Voltage-Controlled Josephson Diode

• Voltage-controlled Josephson diode is a superconducting junction whose forward and reverse critical currents differ, enabling rectification without a finite-voltage state.
• Experiments in InAs nanosheets and EuS-based nanowires demonstrate controlled switching-current asymmetry quantified by metrics like diode efficiency.
• Gate voltage tuning modulates parameters such as Rashba spin–orbit interaction and inversion symmetry, paving the way for reconfigurable superconducting electronics.

A voltage-controlled Josephson diode is a Josephson junction, or a superconducting circuit containing one or more junctions, whose nonreciprocal dissipationless transport is tuned electrically. Its defining property is that the forward and reverse critical or switching currents are unequal, so that supercurrent rectification occurs without a finite-voltage state as long as the smaller threshold is not exceeded. In hysteretic devices, the retrapping currents can also be nonreciprocal. Recent work realizes this behavior by tuning carrier density, Rashba spin–orbit interaction (SOI), inversion asymmetry, transmission, nonlocal Andreev hybridization, or charging asymmetry with gates, and by combining such voltage control with magnetic phase bias, remanent magnetization, or other symmetry-breaking mechanisms (Zhang et al., 2021, Yan et al., 26 Jan 2025, Telkamp et al., 16 Aug 2025).

1. Definition and quantitative characterization

The standard rectification metric is the diode efficiency

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

where $I_c^+$ and $I_c^-$ are the positive- and negative-bias critical currents. In switching-current experiments, the same quantity is commonly evaluated from $I_{SW}^\pm$, often reported as

$$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$

A positive $\eta$ indicates larger forward dissipationless current, while a negative $\eta$ indicates the opposite polarity (Telkamp et al., 16 Aug 2025, Zhu et al., 19 Aug 2025).

Voltage-controlled Josephson diodes are frequently characterized through current-biased $I$–$V$ or $dV/dI$ measurements. Because switching is stochastic, repeated sweeps are used to build distributions and extract averages. In InAs nanosheet junctions, averaged histograms were obtained with $I_c^+$0 in one data set and $I_c^+$1 or $I_c^+$2 in others, and both switching and retrapping asymmetries were analyzed (Yan et al., 26 Jan 2025). In field-free EuS-based nanowires, 100 repeated sweeps were used at fixed field within the superconducting window to determine $I_c^+$3, $I_c^+$4, and $I_c^+$5 (Telkamp et al., 16 Aug 2025).

The broad theoretical framework distinguishes two notions that are often conflated. In an inversion-breaking, voltage-controlled Josephson diode, the electrical control parameter modifies barrier properties such as Rashba coupling $I_c^+$6 or polarization $I_c^+$7, leading to $I_c^+$8. In most gate-defined experiments, however, the gate voltage is held fixed while one measures direction-dependent switching currents at that operating point. Both usages fall under the same general category because the nonreciprocity is tuned by electrical control of the junction or barrier (Zhang et al., 2021).

2. Symmetry requirements and current–phase relations

The Josephson diode effect requires more than a nonzero supercurrent. In general, nonreciprocity emerges when inversion symmetry and time-reversal symmetry are both broken, or when residual combined symmetries that would enforce $I_c^+$9 are removed. The general theory therefore classifies Josephson diodes into inversion-breaking, voltage-controlled devices and time-reversal-breaking, current-controlled devices; voltage-controlled Josephson diodes belong to the first class when the electrical knob changes inversion-breaking fields or asymmetric proximity inside the barrier (Zhang et al., 2021).

At the level of the current–phase relation (CPR), the minimal structures that generate nonreciprocity are an anomalous phase shift or higher harmonics with unequal phase offsets. Common phenomenological forms are

$I_c^-$0

and

$I_c^-$1

In planar spin–orbit Josephson diodes, a convenient form is

$I_c^-$2

with

$I_c^-$3

This makes explicit that diode asymmetry requires both a higher-harmonic contribution and a nontrivial relative phase shift (Shin et al., 2024).

Microscopic descriptions usually express the supercurrent through Andreev bound states. One representative form is

$I_c^-$4

so any gate-induced change in SOI, exchange, channel transmission, or chemical potential that distorts $I_c^-$5 can induce a nonreciprocal CPR (Telkamp et al., 16 Aug 2025). In other platforms, the same logic appears through more specialized mechanisms: finite-momentum pairing in Rashba systems, interference of harmonics in superconducting interferometers, competition between double elastic cotunneling and double-crossed Andreev reflection in Andreev molecules, or asymmetric charging dynamics in small-capacitance junctions (Yan et al., 26 Jan 2025, Souto et al., 2022, Zhu et al., 19 Aug 2025, Misaki et al., 2020).

