Josephson Current-Phase Relation (CPR)

• Josephson CPR is the functional relation between the superconducting phase difference across a weak link and the equilibrium supercurrent, defined via the derivative of free energy.
• The CPR can vary from a simple sinusoidal form to complex multi-harmonic and fractional behaviors depending on junction transparency, geometry, disorder, and material properties.
• Experimental techniques like SQUID interferometry and ring measurements accurately extract CPR, thereby linking microscopic Andreev bound states to macroscopic transport phenomena in superconducting devices.

The Josephson current-phase relation (CPR) is the functional dependence of the equilibrium supercurrent on the gauge-invariant superconducting phase difference across a weak link, $I(\varphi)$. It is the phase-sensitive object that connects microscopic Andreev spectra to macroscopic transport, and it governs switching, interference, anharmonicity, and dissipationless nonreciprocity in Josephson devices. Beyond the elementary sinusoidal form $I_c\sin\varphi$, the CPR can become forward-skewed, strongly multi-harmonic, multivalued, $\pi$-periodic, or contain fractional components, depending on transparency, junction length, disorder, proximity effect, symmetry, and circuit architecture (Kayyalha et al., 2018, Banszerus et al., 2024, Bakurskiy et al., 2017).

1. Formal definition and representations

In equilibrium, the CPR follows directly from the phase dependence of the free energy,

$$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$

and, microscopically, from the phase-dispersing Andreev bound states (ABS),

$$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$

This formulation makes the CPR a direct probe of the ABS spectrum and of the occupation function that weights it (Kayyalha et al., 2018).

A convenient representation is the harmonic expansion

$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$

with Josephson potential

$$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$

Within this language, higher harmonics encode higher-order coherent transport processes and departures from a purely tunnel-like junction. In engineered or unconventional systems, generalized expansions can also include fractional terms such as $A_{1/2}\sin(\varphi/2)$, which are used to parametrize candidate $4\pi$-periodic contributions in spectroscopy problems (Banszerus et al., 2024, Yerin et al., 16 Jun 2026).

The same object can therefore be read in several equivalent ways: as a constitutive relation for the weak link, as the derivative of a phase-dependent energy, or as a mode-resolved sum over ABS dispersions. Which representation is most useful depends on whether the emphasis is on microscopic transport, circuit design, or experimental inversion.

2. Canonical regimes and analytic forms

The tunnel SIS limit is the reference case: for low transparency the CPR is approximately sinusoidal,

$I(\varphi)\approx I_c \sin\varphi.$

As transparency increases, ABS become strongly nonlinear in $I_c\sin\varphi$0, higher harmonics grow, and the maximum current shifts to $I_c\sin\varphi$1, producing a forward-skewed CPR. For a single short ballistic channel with transmission $I_c\sin\varphi$2,

$I_c\sin\varphi$3

which makes the dependence on transparency and thermal smearing explicit (Nanda et al., 2016).

Short ballistic graphene supplies a canonical multi-mode example. Near the Dirac point, in the wide short-junction limit, the CPR is strongly skewed and can be written as

$I_c\sin\varphi$4

The same general tendency toward a more sinusoidal CPR at higher temperature is seen across ballistic graphene, conventional SNS junctions, and proximity structures, because higher-order processes are more thermally suppressed than the first harmonic (Nanda et al., 2016).

Self-consistent treatments show that interface physics can alter even these “standard” expectations. In ballistic graphene SNS junctions, proximity-effect depletion and depairing by current reduce the skewness relative to rigid-boundary Dirac-BdG predictions, and in short junctions can drive the critical phase below $I_c\sin\varphi$5 over a wide range of temperatures and doping levels. This establishes that CPR shape is not determined by transparency alone; it is also controlled by how superconductivity is self-consistently modified near the weak link (Black-Schaffer et al., 2010).

3. Determination of the CPR in experiment

Several phase-sensitive strategies are used to reconstruct $I_c\sin\varphi$6. One direct approach embeds the junction in a superconducting ring and measures the circulating current inductively. In graphene ring interferometry, the phase is set by the loop flux,

$I_c\sin\varphi$7

and current conservation gives the CPR as

$I_c\sin\varphi$8

This configuration provides a direct map of supercurrent versus phase without relying on switching-current asymmetry alone (Chialvo et al., 2010).

A second widely used approach is the asymmetric dc-SQUID. In the favorable limit of a strongly asymmetric reference junction and sufficiently small loop inductance, the SQUID modulation directly tracks the CPR of the weaker junction. Ballistic graphene measurements exploited this regime and showed that the extracted CPR remained reliable under small loop inductance, with RCSJ simulations confirming that inductance did not skew the reconstruction under the reported conditions (Nanda et al., 2016).

