Josephson Junction

Definition

A Josephson junction is two superconductors separated by a thin barrier — typically a 1–2 nanometre insulating oxide — across which Cooper pairs tunnel coherently. Because every electron pair in a superconductor shares one macroscopic quantum wavefunction with a single well-defined phase, what crosses the barrier is not a trickle of independent particles but a phase-locked supercurrent.

Two simple equations describe the entire device. The DC Josephson effect says the supercurrent depends only on the phase difference between the two sides:

I = I_c · sin(Δφ)        (Δφ = φ₂ − φ₁)

The AC Josephson effect says that a voltage makes the phase wind, so the current oscillates:

dΔφ/dt = 2eV/ħ      →      f = 2eV/h

That proportionality constant 2e/h is so large and so precisely known that one millivolt produces a 483.6 GHz oscillation — 483.6 megahertz for every microvolt. This single number turns a Josephson junction into a magnetometer, a qubit, and the legal definition of the volt.

How it works

Start with the heart of superconductivity: below its critical temperature, a superconductor is described by a single complex order parameter ψ = |ψ|·e^{iφ}. Millions of billions of Cooper pairs march in lockstep with one shared phase φ. Two separate superconductors each have their own phase, φ₁ and φ₂. On their own, these phases are independent and meaningless to compare.

Now bring the two superconductors within a nanometre or two of each other. The two wavefunctions decay exponentially into the barrier and overlap slightly. That overlap weakly couples the phases. Solving the coupled equations (Feynman gave a famous two-state derivation) yields exactly the two Josephson relations. The current is set by the difference Δφ, and a voltage V acts as a torque that makes Δφ rotate at rate 2eV/ħ.

The factor of 2e, not e, is the fingerprint of Cooper pairing — the tunneling charge carriers are pairs of electrons, carrying charge 2e. The same 2e shows up in the magnetic flux quantum Φ₀ = h/2e and was historically one of the cleanest confirmations that supercurrent is carried by pairs.

Three regimes matter:

• Below the critical current (I < I_c). A static Δφ between −π/2 and π/2 satisfies I = I_c·sin(Δφ). No voltage, no dissipation — a lossless supercurrent at exactly zero volts.
• At the critical current (I = I_c). The phase pins at Δφ = π/2, where sin is maximal. This is the most supercurrent the junction can carry.
• Above the critical current (I > I_c). No static phase can supply the demanded current. The junction develops a voltage, Δφ starts winding, and the AC effect switches on — the device now radiates and dissipates.

A worked example with concrete numbers

Take a niobium tunnel junction with a critical current I_c = 200 µA. Suppose we force a bias current of 100 µA through it.

Step 1 — find the phase. Solve 100 µA = 200 µA·sin(Δφ), so sin(Δφ) = 0.5 and Δφ = 30° = π/6. The junction sits there with zero voltage across it, carrying 100 µA losslessly. Nudge the current up and the phase climbs toward π/2; the junction never complains until it hits I_c.

Step 2 — cross the critical current. Now ramp the bias to 300 µA > 200 µA = I_c. No static phase can supply it, so a voltage appears. Say it settles at V = 1 mV.

Step 3 — read the frequency. The phase now winds, and the supercurrent oscillates at

f = 2eV/h = (483.6 GHz/mV) × 1 mV = 483.6 GHz

That is a microwave signal at 483.6 GHz emanating from a chip you can barely see — the junction has converted a DC millivolt into a precise oscillation. Drop the voltage to 1 µV and you get 483.6 MHz; raise it to 10 mV and you reach 4.836 THz. The voltage-to-frequency conversion is dead linear and known to better than a part in a billion.

Step 4 — invert for metrology. A volt standard does this backwards. Illuminate the junction with a microwave tone at f = 70 GHz. The junction phase-locks, producing a quantized "Shapiro step" at V = n·(h/2e)·f = n × 70 GHz / 483.6 GHz·mV⁻¹ ≈ n × 144.7 µV. Stack 20,000 junctions in series and the array delivers a rock-solid ~2.9 V reference set only by a frequency and fundamental constants.

Variants and regimes

Type	Barrier	Notation	Where it's used
Tunnel junction	Thin insulating oxide (1–2 nm)	SIS	Qubits, voltage standards, classic SQUIDs
Normal-metal weak link	Non-superconducting metal	SNS	High-I_c devices, RSFQ logic
Constriction / Dayem bridge	Narrow superconducting neck	ScS	Nano-SQUIDs, simple fabrication
Proximity / semiconductor	2D gas, nanowire, graphene	S-Sm-S	Gate-tunable qubits, topological research
DC SQUID	Two junctions in a loop	—	Best field sensitivity, MEG, gradiometers
RF SQUID	One junction in a loop	—	Simpler readout, early magnetometers
Transmon qubit	Junction + large shunt capacitor	—	Quantum computers (IBM, Google)

The RCSJ model (Resistively and Capacitively Shunted Junction) captures real behaviour by adding a resistor R and capacitor C in parallel with the ideal Josephson element. The total current splits three ways: the supercurrent I_c·sin(Δφ), the normal current V/R, and the displacement current C·dV/dt. The Stewart–McCumber parameter β_c = 2e·I_c·R²·C/ħ decides whether the junction's current-voltage curve is single-valued (overdamped, β_c < 1, good for SQUIDs) or hysteretic (underdamped, β_c > 1, used in classic voltage standards). Mathematically the RCSJ model is identical to a driven, damped pendulum — Δφ is the pendulum angle and the bias current is the applied torque.