A recurrent misconception is that any hysteresis in a Josephson junction implies diode behavior. Hysteresis alone does not establish nonreciprocity. What matters is a reproducible directional asymmetry of switching or retrapping thresholds after controlling for stochastic switching, heating, and field-history effects. InAs nanosheet junctions explicitly separate these issues by averaging many switching events and by showing orientation-dependent asymmetry that vanishes for a control field direction (Yan et al., 26 Jan 2025).

3. Spin–orbit semiconductor implementations

Hybrid InAs junctions provide the most direct experimental realization of electrically tunable diode behavior tied to SOI. In Josephson junctions made from MBE-grown InAs nanosheets with Ti/Al contacts, the nanosheet channel at the junction is approximately $I_c^-$6 nm wide and the electrode gap is $I_c^-$7 nm. The nanosheets are transferred onto local back gates insulated by $I_c^-$8 nm HfO$I_c^-$9, with top gates fabricated above $I_{SW}^\pm$0 nm Al$I_{SW}^\pm$1O$I_{SW}^\pm$2; in the reported measurements the top gate is grounded. Under an in-plane magnetic field $I_{SW}^\pm$3, these devices exhibit nonreciprocal switching and retrapping currents. The effect is strongest when $I_{SW}^\pm$4, with extrema around $I_{SW}^\pm$5 mT and diode efficiencies exceeding $I_{SW}^\pm$6, and it is nearly absent when $I_{SW}^\pm$7 (Yan et al., 26 Jan 2025).

The gate dependence in these nanosheet junctions is pronounced. As the back-gate voltage is decreased from $I_{SW}^\pm$8 V toward $I_{SW}^\pm$9 V at fixed $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$0 mT, both $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$1 and $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$2 decrease monotonically and vanish near $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$3 V; $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$4 and $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$5 also go to zero there, even though finite supercurrent remains. The interpretation given is that the gate suppresses the vertical electric field and can quench Rashba SOI, thereby suppressing finite-momentum pairing. The same work reports that around $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$6 mK the remaining supercurrent is approximately $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$7 nA while the JDE metrics drop to zero (Yan et al., 26 Jan 2025).

The finite-momentum-pairing picture is encoded by

$$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$8

at small $$\eta = \left[\frac{I_{SW}^+ - |I_{SW}^-|}{I_{SW}^+ + |I_{SW}^-|}\right]\times 100\%.$$9, and the switching-current asymmetry follows

$\eta$0

with $\eta$1 mT in the fit to the data. This ties the voltage control directly to Rashba-SOI-mediated finite-momentum pairing under Zeeman splitting (Yan et al., 26 Jan 2025).

A related but more elaborate spin–orbit platform is the epitaxial Al–InAs planar Josephson junction. There, local top gates tune electric fields across identical planar junctions integrated into a DC SQUID. With current along the $\eta$2 direction and an in-plane field $\eta$3 perpendicular to the current, the diode efficiency at $\eta$4 is antisymmetric and nonmonotonic in $\eta$5: it peaks at about $\eta$6 around $\eta$7 mT, changes sign around $\eta$8 mT, reaches about $\eta$9 at higher fields, and weakens further at larger fields. At fixed $\eta$0 mT, sweeping $\eta$1 from $\eta$2 to $\eta$3 V drives the anomalous phase through $\eta$4 near $\eta$5 V and reverses the diode polarity. The interpretation is a gate-tuned competition between Rashba and Dresselhaus SOC in a many-subband planar junction, with polarity determined by the sign of $\eta$6 (Shin et al., 2024).

4. Field-free, multiterminal, and nonlocal voltage control

Voltage control is not confined to single spin–orbit junctions. It has also been demonstrated in ferromagnetically proximitized nanowires, structurally symmetric multiterminal interferometers, and nonlocally coupled Andreev molecules.

