Josephson Current-Phase Relation Overview

• The Josephson current-phase relation is the functional relationship between the supercurrent and the superconducting phase difference, encoding key microscopic details of the junction.
• Experimental and theoretical approaches such as SQUID interferometry and Bogoliubov–de Gennes simulations are used to reveal deviations like skewness and higher harmonics in various transport regimes.
• Understanding and engineering the CPR enables advanced quantum circuits, superconducting diode effects, and probes into novel phenomena including topological superconductivity.

The Josephson current-phase relation (CPR) describes the equilibrium relationship between the supercurrent $I(\varphi)$ flowing through a Josephson junction and the superconducting phase difference %%%%1%%%% between the electrodes. The detailed form of the CPR encodes microscopic information about the junction's transport regime, the nature and symmetry of the weak link, and the relevant Andreev bound states or continuum contributions. Accurate knowledge and engineering of the CPR are central to understanding the quantum dynamics of superconducting devices, designing quantum circuits, and probing novel phenomena such as topological superconductivity.

1. Definitions, Physical Origin, and Canonical Forms

The CPR captures the current carried by single-particle and many-particle processes as a function of the phase bias $\varphi$ across the junction. In conventional superconductor–insulator–superconductor (SIS) tunnel junctions, the CPR is nearly sinusoidal: $I(\varphi) = I_c \sin\varphi$, where $I_c$ is the critical current. Other types of junctions—such as superconductor–normal metal–superconductor (SNS), superconductor–graphene–superconductor (SGS), superconductor–ferromagnet–superconductor (SFS), or hybrid devices hosting topological states—can exhibit strongly non-sinusoidal CPRs with additional harmonics or even anomalous components.

These deviations arise from several microscopic mechanisms:

• High-transmission Andreev bound states, leading to skewness and higher harmonics.
• Changes in symmetry, spin structure, or topology of the junction (e.g., triplet pairing, topological modes) introducing $4\pi$ periodicity or anomalous phase shifts.
• Quantum interference or multichannel effects in multiterminal or phase-coherent devices.

The generic Josephson energy is often expanded as

$$E_J(\varphi) = -\sum_{k=1}^{\infty}[E_k\cos(k\varphi) + E_k^{(a)}\sin(k\varphi)]$$

with the current obtained thermodynamically as

$$I(\varphi) = (2e/\hbar)\,\frac{dE_J}{d\varphi} = \sum_{k=1}^{\infty}[C_k \sin(k\varphi) + S_k \cos(k\varphi)]$$

where $C_k$ and $S_k$ are the amplitudes of each harmonic.

2. Microscopic Theories and Experimental Determination

The CPR is determined by the spectrum and occupation of Andreev bound states (ABSs), propagating modes, and scattering states. For each transport regime, microscopic models are constructed:

• Ballistic regime: For graphene or topological surface states, tight-binding Bogoliubov–de Gennes (BdG) or Dirac–BdG models explicitly evaluate the ABS energies and thus the supercurrent.
• Diffusive regime: Usadel or Eilenberger equations, considering ensemble-averaged Green's functions and proximity effects across normal or ferromagnetic regions.
• Multiterminal and hybrid structures: The boundary Green's function (bGF) formalisms, tight-binding simulations with realistic interface models, and full numerical diagonalization.

Experimentally, CPRs are extracted using phase-sensitive interferometry:

• Embedding the junction under test in a SQUID loop with a reference junction allows controlled bias and direct measurement of the CPR, especially in highly asymmetric SQUIDs where one branch fixes the phase (Nanda et al., 2016).
• Non-contact measurement techniques (scanning SQUID, microwave spectroscopy) are used for nanowire and hybrid mesoscopic junctions (Spanton et al., 2017).

Table: Physical Origins and Prototypical Forms of the CPR

Mechanism	Parametric CPR	Distinct Feature
SIS (tunnel)	$I(\varphi) = I_c \sin\varphi$	2π periodic, symmetric
SNS, Ballistic, High $T_n$	$$I(\varphi) \sim \sin\varphi/(1-T_n\sin^2(\varphi/2))^{1/2}$$	Skewed, higher harmonics
Topological, Majorana	$I(\varphi) \propto \sin(\varphi/2)$	4π periodicity
Second-harmonic–dominant (near 0–π transition)	$I(\varphi) \sim \sin2\varphi$	Double-well Josephson potential

3. Junction-Specific Deviations and CPR Engineering

The CPR can be significantly modified beyond the canonical sinusoid. Key instances include:

a) Skewness and Higher Harmonics in Ballistic Graphene JJs

Direct measurements in ballistic graphene junctions with phase-sensitive SQUID interferometry reveal “forward-skewed” CPRs, with the position of maximum current $\varphi_{\mathrm{max}} > \pi/2$ and skewness $S = (2\varphi_{\mathrm{max}}/\pi) - 1 > 0$ (Nanda et al., 2016, English et al., 2013). Skewness is maximized at high n-doping and reduces near the charge neutrality point, with oscillations in anti-phase with Fabry–Pérot resistance due to cavity interference, demonstrating the sensitive dependence on coherent transmission and carrier density.

b) Multiterminal and Periodicity-Tunable Junctions

In four-terminal diffusive SNS junctions, the CPR between a chosen pair of superconducting contacts can be written as a controlled superposition: $$I_s(\varphi_s, \varphi_c) = A(\varphi_c) \sin(\varphi_s/2) + B(\varphi_c) \sin\varphi_s$$ where $A(\varphi_c)$ and $B(\varphi_c)$ are set by the phase on the “control” pair of contacts (Chandrasekhar, 10 Jun 2025). The relative amplitude of $2\pi$ (direct path) and $4\pi$ (indirect, control-mediated path) periodic components can be tuned in situ by adjusting the phase bias $\varphi_c$.