Common pitfalls and misconceptions

• "A DC voltage makes a DC current." The opposite is true and is the whole surprise. Hold the junction at a constant DC voltage and the supercurrent alternates at f = 2eV/h. DC in, AC out.
• "The barrier conducts like a resistor." Below I_c the supercurrent flows at exactly zero voltage — there is no resistance and no Ohm's law. The phase, not a voltage, drives the current.
• "It's single-electron tunneling." It is pair tunneling. The carriers are Cooper pairs of charge 2e, which is why every formula carries a 2e rather than e. Single-electron (quasiparticle) tunneling is a separate, dissipative channel that appears above the gap voltage.
• "Δφ is the phase of one superconductor." Only the difference Δφ = φ₂ − φ₁ is physical and gauge-invariant; the individual phases are not directly measurable.
• "Bigger junction, bigger I_c, better qubit." Qubits want a small Josephson energy relative to the charging energy and strong anharmonicity. A large junction behaves more like a linear inductor with evenly spaced levels — useless as a two-level system.
• "Thermal noise is irrelevant." Junctions only work cold. Thermal energy k_BT must be far below the Josephson coupling energy E_J = ħI_c/2e, which is why qubits and SQUIDs run at millikelvin to a few kelvin.

Applications

• SQUID magnetometers. A loop with one or two junctions has a critical current periodic in applied flux with period Φ₀ = h/2e ≈ 2.07×10⁻¹⁵ Wb. Tracking that interference measures fields to the femtotesla — enough to map the magnetic signature of neurons firing (magnetoencephalography) and the heart (magnetocardiography), and to find ore bodies in geophysics.
• Superconducting qubits. The junction is a nonlinear, lossless inductor whose energy −E_J·cos(Δφ) gives unequally spaced levels. The transmon — a junction shunted by a big capacitor — is the workhorse qubit at IBM and Google. Tunable qubits use a two-junction SQUID loop so magnetic flux sets the frequency.
• The volt standard. Since the 2019 SI redefinition, h and e are exact, so K_J = 2e/h = 483597.8484… GHz/V is exact. Microwave-irradiated junction arrays realise the primary volt directly from a frequency.
• Ultra-fast digital logic. Rapid Single-Flux-Quantum (RSFQ) circuits represent bits as single flux quanta moving between junctions, switching in picoseconds at a fraction of CMOS power — of interest for cryogenic computing.
• Detectors and mixers. SIS tunnel junctions are the most sensitive heterodyne mixers in radio astronomy (used at ALMA), and Josephson parametric amplifiers add near-the-quantum-limit gain for reading out qubits.

Derivation and performance analysis

Feynman's two-state model treats the junction as two coupled quantum states with amplitudes ψ₁ and ψ₂ and a coupling energy K. Writing each amplitude with its density and phase, ψ = √n·e^{iφ}, and inserting into the coupled Schrödinger equations splits into a real and an imaginary part. The imaginary part gives the rate of charge transfer — the supercurrent — as proportional to sin(φ₂ − φ₁), which is exactly the DC relation I = I_c·sin(Δφ). The real part gives the phase evolution: an energy difference 2eV across the barrier makes the phases drift apart at rate (φ̇₂ − φ̇₁) = 2eV/ħ, which is the AC relation.

Integrating the AC relation under a sinusoidal microwave drive V(t) = V₀ + V₁·cos(2πf_d t) shows the phase locks onto rational multiples of the drive — the time-average DC current develops Shapiro steps at quantized voltages V = n·(h/2e)·f_d. These flat steps are why the standard is so robust: the voltage depends only on a counted integer n and a measured frequency.

How accurate is the link? K_J = 2e/h is now defined exactly, and metrology arrays reproduce voltages with relative uncertainties around 1 part in 10⁹ to 10¹⁰ — far beyond any classical resistor-and-cell reference. For magnetometry, the flux quantum is tiny (2.07 fWb), so a SQUID resolves a small fraction of Φ₀ and reaches field sensitivities of order 1 fT/√Hz, several orders of magnitude beyond room-temperature sensors.

Quantity	Symbol	Value	Why it matters
Josephson constant	K_J = 2e/h	483.5979 GHz/µV — i.e. 483.6 GHz/mV	Voltage ↔ frequency conversion
Flux quantum	Φ₀ = h/2e	2.067834×10⁻¹⁵ Wb	SQUID interference period
Critical current	I_c	µA to mA (design-set)	Max lossless supercurrent
Josephson energy	E_J = ħI_c/2e	µeV–meV scale	Sets qubit anharmonicity
Barrier thickness	d	~1–2 nm oxide	Tunes coupling exponentially
Volt-standard accuracy	—	~1 part in 10⁹–10¹⁰	Primary SI volt realisation