Platform	Electrical control	Representative behavior
InAs–EuS–Al nanowire junction	Global back gate $\eta$7	$\eta$8 at $\eta$9 V and $I$0 at $I$1 V at $I$2 mT
Four-terminal InAs/Al junction	Gates $I$3	$I$4 in single-loop mode and about $I$5 in double-loop mode
InAs/Al nanowire Andreev molecule	Local and non-local gates $I$6	Gate-modulated $I$7 with a central-peak feature and phase-controlled sign reversal

In hybrid nanowire junctions consisting of an InAs semiconductor core coated with epitaxial EuS and Al shells, the back gate tunes carrier density and the relative importance of SOC versus superconducting proximity. The diode effect appears within a hysteretic superconducting window as a function of axial magnetic field, and the efficiency is strongly gate dependent. At $I$8 mT, repeated sweeps give $I$9 at $V$0 V, whereas at $V$1 V the distributions of $V$2 and $V$3 largely overlap and $V$4, consistent with zero. A controlled demagnetization protocol then establishes field-free operation: superconductivity persists at $V$5 for demagnetization fields $V$6 between roughly $V$7 and $V$8 mT, and the remanent zero-field diode effect remains gate tunable (Telkamp et al., 16 Aug 2025).

In four-terminal Josephson junctions defined in epitaxial InAs/Al heterostructures, electrostatic gates shape the CPRs of several parallel superconducting branches, while on-chip flux-bias lines provide local phase control without a global magnetic field. The device exhibits widely tunable diode efficiency, reaching about $V$9 in a single-loop configuration and about $dV/dI$0 in a double-loop configuration. The gate $dV/dI$1 reconfigures the topology through a wide, short “switch JJ,” while $dV/dI$2, $dV/dI$3, and $dV/dI$4 route supercurrent and change the relative phase sensitivity of different branches. Because the nonreciprocity is generated by multiterminal phase control and nonsinusoidal CPRs rather than by structural asymmetry, the device establishes that large voltage-tunable diode response can occur in structurally symmetric circuits (Coraiola et al., 2023).

Nanowire-based Andreev molecules realize a distinctly nonlocal form of voltage-controlled diode behavior. Two short Al–InAs–Al junctions are coherently coupled through an Al segment shorter than the coherence length, so the left-junction CPR depends on the nonlocal phase $dV/dI$5 of the right junction. The lowest-order molecular Josephson energy contains

$dV/dI$6

which yields

$dV/dI$7

Here the diode sign reverses when $dV/dI$8 crosses $dV/dI$9, reflecting a swap in the relative importance of double elastic cotunneling and double-crossed Andreev reflection. Local and non-local gates modulate the effect further, with a central-peak structure in $I_c^+$00 near symmetric gate settings; in one device the maximum measured $I_c^+$01 is about $I_c^+$02 at $I_c^+$03 V (Zhu et al., 19 Aug 2025).

5. Theoretical routes to electrical control

Several theoretical proposals generalize voltage-controlled Josephson diodes beyond semiconductor Rashba junctions. One route uses loop-current barriers described by Haldane-model physics. In that setting, loop currents break time-reversal symmetry, but the standard Haldane model preserves inversion and therefore does not by itself produce a diode effect. In monolayers, inversion breaking can be introduced by a staggered potential $I_c^+$04 or by a modified Haldane model; in bilayers, it can be produced by opposite loop-current stacking or by an interlayer voltage $I_c^+$05 generated by a perpendicular electric field, with $I_c^+$06. For small $I_c^+$07, the diode efficiency is odd in $I_c^+$08, approximately $I_c^+$09, and the polarity flips with the sign of $I_c^+$10. A notable symmetry result is that zigzag-oriented junctions exhibit JDE, whereas armchair-oriented junctions remain reciprocal because residual symmetries such as $I_c^+$11 or $I_c^+$12 still forbid it (Shen et al., 2024).

A conceptually different field-free route is the Floquet-engineered Kitaev-chain–quantum-dot–Kitaev-chain junction. There, two periodic drives with phase mismatch $I_c^+$13 break inversion symmetry and time-reversal symmetry electrically, without magnetic fields or intrinsic SOC. The quantum-dot level is tuned by

$I_c^+$14

and the diode performance is characterized by

$I_c^+$15

For suitable $I_c^+$16, $I_c^+$17, $I_c^+$18, and $I_c^+$19, the reported maximum rectification is about $I_c^+$20. The same platform also supports anomalous current $I_c^+$21 and Floquet Majorana modes at quasienergies $I_c^+$22 and $I_c^+$23 (Roy et al., 10 Mar 2025).

