius and can approach $$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$8, while the use of a high-$$I(\phi) = \frac{2e}{\hbar}\frac{\partial E}{\partial \phi},$$9 cuprate permits operation at higher temperatures in principle (Singh et al., 1 Nov 2025). Helimagnetic junctions provide another field-free route: in superconductor/helimagnet/superconductor devices the necessary conditions for $E = -\frac{1}{2}\sum_{E_n\ge 0} E_n(\phi),$0-wave pairing are nonzero chemical potential and a conical magnetic configuration, and efficiencies close to $E = -\frac{1}{2}\sum_{E_n\ge 0} E_n(\phi),$1 can be obtained for specific parameter values; for $E = -\frac{1}{2}\sum_{E_n\ge 0} E_n(\phi),$2-wave pairing, nonzero chemical potential is no longer necessary because transport is mediated by equal-spin Cooper pairs (Cheng et al., 28 May 2026).

Across these platforms, several control parameters recur: Zeeman-field magnitude and orientation, magnetic flux, gate voltage, junction length, barrier transparency, electrode separation $E = -\frac{1}{2}\sum_{E_n\ge 0} E_n(\phi),$3, chirality, tilt angle, skyrmion radius, and the presence of higher harmonics or $E = -\frac{1}{2}\sum_{E_n\ge 0} E_n(\phi),$4-periodic channels. A consistent conclusion is that large JDE is often associated with spectral asymmetry and multichannel interference, but a large diode response is rarely a unique fingerprint of a single microscopic origin. In particular, significant JDE can indicate topological superconductivity when combined with a non-vanishing average critical current and other probes such as CPR measurements or spectroscopy, but further confirmation is required because substantial diode efficiencies also occur in trivial regimes (Liu et al., 2023).