However, two later results established stringent caveats. In WTe$I_c\sin\varphi$9-based asymmetric SQUIDs, a high critical-current asymmetry and nominally negligible loop inductance were not sufficient: additional series/kinetic inductances from self-formed PdTe$\pi$0 inside the junction distorted the phase bias and produced an apparent doubled switching period. After self-consistent inversion, the underlying CPR was recovered as a $\pi$1-periodic, strongly skewed short-ballistic form rather than a genuine fractional one (Endres et al., 2022). A complementary analysis of asymmetric dc-SQUID metrology showed that accuracy is controlled by asymmetry in the CPR derivatives $\pi$2, not by the critical-current ratio alone. In that formulation, the reference branch must dominate not only in amplitude but also in phase stiffness; otherwise, the measured oscillation can be a mixture of reference and test CPRs rather than a faithful image of the latter (Babich et al., 2023).

4. CPR across material platforms

Graphene junctions have become a benchmark platform because transparency, carrier density, and cavity interference are all tunable. In fully gate-tunable ballistic graphene SQUIDs, the reconstructed CPR was forward-skewed and gate-dependent, with typical skewness $\pi$3 on the $\pi$4-doped side, $\pi$5 on the $\pi$6-doped side, and suppression to $\pi$7 near the charge-neutrality point. On the $\pi$8 side, skewness oscillated in anti-phase with Fabry–Pérot resistance oscillations, linking the CPR directly to cavity transmission. By $\pi$9 K the CPR became sinusoidal (Nanda et al., 2016). Earlier direct phase-sensitive measurements on short but diffusive graphene junctions likewise found a skewed CPR whose skewness increased approximately linearly with the critical current, regardless of whether $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$0 was tuned by gate voltage or temperature (Chialvo et al., 2010).

Topological-insulator-based junctions display similarly strong deviations from sinusoidality, but with additional geometric and surface-state structure. In Nb–BSTS–Nb devices, the normalized CPR at $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$1 mK peaked at $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$2, the total harmonic distortion was $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$3, and the Fourier spectrum contained appreciable weight up to the sixth harmonic. The same study attributed the low-temperature growth of $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$4 without saturation to low-energy modes associated with finite-width electrodes and topological surface-state wavefunctions extending around the full TI perimeter (Kayyalha et al., 2018). In a short multi-mode Bi$$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$5Se$$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$6 nanobelt junction, the extracted CPR had $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$7 and $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$8; fits assigned about $$I(\varphi)=\frac{2e}{\hbar}\,\frac{\partial F(\varphi)}{\partial \varphi},$$9 nA of a total $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$0 nA critical current to ballistic topological surface states and about $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$1 nA to diffusive bulk and perimeter channels (Surendran et al., 2023). In Nb/HgTe/Nb, forward skew persisted from $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$2 nm to $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$3 nm, indicating that high-transmission helical ABS survive even when $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$4 (Sochnikov et al., 2014).

Semiconductor nanowires expose the CPR to few-mode quantization, gate-controlled transparency, and interactions. In InAs/Al nanowires, multimode fits and Schrödinger–Poisson–Bogoliubov–de Gennes simulations resolved a single-mode regime with representative $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$5 and $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$6, a fluctuating multimode regime, and interaction-induced shoulders near $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$7 interpreted as precursors of a $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$8 transition (Hart et al., 2019). A separate few-mode InAs nanowire study found that near depletion the CPR became consistent with resonant tunneling through a single, highly transmitting mode, whereas at higher density the CPR indicated $$I(\varphi)=\frac{2e}{\hbar}\,\sum_n \frac{\partial E_n(\varphi)}{\partial \varphi}\,f(E_n).$$9 modes with $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$0 (Spanton et al., 2017). In InSb–Al nanowire junctions under parallel magnetic field, narrow gate intervals displayed $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$1-shifted CPRs, suppressed supercurrent, and doubled-frequency intermediate regimes, all reproduced by a model in which supercurrent interferes between a direct transmission path and a resonant localized state (Levajac et al., 2023).

5. Engineered, higher-harmonic, and unconventional CPRs

The CPR can also be shaped deliberately by geometry and material choice. In a variable-thickness SN-S-SN junction made from a highly disordered superconducting layer and a low-resistive normal metal, the CPR is single-valued and can be close to sinusoidal when $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$2, $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$3, and the constriction length $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$4 are of order $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$5, specifically for $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$6 and $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$7. For thicker superconducting banks the CPR becomes multivalued at low temperature, resembling Dayem bridges. In the favorable regime the characteristic voltage can reach $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$8, while the N layer improves heat evacuation and supports non-hysteretic current–voltage behavior (Marychev et al., 2020).

Multi-junction interference offers still more direct harmonic synthesis. The hybrid Josephson rhombus, built from four gate-tunable semiconductor–superconductor junctions, uses magnetic frustration and junction-by-junction tuning to control the harmonic content of the effective CPR. At full frustration and balanced couplings, odd harmonics destructively interfere, producing a $I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi),$9-periodic $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$0 Josephson potential and a CPR dominated by $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$1; the associated Shapiro response contains half-integer steps at $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$2, consistent with coherent charge-$$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$3 transport. Detuning away from the balanced point yields a superconducting diode effect with efficiency exceeding $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$4 (Banszerus et al., 2024).