On topological-insulator surfaces, voltage control enters through a narrow electrostatic barrier of height $I_c^+$24 in an S–N–S junction. An in-plane Zeeman field along $I_c^+$25 generates a channel-dependent Doppler shift $I_c^+$26, producing angle-resolved CPR asymmetry. The gate selectively suppresses oblique channels while preserving near-normal-incidence transmission through Klein tunneling. In long junctions, this reweighting can change not only the magnitude but also the sign of the diode quality factor $I_c^+$27, so $I_c^+$28 can reverse polarity when $I_c^+$29 (Lu et al., 2022).

At the circuit-theory level, nonreciprocity can arise even without microscopic SOI or magnetic textures if the charging energy itself is inversion asymmetric. The effective charging energy is expanded as

$I_c^+$30

so gate voltages tune the asymmetric coefficients $I_c^+$31 and $I_c^+$32. This makes the differential capacitance direction dependent and leads to a generalized RCSJ dynamics with nonreciprocal switching, hysteresis, Bloch thresholds, and Landau–Zener probabilities (Misaki et al., 2020).

A further field-free proposal combines a gate-tunable 2DEG, Rashba SOC, and a skyrmion crystal underneath a planar $I_c^+$33-wave Josephson junction. The skyrmion texture supplies the symmetry breaking, while the gate shifts the chemical potential $I_c^+$34 and thereby changes the anomalous phase shift and higher harmonics of the CPR. In the BdG+RCSJ analysis, diode efficiencies reach roughly $I_c^+$35 for favorable $I_c^+$36, $I_c^+$37, and skyrmion radius, suggesting a route to higher-temperature operation when the proximitizing leads are high-$I_c^+$38 cuprates (Singh et al., 1 Nov 2025).

6. Applications, interpretive issues, and limitations

Voltage-controlled Josephson diodes are relevant to superconducting electronics because they offer gate-programmable nonreciprocal elements within the dissipationless state. The experimental and theoretical literature explicitly points to rectifiers, memory, logic, phase-biased circuitry, phase batteries, and reconfigurable cryogenic systems. Several works also emphasize their value as probes: in Rashba-SOI semiconductors the gate dependence of $I_c^+$39 can serve as an indirect probe of SOI, while in EuS-based devices $I_c^+$40 diagnoses exchange-induced spin splitting and domain configurations; in broader contexts, diode behavior is proposed as a probe of exotic or unconventional superconducting states (Yan et al., 26 Jan 2025, Telkamp et al., 16 Aug 2025).

The electrical control mechanism is platform dependent. In Rashba systems the gate primarily tunes carrier density and SOI; in interferometers it reshapes harmonic content by changing channel transmission or a quantum-dot level; in Andreev molecules it modifies local and nonlocal chemical potentials and hence dEC/dCAR balance; in charging-energy theories it tunes nonlinear quantum capacitance; and in loop-current or bilayer models it acts directly as an inversion-symmetry knob through $I_c^+$41 (Souto et al., 2022, Zhu et al., 19 Aug 2025, Misaki et al., 2020, Shen et al., 2024). This diversity has an important interpretive consequence: “voltage-controlled” does not denote a single microscopic mechanism.

Several limitations recur across the literature. Many experimental realizations still require auxiliary symmetry-breaking fields or phase biases. In the InAs nanosheet diode, the JDE is maximized for $I_c^+$42 and nearly vanishes for $I_c^+$43, so precise in-plane alignment is essential (Yan et al., 26 Jan 2025). In EuS-based field-free nanowires, the effect depends on magnetic history and is strongest near boundaries of the superconducting window, while the efficiency tends to decrease where the absolute switching current is maximal (Telkamp et al., 16 Aug 2025). In loop-current barriers, armchair junctions remain symmetry-protected against JDE, underscoring the role of crystallographic orientation (Shen et al., 2024). In high-$I_c^+$44 skyrmion proposals, strong proximity-induced $I_c^+$45-wave pairing and controlled texture stability are nontrivial materials constraints (Singh et al., 1 Nov 2025).

A final issue is the distinction between genuine nonreciprocity and artifacts. In the InAs nanosheet experiments, antisymmetry of $I_c^+$46, insensitivity to field-sweep direction, nearly zero response for $I_c^+$47, and large-$I_c^+$48 averaging were used to exclude residual out-of-plane-field artifacts and stochastic-heating effects (Yan et al., 26 Jan 2025). More generally, switching-current diode metrics depend on dissipative escape dynamics as well as on the equilibrium CPR. This does not invalidate the diode concept, but it means that comparisons between platforms must distinguish equilibrium nonreciprocal CPRs, switching asymmetries, and retrapping asymmetries rather than treating them as interchangeable observables (Zhang et al., 2021, Coraiola et al., 2023).