c) Charge-4e Supercurrents and $\pi$-periodic Potentials

In the hybrid Josephson rhombus—four voltage-tunable Josephson junctions arranged in a loop—gate and flux frustration are used to suppress the first harmonic, yielding a $\pi$-periodic $\cos(2\varphi)$ potential and current $I(\varphi) \sim \sin(2\varphi)$, indicative of coherent charge-4e transport (Banszerus et al., 28 Jun 2024). Gate imbalance or flux detuning allows interpolation between dominated 2e and 4e processes and can also induce superconducting diode effects.

d) Second Harmonic and $0$–$\pi$ Transitions

In hybrid semiconductor–superconductor–ferromagnetic insulator devices and SIsFS junctions, the CPR may be dominated by the second harmonic near a $0$–$\pi$ transition or when the fundamental is suppressed by ABS crossings. In the quantum dot regime, the condition $C_1\approx0$ leads to $I(\varphi)\approx C_2\sin2\varphi$, and the device exhibits a double-well Josephson potential (Maiani et al., 2023, Bakurskiy et al., 2017).

e) Nonsinusoidal and Anomalous CPRs in Spinful and Topological Systems

In S–TS junctions (conventional $s$-wave–topological superconductor), the current is blockaded in the ideal Kitaev limit but becomes finite due to $s$-wave pairing remnants in more realistic spinful models. Quantum dot coupling and parity-mixing yield anomalous $\varphi_0$-junction behavior (Zazunov et al., 2018). In topological Josephson junctions formed on 3D TI surfaces, both Andreev bands and scattering states must be considered to accurately model observed non-sinusoidal phase dependencies (Backens et al., 2020).

4. Dependence on Material, Geometry, and External Control

The magnitude and shape of the CPR are functions of device and environment parameters:

• Carrier density and temperature: In graphene JJs, higher carrier density and lower temperature increase both the critical current and CPR skewness, reflecting the participation of high-transmission microscopic channels.
• Magnetic field and frustration: Magnetic flux in rhombus geometries or 0–$\pi$ junctions drives transitions of the CPR between different periodicities and shapes, enabling “magnetic field tunable” Josephson elements (Lipman et al., 2012).
• Spin-orbit and exchange fields: In semiconductor-superconductor-ferromagnetic insulator hybrids, spin-orbit coupling and noncollinear magnetization can introduce anomalous or diode-like CPRs.
• Gate voltage and electrostatic tuning: Gate control of carrier density or electrostatic barriers in 2D and nanowire-based devices allows dynamic adjustment of mode number, transmission, and the harmonic content of the CPR, notably including switches between single-mode forwarding skew and nearly perfect transmission (Nanda et al., 2016, Spanton et al., 2017).

5. Applications: Quantum Circuits, Protected Qubits, and Diode Effects

The engineering of the CPR enables novel device functionalities:

• Hamiltonian tailoring for qubits: By shaping the Josephson energy, particularly by suppressing the first harmonic, qubits with enhanced anharmonicity or inherent parity protection can be designed (e.g., hybrid Josephson rhombi with $\pi$-periodic potentials for charge-4e qubits) (Banszerus et al., 28 Jun 2024).
• Nonreciprocal elements: Junctions with broken inversion or time-reversal symmetry can exhibit the Josephson diode effect, enabling rectification of the supercurrent, with gate-tunable efficiencies up to 30% in hybrid S/SSM/FI devices (Maiani et al., 2023).
• Dynamic control in multiterminal circuits: Devices with in situ tunable periodicity ($2\pi$ vs $4\pi$), such as multiterminal diffusive junctions, allow for adaptive control of qubit couplings and quantum interference elements (Chandrasekhar, 10 Jun 2025).
• Probing topological phenomena: CPR measurements serve as indirect probes for Majorana bound states, $4\pi$ periodicity, or parity anomalies, with warnings that circuit inductance and extrinsic effects must be accounted for to avoid misinterpretation (Endres et al., 2022).

6. Misconceptions and Proper Interpretation of Experimental CPR

Interpreting observed CPRs must account for possible artifacts:

• Apparent $4\pi$ periodicity or non-sinusoidal switching current may arise from circuit inductances or screening currents, not just intrinsic topological phenomena (Endres et al., 2022).
• Correct extraction of the CPR requires non-hysteretic operation ($\beta_L < 1$) in SQUID measurements and careful numerical modeling of screening contributions, especially in devices with self-formed superconducting phases or nontrivial geometry.
• In multiterminal and phase-interferometric setups, the periodicity and amplitude of the CPR depend nonlocally on all controlled phases and supercurrent paths.

7. Summary and Outlook

The Josephson current-phase relation is a sensitive and versatile probe of the quantum-coherent processes in hybrid superconducting structures. Its shape—ranging from simple sinusoidal forms to highly anharmonic, multicomponent, or anomalous relations—encodes detailed information about Andreev bound states, system symmetries, and the interplay of material and device parameters. Recent progress enables deterministic control over the CPR for applications ranging from quantum metrology to noise-resilient qubits and nonreciprocal circuit elements, with the future direction likely to exploit the full topological, multiterminal, and tunable landscape that CPR engineering affords.