Unconventional topological junctions supply another class of nonstandard CPRs. In long 2D TRITOPS–TRITOPS junctions, the low-energy edge-mode mass is proportional to $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$5, so the edge contribution produces a cusp-like singularity near $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$6; in TRITOPS–S junctions the mass is proportional to $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$7, yielding a singular correction near $$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$8 and a spontaneous phase shift that breaks time-reversal symmetry at the junction (Ruiz et al., 2022). A related helical-metal calculation for TI surface line junctions found, in the short single-mode limit,

$$U(\varphi)= -\sum_{n\ge1} E_n \cos(n\varphi), \qquad I_n=\frac{2e}{\hbar}\,n\,E_n.$$9

together with a robust zero-energy crossing at $A_{1/2}\sin(\varphi/2)$0 for $A_{1/2}\sin(\varphi/2)$1. The equilibrium dc CPR remains $A_{1/2}\sin(\varphi/2)$2-periodic because the occupied branch switches at the crossing, but parity-conserving ac dynamics yields the familiar $A_{1/2}\sin(\varphi/2)$3-periodic fractional Josephson effect (Olund et al., 2012).

6. Multivaluedness, hysteresis, nonreciprocity, and CPR spectroscopy

Large second harmonics, explicit phase discontinuities, and nonlinear circuit embedding can all turn the CPR into a genuinely multibranch object. For a short junction with a phase discontinuity $A_{1/2}\sin(\varphi/2)$4, the effective point-like CPR is

$A_{1/2}\sin(\varphi/2)$5

with $A_{1/2}\sin(\varphi/2)$6 and $A_{1/2}\sin(\varphi/2)$7. Over a broad parameter range this realizes a $A_{1/2}\sin(\varphi/2)$8 junction with a unique ground-state phase in each $A_{1/2}\sin(\varphi/2)$9 interval, while near $4\pi$0 and for $4\pi$1 close to zero it becomes a $4\pi$2 junction with two degenerate minima. In the same framework, the near-critical barrier and plasma frequency retain the standard scalings $4\pi$3 and $4\pi$4, but with $4\pi$5-dependent prefactors relevant for escape histograms and macroscopic quantum tunneling (Goldobin et al., 2015).

Diffusive SIsFS junctions near the $4\pi$6 transition furnish a different multivalued regime. There, the second harmonic of the sFS part can become so strong that the total CPR develops hysteretic, multibranch structure, multiple ground states, and, notably, an almost constant critical current through the $4\pi$7 transition even though the branch topology changes qualitatively. The earlier analytic and numerical theory already identified distinct transport regimes—SIS-like, SFS-like, and SInFS-like—with the strongest departures from sinusoidality in the crossover regime; later work emphasized that the CPR in the transition region can have multiple branches with distinct ground states (Bakurskiy et al., 2013, Bakurskiy et al., 2017).

A recent dynamical extension is ac-flux-driven SQUID diode spectroscopy. In that method, a fast ac flux dresses each CPR harmonic with a distinct Bessel factor, so $4\pi$8, $4\pi$9, and $I(\varphi)\approx I_c \sin\varphi.$0 generate different nodal families in diode-efficiency maps $I(\varphi)\approx I_c \sin\varphi.$1: weak arcs for conventional $I(\varphi)\approx I_c \sin\varphi.$2, sparse arcs for fractional $I(\varphi)\approx I_c \sin\varphi.$3, dense arcs for $I(\varphi)\approx I_c \sin\varphi.$4, and intermodulated patterns for mixtures. The proposal supplies a transport-based route to harmonic discrimination that does not rely on static interference patterns alone (Yerin et al., 16 Jun 2026).

Nonreciprocal CPRs show that odd symmetry can also fail in equilibrium transport once inversion and time-reversal symmetry are both broken. In planar NbSe$I(\varphi)\approx I_c \sin\varphi.$5/Bi$I(\varphi)\approx I_c \sin\varphi.$6Se$I(\varphi)\approx I_c \sin\varphi.$7 junctions, a perpendicular field imposes a phase chirp across the width, and when the current density $I(\varphi)\approx I_c \sin\varphi.$8 is edge-amplified and sloped, the integrated CPR acquires both sine and cosine components. Near one flux quantum, this produced a peak–dip CPR and a finite Josephson diode effect quantified by

$I(\varphi)\approx I_c \sin\varphi.$9

The measured anomalies were reproduced by the nonuniform current profile without invoking Majorana bound states, underscoring that CPR structure is often controlled as much by spatial current distribution and inductive embedding as by the intrinsic ABS spectrum (Kudriashov et al., 12 Feb 2025).